editors in chief - alexandru ioan cuza universityjromai/romaijournal/arhiva/...spiritul de...

Editors in Chief :

Adelina Georgescu, Professor, Member of Academy of Romanian Scientists, 54, Splaiul Independentei, 050094, Bucharest, ROMANIA

Ilie Burdujan, Professor, University of Agricultural Sciences and Veterinary Medicine “Ion Ionescu de la Brad” Iaşi Mihail Sadoveanu, 3, 700490 Iaşi, ROMANIA

Editorial Board:

Nuri Aksel, Professor, Faculty of Applied Sciences, Department of Applied Mechanics and Fluid Dynamics, University of Bayreuth, D‐95440 Bayreuth, GERMANY

Dumitru Botnaru, Professor, Chair of Superior Mathematics, Technical University of Moldova, Bv. Stefan cel Mare, 168, MD 2004, Chişinău, REPUBLIC OF MOLDOVA

Tassos Bountis, Professor , Centre for Research and Applications of Nonlinear Systems and Department of Mathematics, University of Patras, 26500, Patras, GREECE Sanda Cleja‐Țigoiu, Professor, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei, 14, 010014, Bucharest, ROMANIA

Mitrofan Choban, Professor, Member of Academy of Sciences of Moldova, State University of Tiraspol, Iablocikin, 5, MD 2069, Chişinău, REPUBLIC OF MOLDOVA Constantin Fetecău, Professor, Faculty of Mechanical Engineering, The Gh. Asachi Technical University of Iaşi, Bv. Dimitrie Mangeron, 61‐63, 700050, Iaşi, ROMANIA

Anca Veronica Ion, Senior Researcher Institute of Mathematical Statistics and Applied Mathematics of The Romanian Academy, Calea 13 Septembrie, 13, 050711, Bucharest, ROMANIA Boris V. Loginov, Professor, Faculty of Natural Sciences, Ulyanovsk State Technical University, Severny Venetz, 32, 432027, Ulyanovsk, RUSSIA Toader Morozan, Senior Researcher, Institute of Mathematics Simion Stoilow of The Romanian Academy, Calea Griviței, 21, 010702, Bucharest, ROMANIA

Peter Knabner, Professor, Chair for Applied Mathematics, Faculty of Science, Friedrich Alexander University Erlangen‐Nuremberg, Martensstr. 3, 91058, Erlangen, GERMANY Nenad Mladenovici, Professor, Mathematical Sciences, John Crank 210, Brunel University, Uxbridge, UB8 3PH, UNITED KINGDOM Emilia Petrişor, Professor, Department of Mathematics, University Politehnica of Timişoara, Victoriei Square, 2, 300006, Timişoara, ROMANIA

Mihail Popa, Professor, Senior Researcher, Institute of Mathematics and Computer Science of The Academy of Sciences of Moldova, Academiei, 5, MD 2028, Chişinău, REPUBLIC OF MOLDOVA Mefodie Rațiu, Professor, Senior Researcher, Institute of Mathematics and Computer Science of The Academy of Sciences of Moldova, Academiei, 5, MD 2028, Chişinău, REPUBLIC OF MOLDOVA Nicolae Suciu, Senior Researcher, Tiberiu Popoviciu Institute of Numerical Analysis, P.O. Box 68‐1, 400110, Cluj‐Napoca, ROMANIA Vladilen A. Trenogin, Professor, Moscow Institute of Steel and Alloys, B‐49, Leninsky Prospect, 4, 119049, Moscow, RUSSIA

Kumbakonam Rajagopal, Professor, Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843‐3368, UNITED STATES OF AMERICA Mirela Ştefănescu, Professor, Faculty of Mathematics and Computer Science, Ovidius University, Bv. Mamaia, 124, 900527, Constanța, ROMANIA Kiyoyuki Tchizawa, Professor, Kanrikogaku Kenkyusho, Ltd., 2‐2‐2 Sotokanda, Chiyoda‐ku, 101‐0021, Tokyo, JAPAN Constantin Vârsan, Senior Researcher, Institute of Mathematics Simion Stoilow of The Romanian Academy, Calea Griviței, 21, 010702, Bucharest, ROMANIA

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In memoriam Adelina Georgescu

April 25, 1942 – May 1, 2010

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Am cunoscut pe marea noastră matematiciană Profesor Dr. Adelina Georgescu şi am admirat-o din inimă atât pentru importanta creaţie ştiinţifică şi frumoasa activitate didactică, cât şi pentru energia inepuizabilă cu care a luptat pentru neamul românesc. Cu negrăită emoţie îmi amintesc tot ce a reuşit pentru unitatea noastră, de la primele manifestări de după 1989, Sesiuni ştiinţifice organizate în 1991-1992 şi în continuare, pentru matematicienii români de la răsărit şi apus de Prut şi lupta pentru o cât mai strânsă colaborare între ei. Spiritul de iniţiativă şi talentul deosebit de organizare stau la temelia importantei Societăţi ROMAI, pe care a creat-o în 1992 şi a seriei de Conferinţe Internaţionale CAIM iniţiată în 1993, care a ajuns în 2010 la cea de a 18-a ediţie. Doresc ca această Societate să rodească mai departe pentru înflorirea matematicii în România şi peste hotare, mai ales în Republica Moldova. La aceste minunate realizări se adaugă cei doi fii Andrei şi Sergiu Moroianu, pe care i-a crescut în dragoste pentru matematică şi care sunt astăzi la rândul lor creatori de renume internaţional în această ştiinţă şi totodată întemeietori de rodnice familii. Li se adaugă doctorii în ştiinţe matematice precum şi alţi tineri călăuziţi de Doamna Profesor pe căile acestei ştiinţe. Retrăind amintiri despre Adelina, gândul mă duce la Doamna Oltea, mama lui Ştefan cel Mare, simbol al femeii române luptând pentru dreptatea şi înflorirea României. Fie ca tradiţia făurită de Doamna Profesor Adelina Georgescu să fie continuată şi dezvoltată strălucit şi mai departe! Prof. univ. Cabiria Andreian Cazacu

Cuvant tinut in cadrul CAIM 2010 la Iasi, la lansarea cartii Matematica si viata, scrisa de Adelina Georgescu si Lucia Dragotescu.

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I have known our great mathematician Professor Adelina Georgescu, and I admired her from all my heart both for her important scientific creation and for the beautiful didactical activity as well as for the inexhaustible energy with which she fought for the Romanian people.

With unspeakable emotion I remember all she managed to do for our unity, from the first manifestations after 1989, the sessions organized in 1991-1992, and after, mainly for the Romanian mathematicians from the East and West of Prut and the fight for their close collaboration. Her vivid spirit and organizing talent lie at the base of the important Mathematical Society ROMAI, which she founded in 1992, and of the series of Conferences on Applied and Industrial Mathematics (CAIM) that she initiated in 1993, and that reached in 2010 the 18th edition.

I wish that this Society to be also fruitful from now on, for the flourishing of the mathematics from Romania and abroad, especially from Republic of Moldova.

To this wonderful achievements we must add her two sons, Andrei and Sergiu Moroianu, that she grew up in the spirit of love for mathematics and that are now, at their turn, well-known mathematicians and also heads of beautiful families.

To them we must add the doctors in mathematics and many other young persons that were lead by Professor Adelina Georgescu on the ways of this science.

While reminding Adelina, the thought takes me to Doamna Oltea, mother of Stephen the Great, the symbol of Romanian women fighting for justice and for the flourishing of our country, Romania. Let the tradition developed by Professor Adelina Georgescu be brightly continued and developed further!

Professor Cabiria Andreian Cazacu.

Speech held by Professor Cabiria Andreian Cazacu, at CAIM 2010, in Iasi, when the book Mathematics and Life, written by Adelina Georgescu and Lucia Dragotescu was launched.

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A life dedicated to the whole "Knowledge" Liliana RESTUCCIA

Department of Mathematics, University of Messina, Italy

Considerate la vostra semenza: fatti non foste a viver come bruti,

ma per seguir virtute e conoscenza. (Dante Alighieri, Divina Commedia, Inferno, Canto XXVI, versi 118-120)

This contribution contains the speech delivered by the author with the occasion of the Mini‐symposium, organized in collaboration with D. Jou, M. S. Mongiovì and W. Muschik, as a part of the 2010 Conference of SIMAI (Italian Society of Applied and Industrial Mathematics), which was held between 21‐25 June in Cagliari, Sardegna, Italia. This Mini‐symposium, entitled "Recent Ideas in Non‐Equilibrium Thermodynamics and Applications" was dedicated to the memory of Prof. Dr. Adelina Georgescu, co‐researcher and a great author’s friend.

We dedicate this Mini-symposium, regarding the latest ideas in Non-Equilibrium Thermodynamics and Applications to physical-mathematical models of complex materials, to Prof. Adelina Georgescu, a member of the Scientists’ Academy of Romania, a great master and an eminent scientist. Her death, on May 1st 2010, leaves behind a very great emptiness in the field of Applied Mathematics and not less in the lives of those who knew and worked with her.

I am deeply grateful for her very precious advices in the organization of my research and the very enlightening discussions and remarks regarding a lot of mathematical concepts and tools, with their practical applications in different fields of science.

She was my great and unforgettable friend, invaluable guide and scientific collaborator. Since 2002 I invited her as visiting professor at the University of Messina for giving lectures and seminars on: Non-linear Dynamics, Bifurcation theory, Variational problems in Mathematical-Physics, Thermodynamics of fluids, Asymptotic methods with applications to waves and shocks, Complex fluid flows, Modeling in Meteorology, Turbulence, Modeling in cancer dynamics.

I always thanked her for her profound participation and for her very precious contribution in spreading her research methodologies (a scientific literature on foundations and recent progress in Mathematical-Physics due to the Romanian and Russian schools) and, especially for her publications (very useful due to their synthetic style and clarity in explaining all concepts) at the University of Messina.

Some of her books and publications have been already translated from Romanian to Italian, in order to be known in Italy for their invaluable qualities.

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She also was a corresponding member of "Accademia Peloritana dei Pericolanti" of Messina; a special issue of the Academic Acts (on-line) of this institution, will be published, containing a collection of the lectures held by Prof. Adelina Georgescu during her visits at Messina.

She was the founder and president of ROMAI (Romanian Society of Applied and Industrial Mathematics) since 1992. Furthermore, she founded the Institute of Applied Mathematics of Romanian Academy (IAM), which she led as director until 1995. In 1997 she became full professor at the University of Piteşti (Romania), where she founded the Chair of Applied Mathematics (the head of which she was for several years), where she created the group of research on Dynamical systems and Bifurcation. She was the editor in chief of ROMAI Journal since 2005.

Since 1993, year by year she has organized the CAIM International Conferences, devoted to contributions in Applied and Industrial Mathematics, the Romanian analogous to SIMAI International Conferences in Italy.

She delivered about 200 conferences at several universities and research units mostly from abroad. Moreover, she sustained short or plenary conferences at about 150 scientific meetings. She published (some are still in press) about 20 books (research monographies, three of them at Kluwer and Chapman & Hall and one at World Scientific, dictionaries, universitary texts), alone or in collaboration, and more than 200 papers in scientific journals most of them in ISI journals or refereed in MR and ZBM. She was awarded the Romanian Academy prize and many other distinctions. She was honored as a Doctor Honoris Causa of the State University of Tiraspol (Rep. of Moldova). She was a corresponding member of the Academy of Nonlinear Sciences (Moscow).

The main topics of Prof. Adelina Georgescu's researches were in Hydrodynamics and applications to complex fluid flows, Turbulence, Meteorology, Perturbation theories for partial differential equations, Boundary layer theory, Hydrodynamics stability, Study of solutions of partial differential equations, Non-linear dynamics, Bifurcation theory, Variational problems in Mathematical-Physics, Modeling in cancer dynamics, Biomathematics, Synergetics.

She dedicated to the progress of science her very generous and honest and highly moral and responsible life, studying very deeply mathematical and physical theories with applications and giving them new impulse.

Her courageous ideas in a difficult politic framework had been well known by whole scientific world.

She was a gifted and talented woman with a very great humanity and

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Professors Adelina Georgescu and Liliana Restuccia, at University of Messina, 2007

very broad interests. Besides her basic research items like Physics and Mathematics (Applied Mathematics, in particular), she was interested in politics, history, geography, languages, religion, literature, medicine.

At the University of Messina we will be open a Study Center "Adelina Georgescu" of Applied Mathematics, dedicated to her memory, for her work be known by the present generations and those to come. This Center has as objective to spread at international level her books and publications, her research methodologies, her ideas, in general, and in particular those in Synergetics (the new synthesis of the science) and in the directions to be followed in order to find the science truths. Furthermore, this Center has also the goal of organizing international conferences, meetings, study days, lectures and seminars for discussing issues as: foundations and recent progresses in different fields of Applied Mathematics, with the participa-tion of speakers of high scientific level, researchers and students.

I think that if there is a place where there is now Adelina Georgescu, well, then this is the Paradise, the Dante Alighieri’s Paradise of the whole "Knowledge", to which Adelina devoted her entire life.

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AA.. BBOOOOKKSS

1. Adelina Georgescu, Bifurcatia, fractali si haos determinist, chapter of Enciclopedia matematica, edited by M. Iosifescu, O. Stănăşilă, D. Ştefănoiu, Editura AGIR, 2010.

2. Adelina Georgescu, L. Palese, G. Raguso, Biomatematica. Modelli, dinamica e biforcazione, Cacucci Editore, Bari, 2009.

3. Adelina Georgescu, Lidia Palese, Stability criteria of fluid flows, Series on Advances in Mathematics for applied sciences 81, World Scientific, Singapore, 2009.

4. Adelina Georgescu, George-Valentin Cârlig, Cătălin-Liviu Bichir, Ramona Radoveneanu, Matematicieni români de pretutindeni, ed. a II-a, , Seria Mat. Apl. Ind. 24, Ed. Pamantul, Pitesti, 2006.

5. Adelina Georgescu, C.-L. Bichir, G.V. Cîrlig, Matematicieni români de pretutindeni, Seria de Mat. Apl. Ind. 18, Ed. Univ. Piteşti , Piteşti, 2004.

6. C. Rocşoreanu, Adelina Georgescu, N. Giurgiţeanu, FitzHugh-Nagumo model: bifurcation and dynamics, Kluwer, Dordrecht, 2000.

7. B.-N. Nicolescu, N. Popa, Adelina Georgescu, M.Boloşteanu, Mişcări ale fluidelor cavitante.Modelare şi soluţii, Seria Mat. Apl. Ind, 3, Ed. Univ. Piteşti, Piteşti, 1999.

8. N. Popa, B.-N. Nicolescu, Adelina Georgescu, M. Boloşteanu, Modelări matematice în teoria lubrificaţiei aplicate la etanşările frontale, Seria Mat. Apl. Ind., 2, Ed. Univ. Piteşti, Piteşti, 1999.

9. Adelina Georgescu, M. Moroianu, I. Oprea, Teoria bifurcaţiei. Principii şi aplicaţii, Seria Mat. Apl. Ind. 1, Ed. Univ. Piteşti, Piteşti, 1999.

10. Adelina Georgescu, Teoria stratului limită. Turbulenţă, Ed. Gh. Asachi, Iaşi, 1997.

11. Adelina Georgescu, Asymptotic treatment of differential equations, Applied Mathematics and Mathematical Computation, 9, Chapman and Hall, London, 1995.

12. Adelina Georgescu, I. Oprea, Bifurcation theory from application viewpoint, Tipografia Univ. Timişoara, 1994, (Monografii Matematice 51).

13. Adelina Georgescu, Sinergetica. Solitoni. Fractali. Haos determinist. Turbulenţă, Tipografia Univ. Timişoara, 1992.

14. H.Dumitrescu, Adelina Georgescu, V.Ceangă, GH.Ghiţă, J.Popovici, B.Nicolescu, Al.Dumitrache, Calculul elicei, Ed.Academiei, Bucureşti, 1990.

15. Adelina Georgescu, Aproximaţii asimptotice, Ed.Tehnică, Bucureşti, 1989.

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16. Adelina Georgescu, Sinergetica-o nouă sinteză a ştiinţei, Ed.Tehnică, Bucureşti, 1987.

17. Adelina Georgescu, Hydrodynamic stability theory, Mechanics: Analysis, 9, Kluwer, Dordrecht, 1985.

18. St.N.Săvulescu, Adelina Georgescu, H.Dumitrescu, M.Bucur, Cercetări matematice în teoria modernă a stratului limită, Ed. Academiei, Bucureşti, 1981.

19. Adelina Georgescu, Teoria stabilităţii hidrodinamice, Ed. Ştiinţifică şi Enciclopedică, Bucureşti, 1976.

BB.. PPAAPPEERRSS

2010

1. Limit cycles by Finite Element Method for a one - parameter dynamical system associated to the Luo - Rudy I model, ROMAI J., this issue, 27-41. (with C. L. Bichir, B. Amuzescu, Gh. Nistor, et al.).

2. A linearization principle for the stability of the chemical equilibrium of a binary mixture, ROMAI J., this issue, 131-138 (with L. Palese).

3. Asymptotic waves as solutions of nonlinear PDEs deduced by double-scale method in viscoanelastic media with memory, ROMAI J., this issue, 139-154 (with L. Restuccia).

4. An application of double-scale method to the study of non-linear dissipative waves in Jeffreys media, submitted to Math. and Its Appl. (with L. Restuccia).

5. On the nonlinear stability of a binary mixture with chemical surface reactions, submitted to Math. and Its Appl. (with L. Palese).

6. Stability and self-sustained oscillations in a ventricular cardiomyocyte model, submitted to Interdisciplinary Sciences – Computational Life. (with C. L. Bichir, B. Amuzescu, M. Popescu, et al.).

7. Limit points and Hopf bifurcation points for a one - parameter dynamical system associated to the Luo - Rudy I model, to be published (with C. L. Bichir, B. Amuzescu, Gh. Nistor, et al.).

2009

8. Further results on approximate inertial manifolds for the FitzHugh-Nagumo model, Atti dell’ Accad. Peloritana dei Pericolanti, Scienze FMN, vol LXXXVII, nr. 2. (2009). (with C.-S. Nartea).

9. On the stability bounds in a problem of convection with uniform internal heat source, Atti dell’ Accad. Peloritana dei Pericolanti, Scienze FMN, acceptata. (with F.-I. Dragomirescu).

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10. Degenerated Bogdanov-Takens bifurcations in an immuno-tumor model, Atti dell’ Accad. Peloritana dei Pericolanti, Scienze FMN, vol LXXXVII, nr. 1. (2009). (with M. Trifan).

11. Stability criteria for quasigeostrofic forced zonal flows ; I. Asymptotically vanishing linear perturbation energy, ROMAI J. 5, 1 (2009). 63-76 (with L. Palese).

12. Linear stability results in a magnetothermoconvection problem, An St. Univ. Ovidius Constanta, 17, 3 (2009), 119-129. (with F.-I. Dragomirescu).

13. Polynomial based methods for linear nonconstant coefficients eigenvalue problems, Proceedings of the Middle Volga Mathematical Society, 11, 2 (2009), 33-41. (with F.-I. Dragomirescu).

14. Thermodynamics of fluids, SIMAI e-Lecture Notes, Vol. 2 (2009), 1-26.

2008

15. A closed-form asymptotic solution of the FitzHugh-Nagumo model, Bul. Acad. St. Rep. Moldova, Seria Mat., 2 (2008), 24-34. (with Gh. Nistor, M.-N. Popescu, D. Popa).

16. A closed-form asymptotic solution of the FitzHugh-Nagumo model, Bul. Acad. St. Rep. Moldova, Seria Mat., 2 (2008), 24-34. (with Gh. Nistor, M.-N. Popescu, D. Popa).

17. Application of two spectral methods to a problem of convection with uniform internal heat source, Journal of Mathematics and Applications, 30 (2008), 43-52. (with F.-I. Dragomirescu).

18. A linear magnetic Bénard problem with Hall effect. Application of the Budiansky-DiPrima method, Trudy Srednevoljckogo Matematichescogo Obshchestva, Saransk, 10, 1 (2008), 294-306. (with L. Palese).

19. Determination of asymptotic waves in Maxwell media by double-scale method, Technische Mechanik, 28, 2 (2008), 140-151. (with L. Restuccia).

20. Approximate limit cycles for the Rayleigh model, ROMAI J. 4, 2 (2008), 73-80. (with M. Sterpu, P. Băzăvan).

21. Lyapunov stability analysis in Taylor-Dean systems, ROMAI J. 4, 2 (2008), 81-98. (with F.-I. Dragomirescu).

2007

22. Analytical versus numerical results in a microconvection problem, Carpathian J. Math., 23, 1-2 (2007), 81-88. MR 2305839 (with F.-I. Dragomirescu) .

23. Relaxation oscillations and the “canard” phenomenon in the FitzHugh-Nagumo model, Cap. 4 in Recent Trends in Mechanics, 1, Ed. Academiei Romane, Bucuresti, 2007, 82-106 (with M.-N. Popescu, Gh. Nistor, D. Popa)

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24. Dynamical approach in biomathematics, ROMAI J., 2, 2 (2007), 63-76. (with L. Palese, G. Raguso).

2006

25. Some results on dynamics generated by Bazykin model, Atti Accad. Peloritana dei Pericolanti, Scienze FMN, 84, C1A0601003 (2006), 1-10.(with R. - M. Georgescu).

26. A linear magnetic Bénard problem with tensorial electrical conductivity, Bolletttino U.M.I. (8) 9-B (2006), 197-214. (Prima data publicata ca Rapp. Int. Dipto. Mat. Univ. Bari, 17 / 03.) (with L. Palese, A. Redaelli)

27. Further study of a microconvection model using the direct method, ROMAI J., 2, 1 (2006), 77-86. (with I. Dragomirescu).

28. Bifurcations and dynamics in the lymphocytes-tumor model, The 7th

Congress of SIMAI 2004, Venezia, Italy, 2004. International Conference on Mathematical Models and Methods in Biology and Medicine -MMMBM 2005, Bedlewo, Poland, May 29-June 3, 2005, trimisa la Atti Accad. Peloritana dei Pericolanti, Scienze FMN. (with M. Trifan) .

29. Asymptotic waves from the point of view of double-scale method, Atti Accad. Peloritana dei Pericolanti, Scienze FMN, 84, C1A0601005 (2006), 1-9. (with L. Restuccia) .

30. Linear stability bounds in a convection problem for variable gravity field, Bul. Acad. St. Rep. Moldova, Mat., 3(2006), 51-56. (with I. Dragomirescu) .

31. Neutral manifolds in a penetrative convection problem. I. Expansions in Fourier series of the solutions, Sci. Annals of UASVM Iasi, 49, 2 (2006), 19-32. MR 2300509 (with A. Labianca) .

32. Mathematical models in biodynamics, Sci. Annals of UASVM Iasi, 49, 2 (2006), 361-371.

2005

33. The static bifurcation in the Gray – Scott model, Rev. Roum. Sci. Tech. – Méc. Appl., 50, 1-3 (2005), 3-13. (with R. Curtu) .

34. A nonlinear hydromagnetic stability criterion derived by a generalized energy method, Bul. Acad. St. Rep. Moldova, Seria Mat., 1(47) (2005), 85-91. (with C.-L. Bichir, L. Palese) .

35. A direct method and its application to a linear hydromagnetic stability problem, ROMAI J., 1, 1 (2005), 67-76. (with L. Palese, A. Redaelli) .

36. Sets governing the phase portrait (approximation of the asymptotic dynamics), ROMAI J., 1, 1 (2005), 83-94. (with S.-C. Ion).

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37. Application of the direct method to a microconvection model, Acta Universitatis Apulensis, Alba Iulia, Mathematics - Informatics, 10 (2005), 131-142. ZB 1113.65076, MR 2240129. (with F.- I. Dragomirescu).

38. Normal forms and unfoldings: a comparative study, Sci. Annals of UASVM Iaşi, 48, 2 (2005), 15-26 (with E. Codeci) .

39. Bifurcation of planar vector fields. I. Normal forms "at the point". One zero eigenvalue, Bul. Şt. Univ. Piteşti, Ser. Mec. Apl., 1 (12) (2005), 41-47. (with D. Sârbu ) .

40. Bifurcation of planar vector fields. II. Normal forms "at the point". Hopf and cups cases, Bul. Şt. Univ. Piteşti, Ser. Mec. Apl., 1 (12) (2005), 49-55. (with D. Sârbu ).

41. Continuity of characteristics of a thin layer flow driven by a surface tension gradient, ROMAI J., 1, 2 (2005), 11-16. (with E. Borsa).

2004

42. Bifurcation in the Goodwin model I, Rev. Roum. Sci. Tech. – Méc. Appl., 49, 1-6 (2004), 13-16. (with N. Giurgiţeanu, C. Rocşoreanu).

43. Curba valorilor de bifurcaţie Hopf pentru sisteme dinamice plane, Bul. St., Seria Mec. Apl., 10 (2004), 55-62. (with E. Codeci) .

44. Dynamic bifurcation diagrams for some models in economics and biology, Acta Universitatis Apulensis, Alba Iulia, Mathematics-Informatics, 8 (2004), 156-161.

45. On instability of the magnetic Bénard problem with Hall and ion-slip effects, Intern. J. Engng. Sci., 42 (2004), 1001-1012. (with L. Palese) .

46. Liapunov method applied to the anisotropic Bénard problem, Math. Sci. Res. J., 8, 7 (2004), 196-204. (with Lidia Palese) .

2003

47. A Lie algebra of a differential generalized FitzHugh – Nagumo system, Bul. Acad. Şt., Rep. Moldova, Seria Mat. 1 (41) (2003), 18-30. (with M. Popa, C. Rocşoreanu)

48. Approximation of pressure perturbations by FEM, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 9 (2003), 31-36. (with C. - L. Bichir) .

49. Global bifurcations for FitzHugh-Nagumo model, Dynamical Systems and Applications, Trends in Mathematics: Bifurcations, Symmetry and Patterns, Birkhäuser, Basel, 2003, 197-202. (with C. Rocşoreanu, N. Giurgiţeanu) .

50. Static and dynamic bifurcation of nonlinear oscillators, Bul. Şt. Univ. Piteşti, Seria Mec. Apl., 1, 7 (2003), 133-138.

51. A Lorenz-like model for the horizontal convection flow, Int. J. Non-Linear Mech., 38 (2003), 629-644. (with E. Bucchignani, D. Mansutti) .

52. Bifurcation in biodynamics, Sci. Annals of UASVM Iasi, 46, 2 (2003), 15-34.

53. Numerical integration of the Orr-Sommerfeld equation by wavelet methods, Bul. St. Univ. Pitesti, Seria Mat.-Inf., 9 (2003), 25-30. (with L. Bichir).

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2002

54. Codimension - one bifurcations for a Rayleigh model, Bul. Acad. Şt. Rep. Moldova, Seria Mat., 1 (38), (2002), 69 – 76. (with M Sterpu).

55. Static bifurcation diagram for a microeconomic model, Bul. Acad. Şt. Rep. Moldova, 3 (40) (2002), 21-26. (with L. Ungureanu, M. Popescu).

56. Improved criteria in convection problems in the presence of thermodiffusive conductivity, Analele Univ. Timişoara, 40, 2 (2002), 49-66. (with L. Palese) .

57. Existence and regularity of the solution of a problem modelling the Bénard problem, Mathematical Reports, 4 (54), 1 (2002), 87-102. (with A. – V. Ion).

58. Degenerated Bogdanov – Takens points in an advertising model, Analele Universităţii de Vest, Seria Economie; Sudiile de Economie, Timişoara, (with L. Ungureanu).

59. Global bifurcations in an advertising model, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., (with Laura Ungureanu, Liviu Ungureanu) .

60. Stability criteria for quasigeostrophic forced zonal flows. I. Asymptotically vanishing linear perturbation energy, Magnetohydrodynamics: an International J. (with L. Palese).

61. Domains of attraction for a model in enzimology, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 8 (2002), 49-58. (with R. Curtu) .

62. Heteroclinic bifurcations for the FitzHugh – Nagumo system, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 8 (2002). (with C. Rocşoreanu, N. Giurgiţeanu).

63. Topological type of some nonhyperbolic equilibria in a problem of microeconomic dynamics, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 8 (2002). (with L. Ungureanu, M. Popescu).

64. On the Misra-Progogine-Courbage theory of irreversibility, Mathematica, 44 (67), 2 (2002), 215-231. (with N. Suciu).

65. Non – Newtonian solution and viscoelastic constitutive equations, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 5 (2000). (with C. Chiujdea).

66. Normal form for the degenerated Hopf bifurcation in an economic model, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 5 (2000). (with L. Ungureanu).

67. k > 3 order degenerated Bautin bifurcation and Hopf bifurcation in a mathematical model of economical dynamics, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., (2002). (with L. Ungureanu, M. Popescu).

68. Static bifurcation diagram for a mathematical model governing the capital of a firm, Bul. Şt. Univ. Piteşti, Seria Mat.- Inf, 8 (2002), 177-181. (with L. Ungureanu).

69. Concavity of the limit cycles in the FitzHugh-Nagumo model, An. Şt. Univ. Al.I.Cuza, Iaşi, Mat. (N. S.), 47, 2 (2001), 287-298, (2002). (with C. Rocşoreanu, N. Giurgiţeanu).

xxvv

2001

70. Concavity of the limit cycles in the FitzHugh-Nagumo model, Analele Univ. Iaşi, Seria I Matematica, 47, 2 (2001), 287-298. (with C. Rocşoreanu, N. Giurgiţeanu) .

71. Codimension–three bifurcations for the FitzHugh–Nagumo dynamical scheme, Mathematical Reports, 3 (53), 3 (2001), 287 – 292. (with M. Sterpu).

72. Classes of solutions for a nonlinear diffusion PDE, J. of Comput. Appl. Math., 133, 1-2 (2001), 373 - 381. (with H. Vereecken, H. Schwarze, U. Jaekel).

73. On special solutions of the Reynolds equation from lubrication, J. of Comput. Appl. Math., 133, 1-2 (2001), 367 - 372. (with B. Nicolescu, N. Popa, M. Boloşteanu).

74. Conections between saddles for the FitzHugh–Nagumo system, Int. J. Bif. Chaos, 11, 2 (2001), 533 – 540. (with C. Rocşoreanu, N. Giurgiţeanu).

75. The complete form for the Joseph extended criterion, Ann. Univ. Ferrara, Sez. VII (N. S.), Sc. Mat., 47 (2001), 9 – 22. (with L. Palese, A. Redaelli).

76. Determination of neutral stability curves for the dynamic boundary layer by splines, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 7 (2002), 15-22. (with L. Bichir).

77. Numerical integration of the Orr-Sommerfeld equation by wavelet methods, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 7 (2001), 9-14. (with L. Bichir).

78. Degenerated Bogdanov-Takens points in an advertising model, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 7 (2001), 173-177. (with L. Ungureanu).

2000

79. Dynamics generated by the generalized Rayleigh equation. II. Periodic solutions, Mathematical Reports, 2(52), 3 (2000), 367 – 378, 2001. (with M. Sterpu, P. Băzăvan).

80. Neutral curves for the MHD Soret – Dufour driven convection, Rev. Roum. Sci. Tech. – Méc. Appl., 45, 3 (2000), 265 – 275. (with S. Mitran, L. Palese).

81. On a method in linear stability problems. Application to natural convection in a porous medium. J. of Ultrascientist of Physical Sciences, 12, 3 (2000), 324 – 336. (with L. Palese).

82. On the Misra-Prigogine-Courbage theory of irreversibility. 2. The existence of the nonunitary similarity, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 6 (2000), 213 - 222. (with N. Suciu).

83. Codimension – three bifurcation for a FitzHugh-Nagumo like system, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 6 (2000), 193 - 197. (with M. Sterpu).

84. Dynamics and bifurcation in the periodically forced FitzHugh-Nagumo system, Intern. J. of Chaos Theory and Applications, 5, 2 (2000), 63 - 79. (with M. Sterpu).

xxvvii

85. On a new method in hydrodynamic stability theory, Math. Sciences Research Hot – Line, 4, 7 (2000), 1 – 16. (with L. Palese, A. Redaelli).

86. Hopf and homoclinic bifurcations in a biodynamical system, Bul. Şt. Univ. Baia Mare, Seria Mat–Inf., 16, 1(2000), 131–142. (with C. Rocşoreanu, N. Giurgiţeanu).

87. Dynamics of the cavitation spherical bubble. II. Linear and affine approximation, Rev. Roum. Sci. Tech. – Méc. Appl., 45, 2 (2000), 163 – 175. (with B.-N. Nicolescu).

88. Degenerated Hopf bifurcation in the FitzHugh – Nagumo system. II. Bautin bifurcation, Mathematica- Anal. Numér. Theor. Approx., 29, 1 (2000), 97 – 109. (with C. Rocşoreanu, N. Giurgiţeanu).

1999

89. Set of attraction of certain initial data in a nonlinear diffusion problem, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 235-261 (with H. Schwarze, H. Vereecken, U. Jaekel).

90. Coincidence of the linear and nonlinear stability bounds in a horizontal thermal convection problem, Intern. J. Nonlin. Mech., 34, 4 (1999), 603-613. (with D. Mansutti).

91. New types of codimension-one and-two bifurcations in the plane, Inst. Matem. Acad. Rom, Preprint No. 12/1999. (with C. Rocşoreanu, N. Giurgiţeanu).

92. Regimes with two or three limit cycles in the FitzHugh-Nagumo system, ZAMM 79, Supplement 2 (1999). (with C. Rocşoreanu, N. Giurgiţeanu).

93. Asymptotic analysis of solute transport with linear nonequilibrium sorption in porous media, Transp. Porous Media, 36, 2 (1999), 189-210 (with H. Vereecken, U. Jaekel).

94. Symmetry of the solution of the nonlinear Reynolds equation describing mechanical face seals, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 333-343. (with B. Nicolescu, N. Popa).

95. Hopf bifurcation and canard phenomenon in the FitzHugh-Nagumo model, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 217-233. (with C. Rocşoreanu, N. Giurgiţeanu).

96. Convecţia termică cu efect Marangoni. Condiţii de echilibru, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 345-350. (with Gh. Nistor).

97. Applications of coarse-grained and stochastic averages to transport processes in porous media, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 435-445. (with N. Suciu, C. Vamos, U. Jaekel, H. Vereeken).

98. Modelul continuu multiplicator-accelerator. II. Cazul liniar pentru anumite valori negative ale parametrilor şi cazul neliniar, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 263-266. (with C. Georgescu).

xxvviiii

99. Investigation of the normalized Gierer-Meinhardt system by center manifold method, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 277-283. (with A. Ionescu).

100. Dynamics and bifurcations in a biological model, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 4 (1999), 137 – 153. (with N. Giurgiţeanu, C. Rocşoreanu).

101. On an inertial manifold in the dynamics of gas bubbles, Rev. Roum. Sci. Tech. – Méc. Appl., 44, 6 (1999), 629 – 631. (with B. Nicolescu).

1998

102. Neutral thermal hydrodynamic and hydromagnetic stability hypersurfaces for a micropolar fluid layer, Indian J. Pure and Appl. Math.,29, 6 (1998), 575-582. (with M. Gavrilescu, L. Palese).

103. On the Misra-Prigogine-Courbage theory of irreversibility, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 2 (1998), 169-188. (with N. Suciu).

104. On the mechanism of drag reduction in Maxwell fluids, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 2 (1998), 107-114. (with C. Chiujdea).

105. Transport processes in porous media. 1. Continuous modelling, Romanian J. Hydrology Water Resources, 5, 1-2 (1998) 39-56. (with N. Suciu, C. Vamoş, U. Jaekel, H. Vereecken).

106. Equilibria and relaxation oscillations of the nodal system of the heart. 2. Hopf bifurcation, Rev. Roum. Sci. Tech.-Méc. Appl., 43, 3, (1998), 403 – 414 (with C. Rocşoreanu, N. Giurgiţeanu).

107. Neutral surfaces for Soret – Dufour – driven convective instability, Rev. Roum. Sci. Tech. – Méc. Appl., 43, 2 (1998), 251 – 260. (with L. Palese, L. Pascu).

1997

108. On the existence and on the fractal and Hausdorff dimensions of some global attractor, Nonlinear Anal., Theory, Methods & Applications, 30, 8, (1997), 5527-5532. (with A. V. Ion).

109. Stability spectrum estimates for confined fluids, Rev. Roum. Math. Pures Appl., 42,1-2 (1997), 37-51. (with L. Palese).

110. Thermosolutal instability of a compresible Soret-Dufour mixture with Hall and ion-slip currents through a porous medium, Rev. Roum. Sci. Tech.-Méc. Appl., 42, 3-4 (1997), 279-296 (with L. Palese, D. Paşca, D. Bonea).

111. Studiul portretului de fază. II. Punctele de inflexiune ale traiectoriilor de fază ale sistemului dinamic Van der Pol, St. Cerc. Mec. Apl., 56, 1-2 (1997), 15-31. (with N. Giurgiţeanu).

112. Studiul portretului de fază. III. Influenţa liniarizării asupra sistemului dinamic neliniar, St. Cerc. Mec. Apl., 56, 3 - 4 (1997), 141-153. (with N. Giurgiţeanu).

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113. Hydrodynamical equations for one-dimensional systems of inelastic particles, Phys. Rev., E (3), 55, 5 (1997), 6277-6280. (with C. Vamoş, N. Suciu).

114. Modelul continuu multiplicator-accelerator. Cazul liniar, Bul. Şt. Seria Mat.-Inform. Univ. Piteşti, 1 (1997), 95-104. (with C. Georgescu).

115. Bifurcation in the Goodwin model from economics. II, Bul. Şt. Seria Mat.-Inform. Univ. Piteşti, 1 (1997), 105-112. (with N. Giurgiţeanu, C. Rocşoreanu).

116. Studiul portretului de fază IV. Absenţa bifurcaţiei canard, St. Cerc. Mec. Apl., 56, 5-6 (1997), 297-305. (with N. Giurgiţeanu, C. Rocşoreanu).

117. Degenerated Hopf bifurcation in the FitzHugh-Nagumo system. 1. Bogdanov-Takens bifurcation, Analele Univ. din Timişoara, 35, 2 (1997), 285-298. (with C. Rocşoreanu, N. Giurgiţeanu).

1996

118. Neutral stability hypersurfaces for an anisotropic MHD thermodiffusive mixture. III. Detection of false secular manifolds among the bifurcation characteristic manifolds, Rev. Roum. Math. Pures Appl.,41,1-2 (1996), 35-49. (with L. Palese).

119. Balance equations for physical systems with corpuscular structure, Physica A, 227 (1996), 81-92. (with C. Vamoş, N. Suciu, I. Turcu).

120. Balance equations for a finite number of material points, Stud. Cerc. Mat., 48, 1-2 (1996), 115-127. (with C. Vamoş, N. Suciu).

121. A nonlinear stability criterion for a layer of a binary mixture, ZAMM Supplement 2, 76 (1996), 529-530. (with L. Palese).

122. Extension of the Joseph's criterion on the nonlinear stability of mechanical equilibria in the presence of thermodiffusive conductivity, Theoret. Comput. Fluid Dyn., 8, 6 (1996), 403-413.( with L. Palese).

123. Coarse grained averages in porous media, KFA / ICG - 4 Internal Report No. 501296/1996. (with N. Suciu, C. Vamoş, U. Jaekel, H. Vereecken).

124. Asymptotic analysis of nonlinear equilibrium solute transport in porous media, Water Ressources Research, 32, 10, (1996), 3093-3098. (with U. Jaekel, H. Vereecken).

1995

125. Amélioration des estimations de Prodi pour le spectre, C. R. Acad. Sci. Paris Sér. I Math., 320, 7 (1995), 891-896. (with L. Palese).

126. Sulla stabilitá globale del equilibrio meccanico per una miscela binaria in presenza di effetti Soret e Dufour, Rapp. Dipto. Mat., Univ. Bari, 8/1995. (with L. Palese, A. Redaelli).

xxiixx

1994

127. Nonlinear stability criteria for l MHD flows. I. IsothermaI isotropic case, Rev. Roum. Math. Pures Appl., 39, 2 (1994), 131-146, (cu M. Maiellaro, L. Palese).

1993

128. Synergetics and synergetic method to study processes in hierarchical systems, Noesis, 18 (1993), 121-127.

129. The application of the shooting method to the hydrodynamic stability of the Poiseuille flow in channels and pipes, Computing, 4 (1993), 3-6. (with R. Florea).

130. Stability of a binary mixture in a porous medium with Hall and ion-slip effect and Soret-Dufour currents, Analele Univ. din Oradea, 3 (1993), 92-96. (with L. Palese, D. Paşca).

131. Metode de determinare a curbei neutrale în stabilitatea Bénard, St. Cerc. Mec. Apl., 52, 4 (1993), 267-276. (with I. Oprea, D. Paşca).

132. Critical hydromagnetic stability of a thermodiffusive state, Rev. Roum. Math. Pures Appl., 38, 10 (1993), 831-840. (with L. Palese, D. Paşca, M. Buican).

133. Direcţii de cercetare principale în teoria sistemelor dinamice, Stud. Cerc. Mec .Apl., 52, 2 (1993), 153-171.

134. Bifurcation manifolds in multiparametric linear stability of continua, ZAMM, 73, 7-8 (1993), T831-T833. (with D. Paşca, S. Grădinaru, M. Gavrilescu).

135. Balance equations for the vector fields defined on orientable manifolds, Tensor (N. S.), 54 (1993), 88-90. (with C. Vamoş, N. Suciu).

1992

136. Aspecte ale modelării stratului limită al atmosferei. I, Stud. Cerc. Mec. Apl., 51, 1 (1992) ), 25-41.

137. Studiul calitativ al ecuaţiilor diferenţiale, St. Cerc. Mec. Apl., 51, 3 (1992), 317-326.

138. Models of asymptotic approximation governing the atmospheric motion over a low obstacle, Stud. Cerc. Mat., 44, 3 (1992), 237-252. (with G. Marinoschi).

139. Linear stability of a turbulent flow of Maxwell fluids in pipes, Rev. Roum. Math. Pures Appl., 37, 7 (1992), 579-586 (with C. Chiujdea, R. Florea).

140. Bifurcation problems in linear stability of continua, Quaderni. Dipto. di Mat. Univ. Bari, 1 (1992). (with I. Oprea).

141. Aspecte ale modelării stratului limită al atmosferei. II, Stud. Cerc. Mec. Apl., 51, 2 (1992).

xxxx

1991

142. Modelarea matematică în mecanica fluidelor, St. Cerc. Mec. Apl., 50, 3-4 (1991), 295-298.

143. Efectul Toms, St. Cerc. Mec. Apl., 50, 5-6 (1991), 305-321. (with C. Chiujdea).

144. Linear stability of a turbulent flow of Maxwell fluids in pipes, Université de Metz, 21/1991, (with C. Chiujdea, R. Florea).

145. Evolution of the concept of asymptotic approximation, Noesis, 17 (1991), 45-50.

1990

146. Models of asymptotic approximation for synoptical flows, Zeitschrift für Meteorologie, 40, 1 (1990), 14-20. (with C. Vamoş).

147. Neutral stability curves for a thermal convection problem. II. The case of multiple solutions of the characteristic equation, Acta Mechanica, 81 (1990), 115-119. (with I. Oprea).

148. Metode numerice în teoria bifurcaţiei. III. Soluţii periodice, Stud. Cerc. Fiz., 42, 1 (1990), 117-125. (with I. Oprea).

149. Scenarii de turbulenţă în cadrul haosului determinist, Stud. Cerc. Mec. Apl., 49, 4 (1990), 413-417. (with C. Vamoş, N. Suciu).

150. Asimptote oblice din punctul de vedere al aproximaţiei asimptotice, Gaz. Mat. M, 2 (1990), 58-60.

151. On a hydrodynamic-social analogy, Rev. Roum. Philos. Logique, 34, 1-2 (1990), 100-102.

1989

152. Bifurcation manifolds in a multiparametric eigenvalue problem for linear hydromagnetic stability theory, Mathematica - Anal. Numér. Théor. Approx., 18, 2 (1989), 123-138 (with I. Oprea, C. Oprea).

153. Model de aproximaţie asimptotică de ordinul patru pentru ecuaţiile meteorologice primitive când numărul Rossby tinde la zero, St. Cerc. Meteorolgie, 3 (1989), 13-21 (with C. Vamoş).

154. Filtred equations as an asymptotic approximation model, Meteorology and Hydrology, 19, 2 (1989), 21-22 (with C. Vamoş).

155. Boundary layer separation I. Bubbles on leading edges, Rev. Roum. Sci. Tech.-Méc. Appl., 34, 5 (1989), 509-525, (with H. Dumitrescu, Al. Dumitrache).

156. Stabilitatea mişcării lichidelor pe un plan înclinat, Stud. Cerc. Mec. Apl., 48, 5 (1989).

157. Comparative study of the analytic methods used to solve problems in hydromagnetic stability theory, An. Univ. Bucureşti, Mat., 38, 1 (1989), 15-20 (with A. Setelecan).

158. Lagrange and the calculus of variations, Noesis, 15 (1989), 29-35.

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159. Fractalii şi unele aplicaţii ale lor, Stud. Cerc. Fiz., 41, 3 (1989), 269-288. 160. Suprafeţe neutrale bifurcate într-o problemă de inhibiţie a convecţiei

termice datorită unui câmp magnetic, Stud. Cerc. Mec. Apl., 48, 3 (1989), 263-278. (with I. Oprea).

1988

161. Metode numerice în teoria bifurcaţiei. II. Soluţii staţionare în cazul infinit dimensional, Stud. Cerc. Fiz., 40, 1 (1988), 7-18. (with I. Oprea).

162. Bifurcation (catastrophe) surfaces in multiparametric eigenvalue problems in hydromagnetic stability theory, Bull. Inst. Politehn. Bucureşti, Ser. Construc. Maş, 50 (1988), 9-12 (with I. Oprea).

163. The bifurcation curve of characteristic equation provides the bifurcation point of the neutral curve of some elastic stability, Mathematica - Anal. Numér. Théor. Approx., 17, 2 (1988), 141-145 (with I. Oprea).

1987

164. Metode de rezolvare a unor probleme de valori proprii care apar în stabilitatea hidrodinamică liniară, St.Cerc.Fiz., 39, 1 (1987), 3-25 (with A. Setelecan).

165. Notă asupra unor probleme izoperimetrice în calculul elicei de randament maxim, St.Cerc.Mec.Apl., 46, 5 (1987), 478-482.

166. Exact solutions for some instability of Bénard type, Rev. Roum. Phys., 32, 4 (1987), 391-397.

1986

167. Proiectarea aerodinamică a elicei de randament maxim, St.Cerc. Mec.Apl., 45, 2 (1986), 129-141 (with H. Dumitrecsu, Al. Dumitrache).

168. Bifurcaţia stratului limită, BITNAV, 3 (1986), 148-149. 169. Metode numerice în teoria bifurcaţiei, Stud. Cerc. Fiz., 38, 10 (1986), 912-

924 (with I. Oprea). 1984

170. Echilibrul plasmei în sisteme toroidale şi stabilitatea sa macroscopică, St.Cerc.Fiz., 36, 1 (1984), 86-110.

1983

171. Neustanovivseesia ploscoe dvijenie tipa Puazeilia dlia jidkostei Rivlina-Eriksena, PMM, 47, 2 (1983), 342-344. (with S.S. Chetti).

172. Stabilitatea şi ramificarea în contextul sinergeticii, St.Cerc.Mec.Apl., 42, 2 (1983), 174-180.

1982

173. Bifurcation (catastrophe) surfaces for a problem in hydromagnetic stability, Rev. Roum. Math. Pures Appl., 27, 3 (1982), 335-337.

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174. Neutral stability curves for a thermal convection problem, Analele Univ. din Craiova, Secţia Mat. Fiz.-Chim., X (1982), 51-53 (with I. Oprea).

175. Characteristic equations for some eigenvalue problems in hydromagnetic stability theory, Mathematica, Cluj, 24 (47), 1-2 (1982), 31-41.

1981

176. On the nonexistence of regular solutions of a Blasius-like equation in the theory of the boundary layer of finite depth, Rev. Roum. Math. Pures Appl., 26, 6 (1981), 849-854 (with M. Moroianu).

177. . Recent results in fluid mechanics, Preprint 2, Univ. "Babeş - Bolyai", Fac. Mat., Cluj-Napoca, 1981.

178. On a Bénard convection in the presence of dielectrophoretic forces, J. Appl. Mech., 48 , 4 (1981), 980-981 (with O. Polotzka).

1980

179. Neutral stability curves for a thermal convection problem, Acta Mechanica, 37 (1980), 165-168 (with V. Cardoş).

1978

180. Bounds for linear characteristics of Couette and Poiseuille flows, Rev. Roum. Math. Pures et Appl., 23, 5 (1978), 707-720 (with Tr. Bădoiu).

1977

181. Stability of the Couette flow of a viscoelastic fluid. II, Rev. Roum. Math. Pures et Appl., 22, 9 (1977), 1223-1233. (with O. Polotzka).

182. Metode analitice în studiul fenomenologic al stabilităţii mişcării fluidelor vâscoase incompresibile descrise de soluţii generalizate ale ecuaţiilor Navier-Stokes, St.Cerc.Mat., 29, 6 (1977), 603-619.

1976

183. Universal criteria of hydrodynamic stability, Rev. Roum. Math. Pures et Appl., 21, 3 (1976), 287-302.

1973

184. Stability of the Couette flow of a viscoelastic fluid, Rev. Roum. Math. Pures et Appl., 18, 9 (1973), 1371-1374.

185. Teorema lui Squire pentru o mişcare într-un mediu poros, Petrol şi Gaze, 24, 11 (1973).

1972

186. Linear Couette flow stability for arbitrary gap between two rotating cylinders, Rev. Roum. Math. Pures et Appl., 17, 4 (1972).

187. Stability of spiral flow and of the flow in a curved channel, Rev. Roum. Math. Pures et Appl., 17, 3 (1972), 353-357.

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1971

188. Theorems of Joseph's type in hydrodynamic stability theory, Rev.Roum.Math.Pures et Appl., 16, 3 (1971), 355-362.

189. On the Kelvin-Helmholtz instability in presence of porous media, Rev. Roum. Math. Pures et Appl., 16, 1 (1971), 27-39 (with Şt. I. Gheorghiţă).

190. On the neutral stability of the Couette flow between two rotating cylinders, Rev.Roum.Math.Pures et Appl. 16, 4 (1971), 499-502.

191. Instability of two superposed liquids in a circular tube in the presence of a porous medium, Rev.Roum.Math.Pures et Appl., 16, 5 (1971), 677-680, (with Şt. I. Gheorghiţă).

1970

192. Note on Joseph’s inequalities in stability theory, ZAMP, 21, 2 (1970), 258-260.

193. Sur la stabilité linéaire des mouvements plans des fluides, Comptes Rendus, Paris, Série A, 271 (1970), 559-561.

194. . Sufficient conditions for linear stability of two Ladyzhenskaya type fluids, Rev.Roum.Math.Pures et Appl., 15, 6 (1970), 819-823.

195. Contribuţii la studiul stabilităţii liniare a mişcării fluidelor, St. Cerc. Mat., 22, 9 (1970), 1247-1333. (The PhD Thesis of A. G.)

1969

196. Asupra soluţiilor asimptotice ale lui Heisenberg, St. Cerc. Mat., 21, 5 (1969), 747-750.

197. On a relationship between Heisenberg and Tollmien solutions, Rev.Roum.Math.Pures et Appl., 14, 7 (1969), 991-998.

198. Improvement of one of Joseph's theorems and one of its applications, Rev. Roum. Math. Pures et Appl., 14, 8 (1969), 1089-1092.

199. Criterii de stabilitate liniară a mişcării plan paralele a unui fluid nenewtonian, St.Cerc.Mat., 21, 7 (1969), 1027-1036.

200. On the generalized Tollmien solutions of the Rayleigh equation for a general velocity profile, Bull.Math. de la Soc. Sci.Math. de la R.S. de Roumanie, 13 (61), 2 (1969), 147-158.

1966

201. Corecţii de compresibilitate pentru profile von Mises, St.Cerc.Mat., 18, 2(1966), 301-308.

ON SPECIAL DIFFERENTIAL SUBORDINATIONSUSING MULTIPLIER TRANSFORMATION ANDRUSCHEWEYH DERIVATIVE

ROMAI J., 6, 2(2010), 1–14

Alina Alb LupasDepartment of Mathematics and Computer Science,University of Oradea, [email protected]

Abstract In the present paper we define a new operator using the multiplier transformation andRuscheweyh derivative. Denote by RIαn,λ,l the operator given by RIαn,λ,l : A → A,

RIαn,λ,l f (z) = (1−α)Rn f (z)+αI (n, λ, l) f (z), for z ∈ U,where Rn f (z) denote the Ruscheweyhderivative, I (n, λ, l) f (z) is the multiplier transformation and An = f ∈ H(U) : f (z) =

z + an+1zn+1 + . . . , z ∈ U is the class of normalized analytic functions with A1 = A. Acertain subclass, denoted by RIn (δ, λ, l, α) , of analytic functions in the open unit discis introduced by means of the new operator. By making use of the concept of differ-ential subordination we will derive various properties and characteristics of the classRIn (δ, λ, l, α) . Also, several differential subordinations are established regardind theoperator RIαn,λ,l.

Keywords: differential subordination, convex function, best dominant, differential operator, Ruscheweyhderivative, multiplier transformations.2000 MSC: 30C45, 30A20, 34A40. .

1. INTRODUCTIONDenote by U the unit disc of the complex plane, U = z ∈ C : |z| < 1 and H(U)

the space of holomorphic functions in U.Let

A (p, n) = f ∈ H(U) : f (z) = zp +

∞∑

j=p+n

a jz j, z ∈ U,

with A (1, n) = An, A (1, 1) = A1 = A and

H[a, n] = f ∈ H(U) : f (z) = a + anzn + an+1zn+1 + . . . , z ∈ U,

where p, n ∈ N, a ∈ C.Denote by K =

f ∈ A : Re z f ′′(z)

f ′(z) + 1 > 0, z ∈ U, the class of normalized convex

functions in U.If f and g are analytic functions in U, we say that f is subordinate to g, written

f ≺ g, if there is a function w analytic in U, with w(0) = 0, |w(z)| < 1, for all z ∈ U

1

2 Alina Alb Lupas

such that f (z) = g(w(z)) for all z ∈ U. If g is univalent, then f ≺ g if and only iff (0) = g(0) and f (U) ⊆ g(U).

Let ψ : C3 × U → C and h an univalent function in U. If p is analytic in U andsatisfies the (second-order) differential subordination

ψ(p(z), zp′(z), z2 p′′(z); z) ≺ h(z), for z ∈ U, (1)

then p is called a solution of the differential subordination. The univalent function qis called a dominant of the solutions of the differential subordination, or more simplya dominant, if p ≺ q for all p satisfying (1).

A dominant q that satisfies q ≺ q for all dominants q of (1) is said to be the bestdominant of (1). The best dominant is unique up to a rotation of U.

Definition 1.1. (Ruscheweyh [3]) For f ∈ A, n ∈ N, the operator Rn is defined byRn : A→ A,

R0 f (z) = f (z)R1 f (z) = z f ′ (z)

...

(n + 1) Rn+1 f (z) = z(Rn f (z)

)′+ nRn f (z) , for z ∈ U.

Remark 1.1. If f ∈ A, f (z) = z +∑∞

j=2 a jz j, then Rn f (z) = z +∑∞

j=2 Cnn+ j−1a jz j, for

z ∈ U.

Definition 1.2. [6] For f ∈ A(p, n), p, n ∈ N, m ∈ N∪ 0, λ, l ≥ 0, the operatorIp (m, λ, l) f (z) is defined by the following infinite series

Ip (m, λ, l) f (z) := zp +

∞∑

j=p+n

(p + λ ( j − 1) + l

p + l

)m

a jz j.

Remark 1.2. It follows from the above definition that

Ip (0, λ, l) f (z) = f (z),

(p + l) Ip (m + 1, λ, l) f (z) =[p(1 − λ) + l

]Ip (m, λ, l) f (z) + λz

(Ip (m, λ, l) f (z)

)′,

for z ∈ U.

Remark 1.3. If p = 1, n = 1, we have A(1, 1) = A1 = A, I1 (m, λ, l) f (z) = I (m, λ, l)and

(l + 1) I (m + 1, λ, l) f (z) = [l + 1 − λ] I (m, λ, l) f (z) + λz (I (m, λ, l) f (z))′ ,

for z ∈ U.

On special differential subordinations using multiplier transformation... 3

Remark 1.4. If f ∈ A, f (z) = z +∑∞

j=2 a jz j, then

I (m, λ, l) f (z) = z +

∞∑

j=2

(1 + λ ( j − 1) + l

l + 1

)m

a jz j,

for z ∈ U.

Remark 1.5. For l = 0, λ ≥ 0, the operator Dmλ = I (m, λ, 0) was introduced and

studied by Al-Oboudi [5], which reduced to the Salagean differential operator S m =

I (m, 1, 0) [4] for λ = 1. The operator Iml = I (m, 1, l) was studied recently by Cho

and Srivastava [7] and Cho and Kim [8]. The operator Im = I (m, 1, 1) was studiedby Uralegaddi and Somanatha [12], the operator Dδ

λ = I (δ, λ, 0), with δ ∈ R, δ ≥ 0,was introduced by Acu and Owa [1].

Lemma 1.1. (Hallenbeck and Ruscheweyh [1, Th. 3.1.6, p. 71]) Let h be a convexfunction with h(0) = a, and let γ ∈ C\0 be a complex number with Re γ ≥ 0. Ifp ∈ H[a, n] and

p(z) +1γ

zp′(z) ≺ h(z), for z ∈ U,

thenp(z) ≺ g(z) ≺ h(z), for z ∈ U,

where g(z) =γ

nzγ/n∫ z

0 h(t)tγ/n−1dt, for z ∈ U.

Lemma 1.2. (Miller and Mocanu [1]) Let g be a convex function in U and let h(z) =

g(z) + nαzg′(z), for z ∈ U, where α > 0 and n is a positive integer.If p(z) = g(0) + pnzn + pn+1zn+1 + . . . , for z ∈ U is holomorphic in U and

p(z) + αzp′(z) ≺ h(z), for z ∈ U,

thenp(z) ≺ g(z), for z ∈ U,

and this result is sharp.

2. MAIN RESULTSDefinition 2.1. Let α, λ, l ≥ 0, n ∈ N. Denote by RIαn,λ,l the operator given byRIαn,λ,l : A→ A,

RIαn,λ,l f (z) = (1 − α)Rn f (z) + αI (n, λ, l) f (z), for z ∈ U.

Remark 2.1. If f ∈ A, f (z) = z +∑∞

j=2 a jz j, then

RIαn,λ,l f (z) = z +∑∞

j=2

α( 1+λ( j−1)+l

l+1

)n+ (1 − α) Cn

n+ j−1

a jz j, for z ∈ U.

4 Alina Alb Lupas

Remark 2.2. For α = 0, RI0n,λ,l f (z) = Rn f (z), where z ∈ U and for α = 1,

RI1n,λ,l f (z) = I (n, λ, l) f (z), where z ∈ U, which was studied in [3].For l = 0, we obtain RIαn,λ,0 f (z) = RDn

1,α f (z) which was studied in [4] and forl = 0 and λ = 1, we obtain RIαn,1,0 f (z) = Ln

α f (z) which was studied in [2].For n = 0, RIα0,λ,l f (z) = (1 − α) R0 f (z) + αI (0, λ, l) f (z) = f (z) = R0 f (z) =

I (0, λ, l) f (z), where z ∈ U.

Definition 2.2. Let δ ∈ [0, 1), α, λ, l ≥ 0 and n ∈ N. A function f ∈ A is said to be inthe class RIn (δ, λ, l, α) if it satisfies the inequality

Re(RIαn,λ,l f (z)

)′> δ, for z ∈ U. (2)

Theorem 2.1. The set RIn (δ, λ, l, α) is convex.

Proof. Let the functions

f j (z) = z +

∞∑

j=2

a jkz j, for k = 1, 2, z ∈ U,

be in the class RIn (δ, λ, l, α). It is sufficient to show that the function

h (z) = η1 f1 (z) + η2 f2 (z)

is in the class RIn (δ, λ, l, α) with η1 and η2 nonnegative such that η1 + η2 = 1.Since h (z) = z +

∑∞j=2

(η1a j1 + η2a j2

)z j, for z ∈ U, then

RIαn,λ,lh (z) = z+∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

(η1a j1 + η2a j2

)z j, for z ∈ U.

(3)Differentiating (3) we obtain

(RIαn,λ,lh (z)

)′= 1+

∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

(η1a j1 + η2a j2

)jz j−1,

for z ∈ U. Hence

Re(RIαn,λ,lh (z)

)′= 1 + Re

(η1

∑∞j=2 j

α( 1+λ( j−1)+l

l+1

)n+ (1 − α) Cn

n+ j−1

a j1z j−1

)+

+Re(η2

∑∞j=2 j

α( 1+λ( j−1)+l

l+1

)n+ (1 − α) Cn

n+ j−1

a j2z j−1

).

(4)Taking into account that f1, f2 ∈ RIn (δ, λ, l, α) we deduce

Re

ηk

∞∑

j=2

jα

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

a jkz j−1

> ηk (δ − 1) , for k = 1, 2.

(5)


Using (5) we get from (4)

Re(RIαn,λ,lh (z)

)′> 1 + η1 (δ − 1) + η2 (δ − 1) = δ, for z ∈ U,

which is equivalent that RIn (δ, λ, l, α) is convex.

Theorem 2.2. Let g be a convex function in U and let h (z) = g (z)+ 1c+2 zg′ (z) , where

z ∈ U, c > 0.If f ∈ RIn (δ, λ, l, α) and F (z) = Ic ( f ) (z) = c+2

zc+1

∫ z0 tc f (t) dt, for z ∈ U, then

(RIαn,λ,l f (z)

)′ ≺ h (z) , for z ∈ U, (6)

implies (RIαn,λ,lF (z)

)′ ≺ g (z) , for z ∈ U,


Proof. We obtain that

zc+1F (z) = (c + 2)∫ z

0tc f (t) dt. (7)

Differentiating (7), with respect to z, we have (c + 1) F (z) + zF′ (z) = (c + 2) f (z)and

(c + 1) RIαn,λ,lF (z) + z(RIαn,λ,lF (z)

)′= (c + 2) RIαn,λ,l f (z) , for z ∈ U. (8)

Differentiating (8) we have

(RIαn,λ,lF (z)

)′+

1c + 2

z(RIαn,λ,lF (z)

)′′=

(RIαn,λ,l f (z)

)′, for z ∈ U. (9)

Using (9), the differential subordination (6) becomes

(RIαn,λ,lF (z)

)′+

1c + 2

z(RIαn,λ,lF (z)

)′′ ≺ g (z) +1

c + 2zg′ (z) . (10)

If we denotep (z) =

(RIαn,λ,lF (z)

)′, for z ∈ U, (11)

then p ∈ H [1, 1] .Replacing (11) in (10) we obtain

p (z) +1

c + 2zp′ (z) ≺ g (z) +

1c + 2

zg′ (z) , for z ∈ U.

Using Lemma 1.2 we have

p (z) ≺ g (z) , for z ∈ U, i.e.(RIαn,λ,lF (z)

)′ ≺ g (z) , for z ∈ U,

6 Alina Alb Lupas

and g is the best dominant.

Example 2.1. If f ∈ RD1(1, 1, 0, 1

2

), then f ′ (z) + z f ′′ (z) ≺ 3−2z

3(1−z)2 implies F′ (z) +

zF′′ (z) ≺ 11−z , where F (z) = 3

z2

∫ z0 t f (t) dt.

Theorem 2.3. Let h (z) =1+(2δ−1)z

1+z , where δ ∈ [0, 1) and c > 0.If α, λ, l ≥ 0, n ∈ N and Ic ( f ) (z) = c+2

zc+1

∫ z0 tc f (t) dt, for z ∈ U, then

Ic [RIn (δ, λ, l, α)] ⊂ RIn(δ∗, λ, l, α

), (12)

where δ∗ = 2δ − 1 + (c + 2) (2 − 2δ)∫ 1

0tc+1

t+1 dt.

Proof. The function h is convex and using the same steps as in the proof of Theorem2.2 we get from the hypothesis of Theorem 2.3 that

p (z) +1

c + 2zp′ (z) ≺ h (z) ,

where p (z) is defined in (11).Using Lemma 1.1 we deduce that

p (z) ≺ g (z) ≺ h (z) , i.e.(RIαn,λ,lF (z)

)′ ≺ g (z) ≺ h (z) ,

where

g (z) =c + 2zc+2

∫ z

0tc+1 1 + (2δ − 1) t

1 + tdt = 2δ − 1 +

(c + 2) (2 − 2δ)zc+2

∫ z

0

tc+1

t + 1dt.

Since g is convex and g (U) is symmetric with respect to the real axis, we deduce

Re(RIαn,λ,lF (z)

)′ ≥ min|z|=1

Re g (z) = Re g (1) = δ∗ = 2δ−1+(c + 2) (2 − 2δ)∫ 1

0

tc+1

t + 1dt.

(13)From (13) we deduce inclusion (12).

Theorem 2.4. Let g be a convex function, g(0) = 1 and let h be the function h(z) =

g(z) + zg′(z), for z ∈ U.If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

(RIαn,λ,l f (z)

)′ ≺ h(z), for z ∈ U, (14)

thenRIαn,λ,l f (z)

z≺ g(z), for z ∈ U,



Proof. By using the properties of operator RIαn,λ,l, we have

RIαn,λ,l f (z) = z +

∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1


Consider

p(z) =RIαn,λ,l f (z)

z=

z +

∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

a jz j

z=

= 1 + p1z + p2z2 + ..., for z ∈ U.

We deduce that p ∈ H[1, 1].Let RIαn,λ,l f (z) = zp(z), for z ∈ U. By differentiating we obtain

(RIαn,λ,l f (z)

)′= p(z) + zp′(z),

for z ∈ U.Then (14) becomes

p(z) + zp′(z) ≺ h(z) = g(z) + zg′(z), for z ∈ U.

By using Lemma 1.2, we have

p(z) ≺ g(z), for z ∈ U, i.e.RIαn,λ,l f (z)

z≺ g(z), for z ∈ U.

Theorem 2.5. Let h be an holomorphic function which satisfies the inequalityRe

(1 +

zh′′(z)h′(z)

)> −1

2 , for z ∈ U, and h(0) = 1.If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

(RIαn,λ,l f (z)

)′ ≺ h(z), for z ∈ U, (15)

thenRIαn,λ,l f (z)

z≺ q(z), for z ∈ U,

where q(z) = 1z

∫ z0 h(t)dt. The function q is convex and it is the best dominant.

Proof. Let

p(z) =RIαn,λ,l f (z)

z=

z +∑∞

j=2

α( 1+λ( j−1)+l

l+1

)n+ (1 − α) Cn

n+ j−1

a jz j

z

= 1 +

∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

a jz j−1 = 1 +

∞∑

j=2

p jz j−1,

8 Alina Alb Lupas

for z ∈ U, p ∈ H[1, 1].Differentiating, we obtain

(RIαn,λ,l f (z)

)′= p(z)+zp′(z), for z ∈ U and (15) becomes

p(z) + zp′(z) ≺ h(z), for z ∈ U.

Using Lemma 1.1, we have

p(z) ≺ q(z), for z ∈ U, i.e.RIαn,λ,l f (z)

z≺ q(z) =

1z

∫ z

0h(t)dt, for z ∈ U,

and q is the best dominant.

Theorem 2.6. Let g be a convex function such that g (0) = 1 and let h be the functionh (z) = g (z) + zg′ (z), for z ∈ U.

If α, λ, l ≥ 0, n ∈ N, f ∈ A and the differential subordination

zRIαn+1,λ,l f (z)

RIαn,λ,l f (z)

′≺ h (z) , for z ∈ U (16)

holds, thenRIαn+1,λ,l f (z)

RIαn,λ,l f (z)≺ g (z) , for z ∈ U,


Proof. For f ∈ A, f (z) = z +∑∞

j=2 a jz j we have

RIαn,λ,l f (z) = z +∑∞

j=2

α( 1+λ( j−1)+l

l+1

)n+ (1 − α) Cn

n+ j−1


Consider

p(z) =RIαn+1,λ,l f (z)

RIαn,λ,l f (z)=

z +∑∞

j=2

α(1+λ( j−1)+l

l+1

)n+1+ (1 − α) Cn+1

n+ j

a jz j

z +∑∞

j=2

α( 1+λ( j−1)+l

l+1

)n+ (1 − α) Cn

n+ j−1

a jz j

.

We have

p′ (z) =

(RIαn+1,λ,l f (z)

)′

RIαn,λ,l f (z)− p (z) ·

(RIαn,λ,l f (z)

)′

RIαn,λ,l f (z)

and we obtain

p (z) + z · p′ (z) =

zRIαn+1,λ,l f (z)

RIαn,λ,l f (z)

′

.Relation (16) becomes

p(z) + zp′(z) ≺ h(z) = g(z) + zg′(z), for z ∈ U.



p(z) ≺ g(z), for z ∈ U, i.e.RIαn+1,λ,l f (z)

RIαn,λ,l f (z)≺ g(z), for z ∈ U.

Theorem 2.7. Let g be a convex function such that g(0) = 0 and let h be the functionh(z) = g(z) + zg′(z), for z ∈ U.


(n + 1) RIαn+1,λ,l f (z) − (n − 1) RIαn,λ,l f (z)−

α

(n + 1 − l + 1

λ

) [I (n + 1, λ, l) f (z) − I (n, λ, l) f (z)

] ≺ h(z), for z ∈ U (17)

holds, thenRIαn,λ,l f (z) ≺ g(z), for z ∈ U.

This result is sharp.

Proof. Let

p(z) = RIαn,λ,α f (z) = (1 − α)Rn f (z) + αI (n, λ, l) f (z) = (18)

= z +

∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

a jz j = p1z + p2z2 + ....

We deduce that p ∈ H[0, 1].By using the properties of operators RIαn,λ,l, Rn and I (n, λ, l), after a short calcula-

tion, we obtain

p (z) + zp′ (z) = (n + 1) RIαn+1,λ,l f (z)− (n − 1) RIαn,λ,l f (z)−−α

(n + 1 − l+1

λ

)[I (n + 1, λ, l) f (z)− I (n, λ, l) f (z)].

Using the notation in (18), the differential subordination becomes

p(z) + zp′(z) ≺ h(z) = g(z) + zg′(z).


p(z) ≺ g(z), for z ∈ U, i.e. RIαn,λ,l f (z) ≺ g(z), for z ∈ U,


Theorem 2.8. Let h(z) =1+(2β−1)z

1+z be a convex function in U, where 0 ≤ β < 1.

10 Alina Alb Lupas

If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination


−α(n + 1 − l + 1

λ

) [I (n + 1, λ, l) f (z) − I (n, λ, l) f (z)

] ≺ h(z), for z ∈ U, (19)

thenRIαn,λ,l f (z) ≺ q(z), for z ∈ U,

where q is given by q(z) = 2β− 1 + 2(1− β) ln(1+z)z , for z ∈ U. The function q is convex

and it is the best dominant.

Proof. Following the same steps as in the proof of Theorem 2.7 and consideringp(z) = RIαn,λ,l f (z), the differential subordination (19) becomes

p(z) + zp′(z) ≺ h(z) =1 + (2β − 1)z

1 + z, for z ∈ U.

By using Lemma 1.1 for γ = 1 and n = 1, we have p(z) ≺ q(z), i.e.,

RIαn,λ,l f (z) ≺ q(z) =1z

∫ z

0h(t)dt =

1z

∫ z

0

1 + (2β − 1)t1 + t

dt = 2β−1+2(1−β)1z

ln(z+1),

for z ∈ U.


[1 +

zh′′(z)h′(z)

]> −1

2 , for z ∈ U, and h (0) = 0.If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination


α

(n + 1 − l + 1

λ

) [I (n + 1, λ, l) f (z) − I (n, λ, l) f (z)

] ≺ h(z), for z ∈ U, (20)

thenRIαn,λ,l f (z) ≺ q(z), for z ∈ U,

where q is given by q(z) = 1z

∫ z0 h(t)dt. The function q is convex and it is the best

dominant.

Proof. Using the properties of operator RIαn,λ,l and considering p (z) = RIαn,λ,l f (z), weobtain

p(z) + zp′(z) = (n + 1) RIαn+1,λ,l f (z) − (n − 1) RIαn,λ,l f (z)−

α

(n + 1 − l + 1

λ

) [I (n + 1, λ, l) f (z) − I (n, λ, l) f (z)

], for z ∈ U.


Then (20) becomes

p(z) + zp′(z) ≺ h(z), for z ∈ U.

Since p ∈ H[0, 1], using Lemma 1.1, we deduce

p(z) ≺ q(z), for z ∈ U, i.e. RIαn,λ,l f (z) ≺ q(z) =1z

∫ z



Theorem 2.10. Let g be a convex function such that g(0) = 1 and let h be the functionh(z) = g(z) + zg′(z), for z ∈ U.


(n + 1) (n + 2)z

RIαn+2,λ,l f (z) − (n + 1) (2n + 1)z

RIαn+1,λ,l f (z) +n2

zRIαn,λ,l f (z) +

α

z

[(l + 1)2

λ2 − (n + 1) (n + 2)]

I (n + 2, λ, l) f (z)−

α

z

2 (l + 1 − λ)

(l + 1

)

λ2 − (n + 1) (2n + 1)

I (n + 1, λ, l) f (z) +

α

z

[(l + 1 − λ)2

λ2 − n2]

I (n, λ, l) f (z) ≺ h(z), for z ∈ U (21)

holds, then[RIαn,λ,l f (z)]′ ≺ g(z), for z ∈ U.

This result is sharp.

Proof. Let

p(z) =(RIαn,λ,l f (z)

)′= (1 − α)

(Rn f (z)

)′+ α (I (n, λ, l) f (z))′ (22)

= 1 +

∞∑

j=2

α

(1 + λ ( j − 1) + l

l + 1

)n

+ (1 − α) Cnn+ j−1

ja jz j−1 = 1 + p1z + p2z2 + ....

We deduce that p ∈ H[1, 1].By using the properties of operators RIαn,λ,l, Rn and I (n, λ, l), after a short calcula-

tion, we obtain

p (z) + zp′ (z) =(n+1)(n+2)

z RIαn+2,λ,l f (z) − (n+1)(2n+1)z RIαn+1,λ,l f (z) +

+n2

z RIαn,λ,l f (z) + αz

[(l+1)2

λ2 − (n + 1) (n + 2)]

I (n + 2, λ, l) f (z)−−αz

[2(l+1−λ)(l+1)

λ2 − (n + 1) (2n + 1)]

I (n + 1, λ, l) f (z) +

+αz

[(l+1−λ)2

λ2 − n2]

I (n, λ, l) f (z) .

12 Alina Alb Lupas

Using the notation in (22), the differential subordination becomes

p(z) + zp′(z) ≺ h(z) = g(z) + zg′(z).


p(z) ≺ g(z), for z ∈ U, i.e.(RIαn,λ,l f (z)

)′ ≺ g(z), for z ∈ U,


Example 2.2. If n = 1, α = 1, λ = 1, l = 0, f ∈ A, we deduce that f ′(z) + 3z f ′′(z) +

z2 f ′′′(z) ≺ g(z) + zg′(z), which yields that f ′(z) + z f ′′(z) ≺ g(z), for z ∈ U.

Theorem 2.11. Let h(z) =1+(2β−1)z

1+z be a convex function in U, where 0 ≤ β < 1.If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

(n + 1) (n + 2)z

RIαn+2,λ,l f (z) − (n + 1) (2n + 1)z


zRIαn,λ,l f (z) +

α

z

[(l + 1)2

λ2 − (n + 1) (n + 2)]

I (n + 2, λ, l) f (z)−

α

z

2 (l + 1 − λ)

(l + 1

)

λ2 − (n + 1) (2n + 1)

I (n + 1, λ, l) f (z) +

α

z

[(l + 1 − λ)2

λ2 − n2]

I (n, λ, l) f (z) ≺ h(z), for z ∈ U, (23)

then (RIαn,λ,l f (z)

)′ ≺ q(z), for z ∈ U,

where q is given by q(z) = 2β− 1 + 2(1− β) ln(1+z)z , for z ∈ U. The function q is convex

and it is the best dominant.

Proof. Following the same steps as in the proof of Theorem 2.10 and consideringp(z) =

(RIαn,λ,l f (z)

)′, the differential subordination (23) becomes

p(z) + zp′(z) ≺ h(z) =1 + (2β − 1)z

1 + z, for z ∈ U.

By using Lemma 1.1 for γ = 1 and n = 1, we have p(z) ≺ q(z), i.e.,

(RIαn,λ,l f (z)

)′ ≺ q(z) =1z

∫ z

0h(t)dt =

1z

∫ z

0

1 + (2β − 1)t1 + t

dt = 2β−1+2(1−β)1z

ln(z+1),

for z ∈ U.



[1 +

zh′′(z)h′(z)

]> −1

2 , for z ∈ U, and h (0) = 1.If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

(n + 1) (n + 2)z

RIαn+2,λ,l f (z) − (n + 1) (2n + 1)z


zRIαn,λ,l f (z) +

α

z

[(l + 1)2

λ2 − (n + 1) (n + 2)]

I (n + 2, λ, l) f (z)−

α

z

2 (l + 1 − λ)

(l + 1

)

λ2 − (n + 1) (2n + 1)

I (n + 1, λ, l) f (z) +

α

z

[(l + 1 − λ)2

λ2 − n2]

I (n, λ, l) f (z) ≺ h(z), for z ∈ U, (24)

then (RIαn,λ,l f (z)

)′ ≺ q(z), for z ∈ U,

where q is given by q(z) = 1z

∫ z0 h(t)dt. The function q is convex and it is the best

dominant.

Proof. Using the properties of operator RIαn,λ,l and considering p (z) =(RIαn,λ,l f (z)

)′,

we obtain

p(z)+zp′(z) =(n + 1) (n + 2)

zRIαn+2,λ,l f (z)− (n + 1) (2n + 1)

zRIαn+1,λ,l f (z)+

n2

zRIαn,λ,l f (z) +

α

z

[(l + 1)2

λ2 − (n + 1) (n + 2)]

I (n + 2, λ, l) f (z)−

α

z

2 (l + 1 − λ)

(l + 1

)

λ2 − (n + 1) (2n + 1)

I (n + 1, λ, l) f (z) +

α

z

[(l + 1 − λ)2

λ2 − n2]

I (n, λ, l) f (z) , for z ∈ U.

Then (24) becomes

p(z) + zp′(z) ≺ h(z), for z ∈ U.

Since p ∈ H[1, 1], using Lemma 1.1, we deduce

p(z) ≺ q(z), for z ∈ U, i.e.(RIαn,λ,l f (z)

)′ ≺ q(z) =1z

∫ z



14 Alina Alb Lupas

References[1] M. Acu, S. Owa, Note on a class of starlike functions, RIMS, Kyoto, 2006.

[2] A. Alb Lupas, On special differential subordinations using Salagean and Ruscheweyh operators,Math. Inequ. Appl., 12, 4(2009), 781-790.

[3] A. Alb Lupas, A special comprehensive class of analytic functions defined by multiplier transfor-mation, J. Comput. Anal. Appl., 12, 2(2010), 387-395.

[4] A. Alb Lupas, On special differential subordinations using a generalized Salagean operator andRuscheweyh derivative, J. Comput. Anal. Appl., 2011 (accepted).

[5] F.M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Ind. J.Math. Math. Sci., 27(2004), 1429-1436.

[6] A. Catas, On certain class of p-valent functions defined by new multiplier transformations, Pro-ceedings Book of the International Symposium on Geometric Function Theory and Applications,August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250.

[7] N.E. Cho, H.M. Srivastava, Argument estimates of certain analytic functions defined by a classof multiplier transformations, Math. Comput. Modelling, 37, 1-2(2003), 39-49.

[8] N.E. Cho, T.H. Kim, Multiplier transformations and strongly close-to-close functions, Bull. Ko-rean Math. Soc., 40, 3(2003), 399-410.

[9] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Marcel DekkerInc., New York, 2000.

[10] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115.

[11] G. St. Salagean, Subclasses of univalent functions, Lecture Notes in Math., Springer, Berlin,1013(1983), 362-372.

[12] B.A. Uralegaddi, C. Somanatha, Certain classes of univalent functions, Current topics in analyticfunction theory, World. Sci. Publishing, River Edge, N.Y., (1992), 371-374.

A PRODUCT FORMULA APPROACH TO ANINVERSE PROBLEM GOVERNED BYNONLINEAR PHASE-FIELD TRANSITIONSYSTEM. CASE 1D

ROMAI J., 6, 2(2010), 15–26

Tommaso Benincasa1, Costica Morosanu2

1University of Bologna, Italy2University ”Alexandru Ioan Cuza” of Iasi, Romania

Abstract In this paper we study an inverse problem, in one space dimension case, connectedwith the industrial solidification process called casting wire, as an optimal control prob-lem governed by nonlinear phase-field system with nonhomogeneous Cauchy-Neumannboundary conditions. We prove the convergence of an iterative scheme of fractionalsteps type for the optimal control problem. Moreover, necessary optimality conditionsare established for the approximating process. The advantage of such approach leads tothe idea of a numerical algorithm in order to approximate the original optimal controlproblem.

Keywords: nonlinear parabolic systems, phase-field models, optimality conditions, applications, time-dependent initial-boundary value problems, fractional steps method, inverse problems.2000 MSC: 35K55, 49N15, 62P30, 65M12, 65M32.

1. INTRODUCTION; SETTING OF THEPROBLEM

Phase-field models, strongly studied in recent years, describe the phase transitionsbetween two different phases in a pure material by a system of nonlinear parabolicequations. These models can be viewed as extensions of the classical Stefan problemin two phases. Consequently, the interface boundary between the phases can be con-structed from the so-called phase function and, phenomena associated with surfacetension and supercooling are incorporated into the model.

The mathematical literature concerning the optimal control problems associatedwith such models is in a deep process of development, as the models are suitable formany modern applications. One of them is the subject of this paper.

1.1. PROBLEM FORMULATIONDenote by Ω = (0, b1) ⊂ R, 0 < b1 < +∞. Let T > 0 and we set:Q = (0, T ) ×Ω,Σ0 = (t, x) ∈ Q, t = f (x),Σ = (0,T ) × b1,

15

16 Tommaso Benincasa, Costica Morosanu

where t = f (x) is considered to be the equation of the moving boundary separatingthe liquid and solid phases, 0 = f (b0), 0 < b0 < b1 (see Figure 1.1).Consider the following nonlinear parabolic system in one space dimension:

ρc ut +`

2ϕt = kuxx, in Q,

τϕt = ξ2ϕxx +1

2a(ϕ − ϕ3) + 2u, in Q,

(1.1)

subject to the non-homogeneous Cauchy-Neumann boundary conditions

ux + hu = w(t), ϕx = 0 on Σ, (1.2)

ux = 0, ϕx = 0 on Σ0, (1.3)

and to the initial conditions

u(0, x) = u0(x), ϕ(0, x) = ϕ0(x) on Ω0 = [b0, b1], (1.4)

where u is the reduced temperature distribution, ϕ is the phase function used to distin-guish between the phase of Ω, u0, ϕ0 : Ω −→ R are given functions,w : [0,T ] −→ R is the boundary control (the temperature surrounding at x = b1),w ∈ U where U = v ∈ L∞([0,T ]), −R ≤ v(t) ≤ 0 a.e. t ∈ [0,T ]; the positive pa-rameters ρ, c, τ, ξ, `, k, h, a, have the following physical meaning: ρ - is the density, c- is the heat capacity, τ - is the relaxation time, ξ - is the length scale of the interface,` - denotes the latent heat, k - the heat conductivity, h - the heat transfer coefficientand a is an probabilistic measure on the individual atoms (a depends on ξ).

Figure 1.1.Geometrical image of the elements in inverse problem (Pinv).

The mathematical model (1.1), introduced by Caginalp [3], has been established inthe literature as an extension of the classical two phase Stefan problem to capture theeffects of surface tension, supercooling, and superheating.

A product formula approach to an inverse problem... 17

As regards the existence, it is known that under appropriate conditions on u0, ϕ0and w, the state system (1.1)-(1.4) has a unique solution u, ϕ ∈ W = W2,1

p (Q)∩L∞(Q),p > 3/2 (see Proposition 2.1 in [6]).

Given the positive numbers d1, d2, we define:the pure liquid region : (t, x) ∈ Q, u(t, x) > d2 and ϕ(t, x) ≥ 1 + d1,the pure solid region : (t, x) ∈ Q, u(t, x) < −d2 and ϕ(t, x) ≤ −1 − d1,the separating region : (t, x) ∈ Q, |u(t, x)| < d2, |ϕ(t, x)| ≤ 1 + d1.

and we setQ0 =

(t, x) ∈ Q, f (x) ≤ t ≤ T

.

Consider the following inverse problem:

(Pinv) Given Σ0 f ind the boundary control w ∈ L∞([0, T ]) such that Q0 is in theliquid region, Q1 = Q \ Q0 is in the solid region and a neighbourhoodof Σ0 is the separating region between the liquid and the solid region.

(Pinv) is in general ”ill posed” and a common way to treat this inverse problem isto reformulate it as an optimal control problem with an appropriate cost functional.Consequently, we will concern in the present paper with an optimal control problemassociated to the inverse problem (Pinv), namely:

(P) Minimize L0(w) =β

2

∫

Q

[(u(t, x) − δ2)+]2 · χQ0

dtdx+

+12

∫

Q

(ϕ(t, x) − 1 − δ1)2 · χQ0dtdx +

12

T∫

0

w2(t) dt,

on all (u, ϕ) solution of the system (1.1)-(1.4) and for all w ∈ U

β > 0 is a given constant.In the above statement we denoted by u+ the positive part of u, i.e.

u+ =

u, if u > 0,0, if u ≤ 0.

We point out that problem (P) is an optimal problem with boundary control w(t)depending on time variable t ∈ [0, T ], being dictated by the industrial solidificationprocess like casting wire.

1.2. APPROXIMATING PROCESSWe associate to the nonlinear system (1.1)-(1.4) the following approximanting

scheme (ε > 0):

ρcuεt +`

2ϕεt = kuεxx

in Qε0 =

(t, x) ∈ Q, ε ≤ t ≤ T

,

τϕεt = ξ2ϕεxx +1

2aϕε + 2uε,

(1.5)


uεx + huε = w(t), ϕεx = 0 on Σε = [ε,T ] × b1, (1.6)

uεx = 0, ϕεx = 0 on Σε0 = (t, x) ∈ Q, ε ≤ t ≤ T , (1.7)

uε(ε, x) = u0(x) ϕε+(ε, x) = z(ε, ϕε−(ε, x)) on Ωε. (1.8)

where z(ε, ϕε−(ε, x)) is the solution of the Cauchy problem:

z′(s) +12a

z3(s) = 0, s ∈ (0, ε),

z(0) = ϕε−(ε, x), ϕε−(0, x) = ϕ0(x),

(1.9)

and ϕε+(ε, x) = limt↓ε ϕε(t, x), ϕε−(ε, x) = limt↑ε ϕε(t, x).The convergence and weak stability of the approximating scheme (1.5)-(1.9), in a

more general case (w(t, x) in place of w(t)), was studied in the paper [2].Corresponding to the approximating scheme (1.5)-(1.9), we will consider the ap-

proximating optimal control problem:

(Pε) Minimize Lε0(w) =β

2

∫

Q

[(uε(t, x) − δ2)+]2 · χQ0

dtdx+

+12

∫

Q

(ϕε(t, x) − 1 − δ1)2 · χQ0dtdx +

12

T∫

0

w2(t) dt,

on all (uε, ϕε) solution of (1.5)-(1.9) corresponding to w ∈ U.

The main result of the present paper (Theorem 2.1) says that problem (P) canbe approximated for ε → 0 by the sequence of optimal control problems (Pε) andso the computation of the approximate boundary control w(t) can be substituted bycomputation of an approximate control of (Pε).

In Section 2 we prove the convergence results regarding the sequence of optimalcontrol problem (Pε). Such a convergence scheme was studied (for an optimal controlproblem governed by nonlinear parabolic variational inequalities) by Barbu [1]. Forother works in this context see [6] and references therein. Necessary optimality con-ditions for the approximating process (Pε) (Theorem 3.1) and, a conceptual algorithmof gradient type are established in the last Section.

2. THE CONVERGENCE OF PROBLEM (Pε)The main result of this paper is

Theorem 2.1. Let w∗ε be a sequence of optimal controllers for problem (Pε). Then

limε→0

inf Lε0(w) = inf L0(w); w ∈ U (2.1)

andlimε→0

L0(w∗ε) = inf L0(w); w ∈ U. (2.2)


Moreover, every weak limit point of w∗ε is an optimal controller for problem (P).

Remark 2.1. Theorem 2.1 amounts to saying that (Pε) approximates problem (P)and, an optimal controller w∗ε of (Pε) is a suboptimal controller for problem (P).

The main ingredient in the proof of the Theorem 2.1 is the following Lemma.

Lemma 2.1. If w∗ε is a sequence of optimal controllers for problems (Pε) then thereexists εn −→ 0 such that

w∗εn−→ w∗ weakly star in L∞(Σ), (2.3)

u∗εn−→ u∗ strongly in L2((0,T ); H1(Ω)), (2.4)

ϕ∗εn−→ ϕ∗ strongly in L2((0,T ); H1(Ω)), (2.5)

where (u∗εn, ϕ∗εn

,w∗εn) = (u

w∗εnεn , ϕ

w∗εnεn ,w∗εn

) is the solution to (1.5)-(1.8) corresponding tow = w∗εn

and (u∗, ϕ∗,w∗) = (uw∗ , ϕw∗ ,w∗) is the solution to (1.1)-(1.4) correspondingto w = w∗.

Proof. Details on the demonstration of this Lemma can be found in the work [6,Lemma 3.1, pp. 11]. We omit them.

We can now give the proof of Theorem 2.1.

Proof. Let w∗ε be an optimal controller for problem (Pε) and let (u∗ε, ϕ∗ε,w∗ε) be thecorresponding solution of (1.5)-(1.8) with w = w∗ε. Lemma 2.1 above allows us toconclude that there exist w∗ ∈ L∞([0, T ]) and εn such that relations (2.3)-(2.5) arevalid. Since:

u −→ β

2

∫

Q

((u(t, x) − δ2)+)2 · χQ0

dtdx,

ϕ −→ 12

∫

Q

(ϕ(t, x) − 1 − δ1)2 · χQ0dtdx,

w −→ 12

T∫

0

w2(t) dt

are convex continuous functions, it follows that these are weakly lower semicontinu-ous functions. Hence

L0(w∗) ≤ lim infn−→∞ Lεn

0 (w∗εn). (2.6)

Let w∗ be an optimal controller for problem (P). Since w∗εnis an optimal controller

for problem (Pεn) it follows that

Lεn0 (w∗εn

) ≤ L0(wε).


But (see (2.4) and (2.5)) uw∗εn−→ uw∗ ϕw∗

εn−→ ϕw∗ strongly in L2((0,T ); H1(Ω)) and

so, the latter inequalities implies

limn−→∞ Lεn

0 (wε) ≤ L0(wε). (2.7)

From (2.6)-(2.7) we get

L0(w∗) ≤ lim infn−→∞ Lεn

0 (w∗εn) ≤ lim sup

n−→∞Lεn

0 (w∗εn) ≤ L0(w∗).

Hencelim infεn−→0

Lεn0 (w∗εn

) = L0(w∗) = infL0(w), w ∈ U

and then (2.1) holds.To prove (2.2) we set: uε = uw∗ε , ϕε = ϕw∗ε (we recall that w∗ε is chosen to be optimal

in (Pε)). On a subsequence εn we have

w∗ε −→ w0 weakly star in L∞([0,T ]),uεn −→ u strongly in L2((0, T ),H1(Ω)),ϕεn−→ ϕ strongly in L2((0, T ),H1(Ω)),

where (u, ϕ,w0) satisfy (1.1)-(1.4), i.e., (u, ϕ) = (uw0, ϕw0

). Therefore, we derive

L0(w0) ≤ inf P

and, because εn was chosen arbitrarily, (2.2) follows.Now, taking into account that w∗ε is an optimal controller for problem (Pε), it fol-

lows thatLε0(w∗ε) ≤ Lε0(w) ∀w ∈ U.

On the other part, on the basis of relation (2.6), we can put

L0(w∗) ≤ lim infε→0

Lε0(w∗ε)

and thus, along with previous inequality, we may conclude that

L0(w∗) ≤ limε→0

Lε0(w) ∀w ∈ U.

ConsequentlyL0(w∗) ≤ L0(w) ∀w ∈ U

i.e., the weak limit point w∗ is a suboptimal controller for problem (P). This com-pletes the proof of Theorem 2.1.


3. NECESSARY OPTIMALITY CONDITIONS IN(Pε)

Let (uε, ϕε,w) be the solution of (1.5)-(1.8) and let w ∈ L∞[0,T ]) be arbitrary butfixed and λ > 0. Set wλ = w + λw and let (uλ,ε, ϕλ,ε) be the solution of (1.5)-(1.8)corresponding to wλ, that is:

ρcuλ,εt +`

2ϕλ,εt = kuλ,εxx ,

in Qε0,

τϕλ,εt = ξ2ϕλ,εxx +1

2aϕλ,ε + 2uλ,ε,

(3.1)

subject to non-homogeneous Cauchy-Neumann boundary conditions:

uλ,εx + huλ,ε = wλ, ϕλ,εx = 0 on Σε, (3.2)

uλ,εx = 0, ϕλ,εx = 0 on Σε0, (3.3)

and initial conditions:

uλ,ε(ε, x) = u0(x), ϕλ,ε+ (ε, x) = zλ(ε, ϕ0(x)) on Ωε, (3.4)

where zλ(ε, ϕ0(x)) is the solution of the Cauchy problem:

(zλ(s)

)′+

12a

(zλ(s)

)3= 0, s ∈ (0, ε),

zλ(0) = ϕλ,ε− (ε, x), ϕλ,ε− (0, x) = ϕ0(x).

(3.5)

Subtracting (1.5)-(1.8) from (3.1)-(3.4) and dividing by λ > 0, we get

ρc(uλ,ε − uε

λ

)t+`

2

(ϕλ,ε − ϕελ

)t= k

(uλ,ε − uε

λ

)xx,

in Qε0,

τ(ϕλ,ε − ϕε

λ

)t= ξ2

(ϕλ,ε − ϕελ

)xx

+1

2a

(ϕλ,ε − ϕελ

)+ 2

(uλ,ε − uε

λ

),

(3.6)

(uλ,ε − uε

λ

)x

+ h(uλ,ε − uε

λ

)=

wλ − wλ

,(ϕλ,ε − ϕε

λ

)x

= 0 on Σε, (3.7)

(uλ,ε − uε

λ

)x

= 0,(ϕλ,ε − ϕε

λ

)x

= 0 on Σε0, (3.8)

uλ,ε(ε, x) − uε(ε, x)λ

= 0,ϕλ,ε+ (ε, x) − ϕε+(ε, x)

λ=

zλ(ε, ϕ0(x)) − z(ε, ϕ0(x))λ

on Ωε.

(3.9)


Letting λ tend to zero in (3.6)-(3.9) we get the system in variation (3.10)-(3.13)below

ρcuεt +`

2ϕεt = kuεxx,

in Qε0,

τϕεt = ξ2ϕεxx +12aϕε + 2uε,

(3.10)

uεx + huε = w, ϕεx = 0 on Σε, (3.11)

uεx = 0, ϕεx = 0 on Σε0, (3.12)

uε(ε, x) = 0, ϕε+(ε, x) = η(ε, x) on Ωε. (3.13)

where uε = limλ−→0

uλ,ε − uε

λ, etc., and

η(ε, x) = limλ−→0

zλ(ε, ϕ0(x)) − z(ε, ϕ0(x))λ

=

= zλ(ε, ϕ0(x)) · ϕε−(ε, x) + z(ε, ϕ0(x)) = z(ε, ϕ0(x))

with η(·, x) the solution of the Cauchy problem

η′(s, x) +32a

z2(s, x)η(s, x) = 0, s ∈ (0, ε),

η(0, x) = ϕε−(ε, x),

(3.14)

that is

η(s, x) = exp(−

ε∫

0

32a

z(t, ·)2dt)ϕε−(ε, x). (3.15)

We now introduce the adjoint state system. For this, the system (3.10) can be writtenin the abstract form: (

uε

ϕε

)

t= A

(uε

ϕε

)in Qε

0

where (here ∆ϕ = ϕxx)

A =

1ρc

(k∆− `

τ

)− `

2τρc

(ξ2∆+

12a

)

2τ

1τ

(ξ2∆+

12a

)

,

D(A) =

(ψ, γ) ∈ H2(Ω) × H2(Ω);

∂ψ

∂ν+ hψ ∈ L2(∂Ω),

∂γ

∂ν= 0

.


Then

A∗ =

1ρc

(k∆− `

τ

) 2τ

− `

2τρc

(ξ2∆+

12a

) 1τ

(ξ2∆+

12a

)

,

D(A∗) =

(ψ, γ) ∈ H2(Ω) × H2(Ω);

∂ψ

∂ν+ hψ = 0,

∂γ

∂ν=

2ρc`

∂ψ

∂ν

.

Thus, the adjoint state system is(pε

qε

)

t= −A∗

(pε

qε

)+

( ∂∂uε Lε0(w)∂∂ϕε Lε0(w)

)

i.e.

pεt +kρc

pεxx −`

τρcpε +

2τ

qε = β(uε − d2)+ · χQ0, Qε

0,

pεx + hpε = 0, on Σε,

pε−(ε, x) = 0, pε−(T, x) = 0 x ∈ ΩT ,

(3.16)

qεt −`ξ2

2τρcpεxx−

`

4aτρcpε+

ξ2

τqεxx+

12aτ

qε= (ϕε−1−d1) · χQ0, in Qε

0,

qεx =`

2ρcpεx, on Σε,

qεx = 0, on Σε0,

qε−(ε, x) = exp( ε∫

0

32a

z2(t, ·)dt)qε+(ε, x), qε−(T, x) = 0, x ∈ ΩT .

(3.17)

Let us introduce the cost functional

Lε1(w) = Lε0(w) +12

IU(w)

where, as usually, IU(w) is the indicator function of the set U.If w∗ is an optimal controller of problem (Pε), then

Lε1(w∗ + λw) − Lε1(w∗)λ

≥ 0 ∀λ > 0.

that leads to (letting λ tend to zero)


β∫Q

uε (uε − d2)+ · χQ0dtdx +

∫

Q

ϕε (ϕε − 1 − d1) · χQ0dtdx+ (3.18)

+

T∫

0

w∗ w dt + I′U(w∗, w) ≥ 0 ∀w ∈ TU(w∗).

Multiplying (3.16)1 by uε and (3.17)1 by ϕε, using integration by parts and Green’sformula, we get

∫Qε

0

pεt uεdt dx + kρc

∫Qε

0

pεuεxxdt dx − `τρc

∫Qε

0

pεuεdt dx+

+2τ

∫Qε

0

qεuεdt dx + kρc

∫Σε

(pεuεx − pεxuε

)dt dγ =

= β∫

Qε0

(uε − d2)+ · χQ0uε dtdx,

(3.19)

∫Qε

0

qεt ϕεdt dx − `ξ2

2τρc

∫Qε

0

pεϕεxxdt dx − `4aτρc

∫Qε

0

pεϕεdt dx+

+`ξ2

2τρc

∫Σε

(pεxϕ

ε − pεϕεx)dt dγ +

ξ2

τ

∫Σε

(qεϕεx − ϕεqεx

)dtdγ+

+ξ2

τ

∫Qε

0

qεϕεxxdt dx + 12aτ

∫Qε

0

qεϕεdt dx =∫

Qε0

(ϕε − 1 − d1) · χQ0ϕε dtdx.

(3.20)

Now we multiply (3.11) by pε, (3.16)2 by uε, by subtraction we get

pεxuε − uεx pε = −pεw. (3.21)

Adding (3.19)-(3.20) and taking into account (3.11)2, (3.17)2, (3.21), we obtain∫

Qε0

pεt uεdt dx +∫

Qε0

qεt ϕεdt dx + k

ρc

∫Σε

pεwdt dγ +

+∫

Qε0

pε[ kρc

uεxx −`ξ2

2τρcϕεxx −

`

4aτρcϕε − `

τρcuε

]dt dx +

+∫

Qε0

qε[ξ2

τ ϕεxx + 1

2aτ ϕε + 2

τ uε]dt dx =

= β∫

Qε0

(uε − d2)+ · χQ0uε dtdx +

∫Qε

0

(ϕε − 1 − d1) · χQ0ϕε dtdx,

i.e., making use of equations in (3.10), the last relation leads to∫

Qε0

(pεt uε + pεuεt + qεt ϕ

ε + qεϕεt)dt dx + k

ρc

∫Σε

pεwdt dγ =

= β∫

Qε0


∫Qε

0



By Fubini’s theorem and definition of distributional derivative, the latter relation giveus

kρc

∫

Σε

pεwdt dγ = β

∫

Qε0


∫

Qε0


and then (3.18) becomes

kρc

∫

Σε

pεw dtdγ +

T∫

0

w∗w dt + I′U(w∗, w) ≥ 0 ∀w ∈ TU(w∗)

orT∫

0

[ kρc

pε(s, b1) + w∗(s)]w(s) ds + I′U(w∗, w) ≥ 0 ∀w ∈ TU(w∗).

The last inequality is equivalent with

−r(t) ∈ ∂IU(w∗) a.p.t. (t, x) ∈ [0,T ],

where r(t) = kρc pε(t, b1) + w(t), and thus we can conclude that

w∗(t) =

0, if r(t) > 0,−R, i f r(t) < 0. (3.22)

Summing up, we have proved the following maximum principle for problem (Pε)

Theorem 3.1. Let (u∗,ε, ϕ∗,ε,w∗) be optimal in problem (Pε). Then the optimal con-trol is given by (3.22) where (pε, qε) satisfy along with u∗,ε, ϕ∗,ε the dual system(3.16) − (3.17).

Now we will present a numerical algorithm of gradient type in order to computethe approximating optimal control stated by Theorem 3.1.Algorithm InvPHT1D (Inverse PHase Transition case 1D)

P0. Choose w(0) ∈ U and set iter= 0; Choose ε > 0;P1. Compute z(ε, ·) from (1.9);P2. Compute (uε,iter, ϕε,iter) from (1.5)-(1.8);P3. Compute (pε,iter, qε,iter) from (3.16)-(3.17);P4. For t ∈ [0,T ], compute

riter(t) =kρc· pε,iter(t, b1) + witer;


P5. Set

witer(t) =

0, if riter(t) > 0,−R, if riter(t) < 0.

P6. Compute λiter ∈ [0, 1] solution of the minimization process:

min Lε0(λwiter + (1 − λ)witer, λ ∈ [0, 1];

Set witer+1 = λiterwiter + (1 − λiter)witer;P7. If ‖ witer+1 − witer ‖ ≤ η /* the stopping criterion */

then STOPelse iter:= iter+1; Go to P1.

In the above, the variable iter represents the number of iterations after which thealgorithm InvPHT1D found the optimal value of the cost functional Lε0(w) in (Pε).

Remark 3.1. The stopping criterion in P7 could be ‖ Lε0(witer+1) − Lε0(witer) ‖ ≤ η,where η is a prescribed precision.

4. CONCLUSIONSThe main novelty brought by this work is that the computation of the approximate

solution corresponding to the nonlinear system (1.1) is replaced with calculation ofthe approximate solution for an ordinary equation and a linear system (compare stepP1 in [5] with the steps P1-P2 in present paper).

Numerical implementation of the conceptual algorithm InvPHT1D remain anopen problem. We wish only to draw attention to the type of boundary conditionconsidered here (see (1.2)) namely that they fully cover industrial application pro-posed by us for numerical simulations - a matter for further investigation.

References[1] V. Barbu, A product formula approach to nonlinear optimal control problems, SIAM J. Control

Optim., 26(1988), 496-520.[2] T. Benincasa, C. Morosanu, Fractional steps scheme to approximate the phase-field transition

system with nonhomogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. andOptimiz., 30, 3-4(2009), 199-213.

[3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal.,92(1986), 205-245.

[4] 4. M. Heinkenschloss, F. Troltzsch, Analysis of the Lagrange-SQP- Newton Method for the Con-trol of a Phase Field Equation, Control & Cybernetics, 28, 2(1999), 177–211.

[5] C. Morosanu, Numerical approach of an inverse problem in the phase field equations, An. St.Univ. ”Al.I. Cuza” Iasi, T XXXIX, s. I-a, 4(1993), 419-436.

[6] C. Morosanu, Boundary optimal control problem for the phase-field transition system using frac-tional steps method, Control & Cybernetics, 32, 1(2003), 05-32.

LIMIT CYCLES BY FINITE ELEMENT METHODFOR A ONE - PARAMETER DYNAMICAL SYSTEMASSOCIATED TO THE LUO - RUDY I MODEL

ROMAI J., 6, 2(2010), 27–39

Catalin Liviu Bichir1, Adelina Georgescu2, Bogdan Amuzescu 3,Gheorghe Nistor 4, Marin Popescu 5, Maria-Luiza Flonta 6,Alexandru Dan Corlan 7, Istvan Svab 8

1 Rostirea Maths Research, Galati, Romania,2 Academy of Romanian Scientists, Bucharest, Romania,3, 6, 8 Faculty of Biology, University of Bucharest, Romania,4, 5 University of Pitesti, Romania,7 Bucharest University Emergency Hospital, Romania,[email protected], [email protected], [email protected], [email protected],

[email protected], [email protected], [email protected]

Abstract A one - parameter dynamical system is associated to the mathematical problem gov-erning the membrane excitability of a ventricular cardiomyocyte, according to the Luo-Rudy I model. Limit cycles are described by the solutions of an extended system. Afinite element method time approximation (FEM) is used in order to formulate the ap-proximate problem. Starting from a Hopf bifurcation point, approximate limit cycles areobtained, step by step, using an arc-length-continuation method and Newton’s method.Some numerical results are presented.

Keywords: limit cycle, finite element method time approximation, Luo-Rudy I model, arc-length-continuation method, Newton’s method.2000 MSC: 37N25 37G15 37M20 65L60 90C53 37J25.

1. INTRODUCTIONThe well-known Hodgkin-Huxley model of the squid giant axon [16] represented

a huge leap forward in comparison with the earlier models of excitable systems builtfrom abstract sets of equations or from electrical circuits including non-linear com-ponents, e.g. [33]. The pioneering work of Denis Noble’s group made the transitionfrom neuronal excitability models, characterized by Na+ and K+ conductances withfast gating kinetics, to cardiomyocyte electrophysiology models, a field expandingsteadily for over five decades [23]. Nowadays, complex models accurately reproduc-ing transmembrane voltage changes as well as ion concentration dynamics betweenvarious subcellular compartments and buffering systems are incorporated into de-tailed anatomical models of the entire heart [24]. The Luo-Rudy I model of isolatedguinea pig ventricular cardiomyocyte [21] was developed in the early 1990s starting

27

28 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al

from the Beeler-Reuter model [1]. It includes more recent experimental data relatedto gating and permeation properties of several types of ion channels, obtained inthe late 1980s with the advent of the patch-clamp technique [22]. The model com-prises only three time and voltage-dependent ion currents (fast sodium current, slowinward current, time-dependent potassium current) plus three background currents(time-independent and plateau potassium current, background current), their dynam-ics being described by Hodgkin-Huxley type equations. This apparent simplicity,compared to more recent multicompartment models, renders it adequate for mathe-matical analysis using methods of linear stability and bifurcation theory.

Nowadays, there exist numerous software packages for the numerical study offinite - dimensional dynamical systems, for example MATCONT, CL−MATCONT,CL−MATCONTM [7], [15], AUTO [8]. In [19], [8], [7], [15]; the periodic boundaryvalue problems used to locate limit cycles are approximated using orthogonal col-location method. Finite differences method is also considered. In this paper, limitcycles are obtained for the dynamical system associated to the Luo-Rudy I model byusing finite element method time approximation (FEM).

2. LUO-RUDY I MODELThe mathematical problem governing the membrane excitability of a ventricular

cardiomyocyte, according to the Luo-Rudy I model [21], is a Cauchy problem

u(0) = u0 , (1)

for the system of first order ordinary differential equations

dudt

= F(η, u) , (2)

where u = (u1, . . . , u8) = (V , [Ca]i, h, j, m, d, f , X), η = (η1, . . . , η13) = (Ist, Cm, gNa,gsi, gK p, gb, [Na]0, [Na]i, [K]0, [K]i, PRNaK , Eb, T ), M = R8, F : R13 × M → M,F = (F1, . . . ,F8),

F1(η, u) = − 1η2

[Ist + η3u3u4u35(u1 − ENa(η7, η8, η13))

+η4u6u7(u1 − c1 + c2 ln u2)+gK(η10)Xi(u1)(u1 − EK(η7, η8, η9, η10, η11, η13))u8

+gK1(η10)K1∞(η9, η10, η13, u1)(u1 − EK1(η9, η10, η13))+η5K p(u1)(u1 − EK p(η9, η10, η13)) + η6(u1 − η12)] ,

F2(η, u) = −c3η4u6u7(u1 − c1 + c2 ln u2) + c4(c5 − u2) ,F`(η, u) = α`(u1) − (α`(u1) + β`(u1))u` , ` = 3, . . . , 8 .

For the definition of variables V , [Ca]i, h, j, m, d, f , X, parameters Ist, Cm, gNa,gsi, gK p, gb, [Na]0, [Na]i, [K]0, [K]i, PRNaK , Eb, T , constants c1, . . . , c5, functions

Limit cycles by Finite Element Method for a one - parameter dynamical system... 29

gK , ENa, EK , EK1, EK p, K1∞, Xi, K p, α`, β`, default values of parameters and initialvalues of variables in the Luo-Rudy I model, the reader is referred to [21]. The readeris also referred to [20] for the continuity of the model, and to [4] for the treatment ofthe vector field F singularities. F is of class C2 on the domain of interest.

3. THE ONE - PARAMETER DYNAMICALSYSTEM ASSOCIATED TO THE LUO - RUDYI MODEL

We performed the study of the dynamical system associated with the Cauchy prob-lem (1), (2) by considering only the parameter η1 = Ist and fixing the rest of param-eters. Denote λ = η1 = Ist and η∗ the vector of the fixed values of η2, . . . , η13. LetF : R × M → M, F(λ, u) = F(λ, η∗, u), F = (F1, . . . , F8).

Consider the dynamical system associated with the Cauchy problem (1), (3), where

dudt

= F(λ, u) . (3)

The equilibrium points of this problem are solutions of the equation

F(λ, u) = 0 . (4)

The existence of the solutions and the number were established by graphical repre-sentation in [4], for the domain of interest. The equilibrium curve (the bifurcation di-agram) was obtained in [4], via an arc-length-continuation method [13] and Newton’smethod [12], starting from a solution obtained by solving a nonlinear least-squaresproblem [13] for a value of λ for which the system has one solution. In [4], the re-sults are obtained by reducing (4) to a system of two equations in (u1, u2) = (V, [Ca]i).Here, we used directly (4).

4. EXTENDED SYSTEM METHOD FOR LIMITCYCLES

The extended system in (λ, T, u)

dudτ− T F(λ, u) = 0,

u(0) − u(1) = 0,

1∫0< u(t),

dw(t)dt

> dt = 0,

(5)

was introduced, in [19], [8], [7], in order to locate limit cycles of a general problem(1), (3). Here, T is the unknown period of the cycle, w is a component of a known


reference solution (λ, T ,w) of (5), and, in our case,

< u, v >=

8∑

i=1

uivi and ‖u‖ =

√√√ 8∑

i=1

u2i for u, v ∈ R8.

The system (5) becomes determined in a continuation process. In order to approx-imate and solve it by FEM time approximation, let us obtain the weak form of (5) inthe sequel. For this, we consider the function spaces:

X = x ∈ L2(0, 1;R8);dxdt∈ L2(0, 1;R8),

x = (x1, . . . , x8), xi(0) = xi(1), i = 1, . . . , 8 .V = v ∈ L2(0, 1;R);

dvdt∈ L2(0, 1;R), v(0) = v(1) .

The weak form of (5) is the problem in (λ,T, u) ∈ R × R × X

1∫

0

ui(τ)dv(τ)

dτdτ + T

1∫

0

Fi(λ, u(τ))v(τ)dτ = 0,

∀v ∈ V, i = 1, . . . , 8,1∫

0

< u(t),dw(t)

dt> dt = 0 .

(6)

5. ARC-LENGTH-CONTINUATION METHODFOR (6)

Following the usual practice ([17], [18], [7], [8], [12], [13], [14], [15], [19], [25],[27], [28], [29]), we also use an arc-length-continuation method in order to formulatean algorithm to solve (6) approximatively.

Glowinski ([13], following Keller [17], [18]) and Doedel ([8], where Keller’s nameis also cited) chose a continuation equation written in our case as

1∫

0

‖du(t)ds‖2dt + (

dTds

)2 + (dλds

)2 = 1 , (7)

where s is the curvilinear abscissa.Let (λ0, u0) be a Hopf bifurcation point, ±β0i a pair of purely imaginary eigenval-

ues of of the Jacobian matrix DuF(λ0, u0), and a nonzero complex vector g0 = g0r +ig0

i .(λ0, u0) is located on the equilibrium curve during a continuation procedure using


some test functions ([19], [14], [7]). (λ0, β0, u0, g0r , g

0i ) ∈ R × R × R8 × R8 ×R8 is the

solution of the extended system ([27], [28], [29])F(λ, u)DuF(λ, u)gr + βgiDuF(λ, u)gi − βgrgr,k − 1gi,k

= 0 , (8)

where k is a fixed index of gr and of gi, 1 ≤ k ≤ 8.To solve (6), the extended system formed by (6) and (7), parametrized by s, was

considered. Let 4s be an arc-length step and λn u(λ4s), T n T (n4s), un u(n4s). We have the algorithm (following the cases from [13], [8], [28], [29]):

Algorithm 1. 1. take the Hopf bifurcation point (λ0, u0) and T 0 = 2π/β0;retain g0

r , g0i ;

2. for n = 0, (λ1,T 1, u1) ∈ R × R × X is obtained ([8], [29]) by (13) below

1∫0

8∑i=1

u1i (t)

dφi(t)dt

dt = 0 , (9)

and1∫

0

8∑i=1

(u1i (t) − u0

i (t))φi(t) dt = 4s , (10)

whereφ(t) = sin(2πt)g0

r + cos(2πt)g0i , (11)

by using Newton’s method with the initial iteration

(u1)0(t) = u0 + 4s φ(t) , (T 1)0 = T 0 , (λ1)0 = λ0 . (12)

3. for n ≥ 1, assuming that (λn−1, T n−1, un−1), (λn, T n, un) are known, (λn+1,T n+1, un+1) ∈ R × R × X is obtained from

1∫0

un+1i (τ)

dv(τ)dτ

dτ + T n+1

1∫0

Fi(λn+1, un+1(τ))v(τ)dτ = 0 , (13)

∀v ∈ V, i = 1, . . . , 8,

1∫0

8∑i=1

un+1i (t)

duni (t)dt

dt = 0, (14)


1∫

0

8∑

i=1

(un+1i (t) − un

i (t))un

i (t) − un−1i (t)

4sdt+

+(T n+1 − T n)T n − T n−1

4s+ (λn+1 − λn)

λn − λn−1

4s= 4s, , (15)

by using Newton’s method with the initial iteration

((λn+1)0, (T n+1)0, (un+1)0) = (λn,T n, un). (16)

6. NEWTON’S METHOD FOR THE STEPS OFALGORITHM 1

In (15) (n ≥ 1), let us denote λ∗ = λn, T ∗ = T n, u∗ = un, λ∗∗ = (λn − λn−1)/4s,T ∗∗ = (T n − T n−1)/4s, u∗∗ = (un − un−1)/4s.

We write (13), (9), (10) (the iteration n = 0) in the same general form as (13), (14),(15). So denote u∗ = u0, u∗∗ = φ and consider λ∗ = λ0, T ∗ = T 0, λ∗∗ = 0, T ∗∗ = 0 in(15) and consider u∗ = u0 = φ in (14).

Each step of Algorithm 1, given (λ∗, T ∗, u∗), (λ∗∗, T ∗∗, u∗∗), calculates(λn+1, T n+1, un+1) ∈ R × R × X, n ≥ 0, by (13),

1∫

0

8∑

i=1

un+1i (t)

du∗i (t)dt

dt = 0 , (17)

and

1∫

0

8∑

i=1

(un+1i (t) − u∗i (t))u∗∗i (t) dt + +(T n+1 − T ∗)T ∗∗ + (λn+1 − λ∗)λ∗∗ = 4s . (18)

Newton’s method applied (13), (17) and (18), for n ≥ 0, leads to:

let ((λ1)0, (T 1)0, (u1)0), given by (12), be an initial iteration (m = 0), if n = 0;

let ((λn+1)0, (T n+1)0, (un+1)0), given by (16), be an initial iteration (m = 0), ifn ≥ 1;

calculate (λn+1, T n+1, un+1) as the solution of the algorithm:

for m ≥ 0, ((λn+1)m+1, (T n+1)m+1, (un+1)m+1) = (λm+1, T m+1, um+1) ∈ R × R × X isobtained by


1∫

0

um+1i (τ)

dv(τ)dτ

dτ + T m+1

1∫

0

Fi(λm, um(τ))v(τ)dτ+

+T m

1∫

0

DFi(λm, um(τ))(λm+1, um+1(τ))v(τ)dτ = (19)

= T m

1∫

0

DFi(λm, um(τ))(λm, um(τ))v(τ)dτ , ∀v ∈ V, i = 1, . . . , 8.

1∫

0

8∑

i=1

um+1i (t)

du∗i (t)dt

dt = 0, (20)

1∫

0

8∑

i=1

um+1i (t)u∗∗i (t) dt + T m+1T ∗∗ + λm+1λ∗∗ = (21)

=

1∫

0

8∑

i=1

u∗i (t)u∗∗i (t) dt + T ∗T ∗∗ + λ∗λ∗∗ + 4s .

7. APPROXIMATION OF PROBLEM (19), (20),(21) BY FINITE ELEMENT METHOD TIMEAPPROXIMATION

In order to perform this approximation, let us divide the interval [0, 1] in N + 1subintervals K = K j = [t j, t j+1], 0 ≤ j ≤ N, where 0 = t0 < t1 < . . . < tN+1 = 1. Thesets K represent a triangulation Th of [0, 1].

Let us approximate the spaces V and X by the spaces

Vh = v : [0, 1]→ R; v ∈ C[0, 1], v(0) = v(1), v|K ∈ Pk(K), ∀K ∈ Th,Xh = x : [0, 1]→ R8; x = (x1, . . . , x8), xi ∈ Vh, i = 1, . . . , 8 ,

respectively, where Pk(K) is the space of polynomials in t of degree less than or equalto k defined on K, k ≥ 2.

Let k = 2. An element K ∈ Th has three nodal points. To obtain a functionuh ∈ Xh reduces to obtain a function vh ∈ Vh. In order to obtain a function vh ∈ Vh,we use a basis of functions of Vh. Let JK = 1, 2, 3 be the local numeration for thenodes of K, where 1, 3 correspond to t j, t j+1 respectively and 2 corresponds to a nodebetween t j and t j+1. Let ψi, i ∈ JK be the local quadratic basis of functions on Kcorresponding to the local nodes. Let J = 1, . . . , 2N + 1 be the global numeration


for the nodes of [0, 1]. The two numerations are related by a matrix L whose elementsare the elements j ∈ J. Its rows are indexed by the elements K ∈ Th (by the numberof the element K in a certain fixed numeration with elements from the set 1, . . . ,N)and its columns, by the local numeration i ∈ JK , that is j = L(K, i).

A function vh ∈ Vh is defined by its values v j from the nodes j ∈ J,

vh(t) =∑

K∈ Th

∑

i∈ JK , j=L(K,i)

v j ψi(t) , (22)

and a function uh ∈ Xh is defined by its values u j from the nodes j ∈ J,

uh(t) =∑

K∈ Th

∑

i∈ JK , j=L(K,i)

u j ψi(t) . (23)

So, an unknown function uh = ((uh)1, . . ., (uh)8) is reduced to the unknowns u j,u j = ((u j)1, . . ., (u j)8), j ∈ J.

In (19), (20), (21), approximate (λm+1, T m+1, um+1) ∈ R × R × X by (λm+1, T m+1,um+1

h ) ∈ R × R × Xh. Taking um+1h = uh, uh given by (23), and v = ψ`, for all ` ∈ JK ,

for all K ∈ Th, we obtain the discrete variant of problem (19), (20), (21) as thefollowing problem in (λ, T , u1, . . ., u2N+1) ∈ R × R × R8·(2N+1), written suitable forthe assembly process,

∑

K∈ Th

∑

i∈ JK , j=L(K,i)

(um+1j )n

∫

K

ψi(τ)ψ` (τ)

dτdτ + T m+1

∫

K

Fi(λm, umh (τ)) ψ` dτ +

+ T m∑

i∈ JK , j=L(K,i)

< um+1j ,

∫

K

(DuFn(λm, umh (τ))ψi(τ))ψ` dτ > +

+λm+1∫

K

(DλFn(λm, umh (τ))) ψ` dτ

= (24)

= T m∑

K∈ Th

∫

K

(DuFn(λm, umh (τ))um

h (τ))ψ` dτ +

∫

K

(DλFn(λm, umh (τ))λm) ψ` dτ

,

n = 1, . . . , 8,um+1

0 = um+12N+1 , (25)

∑

K∈ Th

∑

i∈ JK , j=L(K,i)

< um+1j ,

∫

K

ψi(τ)du∗h(τ)

dτdτ >= 0 , (26)


∑

K∈ Th

∑

i∈ JK , j=L(K,i)

< um+1j ,

∫

K

ψi(τ) u∗∗h (τ) dτ > +T m+1T ∗∗ + λm+1λ∗∗ = (27)

=∑

K∈ Th

∫

K

< u∗h(τ), u∗∗h (τ) > dτ + T ∗T ∗∗ + λ∗λ∗∗ + 4s ,

for all ` ∈ JK , for all K ∈ Th.

8. NUMERICAL RESULTSBased on [30] and on the computer programs for [2] and [3], relations (24), (25),

(26), (27) and the algorithm at the end of section 5 furnished the numerical results ofthis section.

Let (λ0, u0) be the Hopf bifurcation point located during the construction of theequilibrium curve by a continuation procedure in [5].

The solution (λ0, β0, u0, g0r , g

0i ) of (8), calculated in [5], is

λ0 = −1.0140472901 , β0 = 0.0162886062,u0 = (−24.3132508542, 0.0034641214, 0.0, 0.0, 0.9176777444, 0.5025242162,4920204612, 0.5071561613),g0

r = (1.0, 0.0000468233, 0.0, 0.0, 0.0093195354, 0.0198748652, −0.0072420216,0.0001706577),g0

i = (0.0, 0.0000029062, 0.0, 0.0, −0.0000171311, −0.0118192370, 0.0136415789,−0.0017802907).

The eigenvalues of the Jacobian matrix DuF(λ0, u0), calculated by the QR al-gorithm, are ± 0.0162886062 i, −8.8611865338, −0.1026761869, −0.0647560667,−0.0024565181, −1.7398266947, −0.2049715178. These data are considered in thestep 1 of the algorithm at the end of section 5.

We took Cm = 1, gNa = 23, gsi = 0.09, gK = 0.282, gK1 = 0.6047, gK p = 0.0183,Gb = 0.03921, [Na]0 = 140, [Na]i = 18, [K]0 = 5.4, [K]i = 145, PRNaK = 0.01833,Eb = −59.87, T = 310.

In order to solve (6) numerically by the algorithm at the end of section 5 and by(24), (25), (26), (27), we performed calculations using 4s = 1.0 and 500 iterations inthe continuation process. Integrals

∫K

f (τ) dτ were calculated using Gauss integration

formula with three integration points.Figure 1 and 2 present some results obtained using 20 elements K (41 nodes)

(N = 20, J = 1, . . . , 41 in section 7). The curves of the projections of the limitcycles, on the planes indicates in figure, are plots generated from values calculated inthe nodes, corresponding to a fixed value of the parameter.

Two projections of some limit cycles and of a part of the equilibrium curve (markedby ”N”) are presented in Fig. 1. The Hopf bifurcation point is marked by ”•”.


−2 −1.5 −1 −0.5 0−80

−60−40

−200

0.4

0.5

0.6

0.7

0.8

0.9

1

Ist

V

f

−2 −1.5 −1 −0.5 0−80−60−40−200

0.1

0.2

0.3

0.4

0.5

0.6

Ist

V

X

Fig. 1. Two projections of limit cycles and of a part of the equilibrium curve (marked by ”N”). TheHopf bifurcation point is marked by ”•”.


−30 −20 −10 0

3

4

5

x 10−3

V

[Ca]

i

−30 −20 −10 0−5

0

5x 10

−3

V

h

−30 −20 −10 0−5

0

5x 10

−3

V

j

−30 −20 −10 0

0.7

0.8

0.9

1

V

m

−30 −20 −10 00.2

0.4

0.6

0.8

V

d

−30 −20 −10 0

0.4

0.6

0.8

V

f

−30 −20 −10 00.45

0.5

0.55

V

X

Fig. 2. Projections of two limit cycles calculated for Ist = −1.2000465026 and for Ist =

−1.2000183729 (marked by ”x”) (20 elements, 41 nodes).


In Fig. 2, there are represented the projections of the plots of two limit cyclescalculated for Ist = −1.2000465026(iteration 148) and for Ist = −1.2000183729 (iteration 248, marked by ”x” in figure).

The results obtained are relevant from a biological point of view, pointing to un-stable electrical behavior of the modeled system in certain conditions, translated intooscillatory regimes such as early afterdepolarizations [32] or self-sustained oscilla-tions [4], which may in turn synchronize, resulting in life-threatening arrhythmias:premature ventricular complexes or torsades-de-pointes, degenerating in rapid poly-morphic ventricular tacycardia or fibrillation [26].

Acknowlegdements: This research was partially supported from grant PNCDI2 61-010 to M-LF by the Romanian Ministry of Education, Research, and Innovation.

References

[1] G. W. Beeler , H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres,J. Physiol. 268(1977), 177-210.

[2] C. L. Bichir, A.Georgescu, Approximation of pressure perturbations by FEM, Scientific Bulletinof the Pitesti University, the Mathematics-Informatics Series, 9 (2003), 31-36.

[3] C. L. Bichir, A numerical study by FEM and FVM of a problem which presents a simple limitpoint, ROMAI J., 4, 2(2008), 45-56, http://www.romai.ro, http://rj.romai.ro.

[4] C. L. Bichir, B. Amuzescu, A. Georgescu, M. Popescu, Ghe. Nistor, I. Svab, M. L. Flonta, A. D.Corlan, Stability and self-sustained oscillations in a ventricular cardiomyocyte model, submittedto Interdisciplinary Sciences - Computational Life Sciences, Springer.

[5] C. L. Bichir, A. Georgescu, B. Amuzescu, Ghe. Nistor, M. Popescu, M. L. Flonta, A. D. Corlan, I.Svab, Limit points and Hopf bifurcation points for a one - parameter dynamical system associatedto the Luo - Rudy I model, to be published.

[6] C. Cuvelier, A.Segal, A.A.van Steenhoven, Finite Element Methods and Navier-Stokes Equa-tions, Reidel, Amsterdam, 1986.

[7] A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet, B. Sautois,MATCONT and CL−MATCONT: Continuation toolboxes in MATLAB, 2006,http://www.matcont.ugent.be/manual.pdf

[8] E. Doedel, Lecture Notes on Numerical Analysis of Nonlinear Equations, 2007,http://cmvl.cs.concordia.ca/publications/notes.ps.gz, from the Home Page of the AUTOWeb Site, http://indy.cs.concordia.ca/auto/.

[9] A.Georgescu, M.Moroianu, I.Oprea, Bifurcation Theory. Principles and Applications, Appliedand Industrial Mathematics Series, 1, University of Pitesti, 1999.

[10] W. J. Gibb , M. B. Wagner, M. D. Lesh, Effects of simulated potassium blockade on the dynamicsof triggered cardiac activity, J. theor. Biol 168(1994), 245-257.

[11] V.Girault, P.-A.Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer,Berlin, 1979.

[12] V.Girault, P.-A.Raviart, Finite Element Methods for Navier-Stokes Equations.Theory and Algo-rithms, Springer, Berlin, 1986.

[13] R.Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York,1984.


[14] W.J.F. Govaerts, Numerical methods for Bifurcations of Dynamical Equilibria, SIAM, Philadel-phia, 2000.

[15] W. Govaerts, Yu. A. Kuznetsov, R. Khoshsiar Ghaziani, H.G.E. Meijer,Cl MatContM: A toolbox for continuation and bifurcation of cycles of maps, 2008,http://www.matcont.ugent.be/doc−cl−matcontM.pdf

[16] A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its applicationto conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.

[17] H.B.Keller, Numerical Solution of Bifurcation Eigenvalue Problems, in Applications in Bifurca-tion Theory, ed. by P.Rabinowitz, Academic, New York, 1977.

[18] H.B.Keller, Global Homotopies and Newton Methods, in Recent Advances in Numerical Meth-ods, ed. by C. de Boor, G.H.Golub, Academic, New York, 1978.

[19] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998.

[20] L. Livshitz, Y. Rudy, Uniqueness and stability of action potential models during rest, pacing, andconduction using problem - solving environment, Biophysical J., 97 (2009), 1265-1276.

[21] C.H. Luo, Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolar-ization, and their interaction, Circ. Res., 68 (1991), 1501-1526.

[22] E. Neher, B. Sakmann, Single-channel currents recorded from membrane of denervated frogmuscle fibres, Nature, 260 (1976), 799-802.

[23] D. Noble, Modelling the heart: insights, failures and progress, Bioessays, 24 (2002), 1155-1163.

[24] D. Noble, From the Hodgkin-Huxley axon to the virtual hear, J. Physiol., 580 (2007), 15-22.Epub 2006 Oct 2005.

[25] T.S.Parker, L.O.Chua, Practical Numerical Algorithms for Chaotic Systems, Springer, New York,1989.

[26] D. Sato, L. H. Xie, A. A. Sovari, D. X. Tran, N. Morita, F. Xie, H. Karagueuzian, A. Garfinkel, J.N. Weiss, Z. Qu , Synchronization of chaotic early afterdepolarizations in the genesis of cardiacarrhythmias, Proc. Natl. Acad. Sci. USA, 106 (2009), 2983-2988. Epub 2009 Feb 2913.

[27] R. Seydel, Numerical computation of branch points in nonlinear equations, Numer. Math., 33(1979), 339-352.

[28] R. Seydel, Nonlinear Computation, invited lecture and paper presented at the Distinguished Ple-nary Lecture session on Nonlinear Science in the 21st Century, 4th IEEE International Workshopson Cellular Neural Networks and Applications, and Nonlinear Dynamics of Electronic Systems,Sevilla, June, 26, 1996.

[29] R. Seydel, Practical Bifurcation and Stability Analysis, Springer, New York, 2010.

[30] C. Taylor, T.G. Hughes, Finite Element Programming of the Navier-Stokes Equations, PineridgePress, Swansea, U.K., 1981.

[31] R.Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam,1979.

[32] D. X, Tran, D. Sato, A. Yochelis, J. N. Weiss, A. Garfinkel, Z. Qu, Bifurcation and chaos in amodel of cardiac early afterdepolarizations, Phys. Rev. Lett., 102:258103 (2009). Epub 252009Jun 258125.

[33] B. Van der Pol, J. Van der Mark, The heartbeat considered as a relaxation oscillation and anelectrical model of the heart, Phil. Mag. (suppl.), 6 (1928), 763-775.

THE FACTORIZATION OF THE RIGHT PRODUCTOF TWO SUBCATEGORIES

ROMAI J., 6, 2(2010), 41–53

Dumitru Botnaru, Alina TurcanuTechnical University of Moldova, Chisinau, Republic of [email protected], alina−[email protected]

Abstract In the category of locally convex spaces the right product of two subcategories is areflective subcategory. In the topological completely regular spaces a similar property isnot always true. The factorization of this product according to a structure of factorizationleads always to a reflective subcategory. Thus, some well known compactifications inthe topology appear as this type of factorization.

Keywords: reflective and coreflective subcategories, the right product of two subcategories, τ-completespaces.2000 MSC: 18A20, 18 B30.

INTRODUCTION

In the sequel we use the following notations:Eu (resp. Mu ) denotes the class of universal epi (resp. mono);Ep, (resp. Mp) the class of precise epi (resp. mono) : Ep= Me

u, Mp= Ebu;C2V - the category of Hausdorff locally convex topological vector space;Th - the category of Tikhonov spaces (the completely regular Hausdorff spaces);

if K (resp. R) is coreflective (resp. reflective) subcategory, then k: C→R (r: C→R)is the coreflector (resp. reflector) functor;

(Epi,M f ) - (the class of epimorphisms, the class of strict monomorphisms) = (theclass of mappings with dense image, the class of topological inclusions with closedimage);

(E f ,Mono) - (the class of strict epimorphisms, the class of monomorphisms)(Eu,Mp)-(the class of universal epimorphisms, the class of precise monomor-

phisms)=(the class of surjective mapping, the class of topological inclusions).For concepts from general topology see [7], from topology of locally convex

spaces see [8], and for those related to factorization structures see also [8].The right product of a coreflective and reflective subcategory was introduced and

examined in the paper [5]. Necessary and sufficient conditions for the product to bea reflective subcategory were identified. In the category C2V of Hausdorff locallyconvex topological vector spaces many cases when this product is a reflective subcat-egory were found.

The examination of the right product of two subcategories is requested by thefollowing situations:

41

42 Dumitru Botnaru, Alina Turcanu

1. The right product appears in natural way when studying the semi-reflexivesubcategories [2].

2. The relative torsions theories, that are so frequent in the category C2V, can beperformed as right product theories.

3. Many reflective subcategories can be explained as the right product of twosubcategories of certain type [3].

In the category C2V, as well as in the category of Tikhonov spaces, there are ex-amples when this product is not a reflective subcategory (Theorem 1.1).

The properties of right product factorization are examined in this paper (Lemma1.2). There are stipulated the conditions when this factorization defines a reflectivesubcategory (Theorem 1.3).

In Section 2 it is proved that τ-complete spaces [6], [11] could be constructed assuch factorizations (Theorem 2.3).

In Section 3 it is shown how the subcategory of τ-complete spaces could be per-formed in two ways: either varying in the product the coreflective subcategory, orvarying the factorization structure.

In Section 4 there are formulated some issues for the category of Tikhonov spaces.

1. THE RIGHT PRODUCT OF TWOSUBCATEGORIES

Let K be a coreflective subcategory, and R - a reflective subcategory of the cate-gory C. For any object X of the category C assume that rX : X → rX is R-replique ofobject X, and kX : kX → X and kkrX : krX → rX are the K-corepliques of respectiveobjects. Then

rXkX = krXg (1)

for some morphism g. Since g = k(rX) we can write the preceding equality

rXkX = krXk(rX). (2)

We assume that in the category C pushout squares exist and we construct the pushoutsquare on the morphisms kX and k(rX):

vXkX = gXk(rX). (3)

Then there exists a morphism uX so that

rX = uXvX , (4)

krX = uXgX . (5)

The factorization of the right product of two subcategories 43

Figure 1.1

We denote by K ∗d R the full subcategory of all objects of category C, isomorphicwith the objects of vX, X ∈| C | form.

Definition 1.1. The subcategory K ∗d R is called the d-product or the right productof subcategories K and R.

Theorem 1.1. Let C be a category with pull-back and pushout squares, R a monore-flective subcategory, K - a epireflective subcategory, and V = K ∗d R. Then, thefollowing affirnations are equivalent:

1. V is a reflective subcategory of category C.2. For any object X of category C the morphism vX is V-replique of object X.3. For any object X of category C the morphism vX is an epi.4. For any object X of category C the morphism uX is R-replique of object vX.5. X ∈| V |⇔ rXkX is K-coreplique of object rX.

The proving is performed as in the case of category C2V ([5], Theorem 2.5).We mention that there are known various cases when subcategory V is reflective

(see [5], Theorems 3.2.-3.4. and [3] Theorem 2.6.).Let us examine the right product of two subcategories in the category Th of Tikhonov

spaces. Let K be a coreflective subcategory. Then it contains the subcategory D ofthe spaces with discrete topology. It follows that it is a monoreflective subcategory,and then it is (Eu

⋂Mono) - coreflective, where Eu is class of universal epimorphisms

(continued and surjective maps).It is obvious that in the category Th the reflective subcategory R is monoreflective

iff when R includes the subcategory of compact spaces: Comp ⊂ R.Let Comp ⊂ R. We reffer to Figure 1.1. Let’s presume that V is a reflective sub-

category. Then uX is Comp-replique of object vX. So, uX is a topological inclusion.And from the equality (5) it results that uX is a surjective application. Therefore, uX

is an isomorphism. Thus, we proved:

Theorem 1.2. Let K be a coreflective subcategory, R - a reflective subcategory inthe category Th and Comp ⊂ R. Then, two cases are possible:1. K ∗d R = R.2. K ∗d R is not a reflective subcategory of Th category.


But some examples of reflective subcategory of Th category show they can beobtained as a simple modification of the right product of two subcategories. In whatfollows we will describe this modification.

Definition 1.2. The class of morphisms A of category C is called right-stable if,because

g′ f = f ′g,

it is a pushout square, and from f ∈ A it follows that f ′ ∈ A.

In the categories Th and C2V the pair (Eu,Mp) = (the class of surjective mor-phisms, the class of topological inclusions) is a factorization structure. Therefore theclass Eu is right-stable.

Lemma 1.1. Let C be a category with pull-back and pushout squares in which theclass Eu is right-stable, K - a monoreflective subcategory, and R - a reflective sub-category. Then for any abject X of the category C, uX is a monomorphism.

Proof. Any monoreflective subcategory is (Eu⋂

Mono)-coreflective. We use the no-tations from the beginning of the section. In the pushout square (3) kX ∈ Eu andaccording to the hypothesis that gX ∈ Eu. In the equality (5) we have krX ∈ Monoand gX ∈ Eu. According to the Lemma 1.3 [5], we deduce that uX ∈Mono.

Corollary 1.1. In the categories Th and C2V for any two subcategories one coreflec-tive and other reflective, uX is always a monomorphism.

Lemma 1.2. Assume in the category C and in the subcategory K and R, for anyobject X, uX is a monomorphism. Then:

1. For any object X of category C the morphism gX is K-coreplique of objectvX : gX = kvX .

2. The correspondence X 7−→ vX defines a covariant functor v : C→ C.3. R is a monoreflective subcategory of the category K ∗d R.4. For any object X of the subcategory K ∗d R the morphism vX is sectionable.

Proof. 1. Consider f : A→ vX, where A ∈| K |. Then

uX f = krXg

for some morphism g.

Figure 1.2


We have

uX f = krXg = uXgXg

i.e.

uX f = uXgXg,

and since uX is a mono, we deduce that

f = gXg.

The uniqueness of the morphism g which satisfies the preceding equality resultsfrom the fact that krX is K-coreplique of object rX.

So, we proved that gX = kvX .2. Let us define the functor v on morphisms. Let f : X → Y ∈ C.

Figure 1.3

ThenrY f = f1rX , (6)

for some morphism f1, where f1 = r( f );

f1krX = krY f2, (7)

for some morphism f2, where f2 = k( f1).We haveuYvY f kX = (from (4), for object Y) = rY f kX =(from (6))= f1rXkX =(from (2))

= f1krXk(rX) = (from (7)) = krY f2k(rX)(= from (5) for object Y) = uYkvY f2k(rX), i.e.

uYvY f kX = uYkvY f2k(rX). (8)


Since uY is a mono, from equality (8) we obtain

(vY f )kX = (kvY f2)k(rX). (9)

From equality (9) and the fact that (3) is an pushout square, we deduce that

vY f = f3vX , (10)

kvY f2 = f3kvX . (11)

We define v( f ) = f3.3. Let us get look again to Fig.1.1. Let X ∈| R |. Then rX , and with him and k(rX)

and vX , are isomorphisms. Therefore R ⊂ K ∗d R.Further on, let be rvX : vX → rvX R replique of object vX. Then

uX = t · rvX

for some morphism t. Since uX is a mono of category C , we deduce that rvX is thesame. The only thing we can add is: the subcategory R is (Epi

⋂Mu)-reflective in

the category K ∗d R.4. Let’s complete the diagram from Fig. 1.1 with an analogous diagram built for

the object vX.

Figure 1.4

We haveuX = tXrvX (12)

for some morphism tX . ThentXkrvX = krXhX (13)

for some morphism hX = k(tX).We have


krXhXk(rvX)= (from (13)) =tXkrvXk(rvX)=(from (2) for object vX) = tXrvXkvX =(from(12))= uXkvX = (from (10) since gX = kvx) = krX , i.e.

krXhXk(rvX) = krX . (14)

ThereforehXk(rvX) = 1. (15)

Then in the pushout squarevvXkvX = kvvXk(rvX) (16)

the morphism k(rvX) is sectional. Therefore the morphism vvX is the same.We can assert that

rvXvX = lXrX (17)

for some morphism lX . We havetXlXrX = (from (17)) = tXrvXvX = (from (1)) = uXvX = (from (4)) = rX i.e.

tXlXrX = rX (18)

ortXlX = 1. (19)

We assume that the conditions of preceding lemma are satisfied, and (P, I) is afactorization structure in the category C. Let L be the full subcategory of categoryC comprising I-subobjects of objects of subcategories K ∗d R. For any object X ofcategory C let

vX = iXlX (20)

be (P, I)-factorization of morphism vX .

Theorem 1.3. The correspondence X 7−→ (lX, lX) defines the category L as aP-reflective subcategory of the category C.

Proof. Let be A ∈| L |, and f : X −→ A ∈ C. We show that the morphism f isextended through morphism lX . According to the hypothesis there exists an objectB ∈| K ∗d R | and a morphism i : A −→ B ∈ I. Then

vB(i f ) = v(i f )vX , (21)

or

(vBi) f = (v(i f )iX)lX . (22)

Since vB is sectional we deduce that vBi ∈ I, and lX ∈ P. Therefore lX ⊥ vBi,i.e.

f = glX , (23)


v(i f )iX = vBig, (24)

for some morphism g. The uniqueness of morphism g, that satisfies equality (23),results from the fact that lX is an epi.

Figure 1.5

2. THE SUBCATEGORY OF τ-COMPLETESPACES

We examine some coreflective subcategories of the category Th of Tikhonov spaces.

Definition 2.1. Let τ be an cardinal.1. In a topological space the intersection of τ open sets is called Gτ -set.2. P−τ -change of the space (X, t) is called the space (X, t−τ ), where the basis of

topology t−τ is formed by Gα-set, α < τ.3. Pτ-change of space (X, t) is called the space (X, tτ), where the basis of topology

tτ is formed by Gτ-sets.

We note with K−(τ) (respectively K(τ)) the full subcategory of the category Thcomprising all spaces (X, t) for which t = t−τ (respectively, t = tτ).

We observe that K(τ) = K−(τ+), where τ+ is the first cardinal which follows τ. Ifτ is limiting cardinal, then K−(τ) , K−(λ), for any cardinal λ. Therefore, it is enoughto examine only the subcategories K−(τ).

It is easily checked that K−(τ) (similar by and K(τ)) are the coreflective subcate-gories of category Th with coreflective functors.

P−τ : Th −→ K−(τ), P−τ (X, t) = (X, t−τ ),

Pτ : Th −→ K(τ), Pτ(X, t) = (X, tτ).

We mention the following properties of the subcategories K−(τ):1. K−(τ) = Th for τ ≤ ω.2. Let α < β be. Then K−(α) ⊇ K−(β).3. Let Disc be the subcategory of discrete spaces. Then∩K−(τ)/τ = ∩K(τ)/τ = Disc.Therefore, we can conclude that Disc = K−(∞) = K(∞), considering that τ < ∞

for every cardinal τ.


Theorem 2.1. ([4], Theorem1.2). Consider ω ≤ α < β. Then:

K−(β) ⊂ K−(α) and K−(β) , K−(α).

Definition 2.2. Let τ be an cardinal. The Tikhonov space X is called Q(τ)-space(respectively Q−(τ)-space), if X is closed in K(τ)-coreplique (respectively K−(τ)-coreplique) of space βX, where βX is Comp-replique of spaces X.

We note with Q(τ) (respectively, Q−(τ) ) - the full subcategory of all Q(τ)-spaces(respectively, Q−(τ) -spaces).

The categories Q(τ) have been studied by A. Cigoghidze [6], and the Q−(τ) by H.Herrlich [11].

Let (X, t) be a Tikhonov space, (Y, u) = β(X, t), and (Y, uτ) − K−(τ) -corepliqueof space (Y, u). Let X be the closure of the set X in the space (Y, uτ). On set X weinduce the topology u′ out of space (Y, u). The topology space (X, u′) we note v−t X,or v−t (X, t).

Figure 2.1

Evidently, Q(τ) = Q−(τ+) for any limiting cardinals τ. But for limiting cardinalsτ, the subcategories Q−(τ) are of other form that Q(τ).

For a subcategory A of category Th we note with PrA the subcategory that con-tains the products of objects of category A, and with M f (A) the subcategory thatcontains M f -subobjects of objects of subcategory A.

Theorem 2.2. ([4], Theorem 2.6). 1. The subcategory Q−(τ) is a monoreflectivesubcategory (therefore also epireflective) of category Th with reflector functor

v−t : Th −→ Q−(τ).

2. Q−(ω) = Q(n) = Comp, n ∈ N.3. Q−(ωI) = Q(ω) = Q - the subcategory of Hewitt spaces.4.Q−(τ) = M f Pr(R(τ)), where R(τ) = [−1; 1]τ \ −1; 1.5. Q−(τ) = M f Pr(E(τ)), where E(τ) = Πα<τR(α).6. α < β and ω < β. Then Q−(α) ⊂ Q−(β) and Q−(α) , Q−(β) .

Corollary 2.1. Let τ be a limiting cardinal. Then

Q(τ) = M f Pr(∪Q(λ) : λ > τ).


Remark 2.1. In [11] is defined the problem of existence of generators for subcate-gories Q−(τ), i.e. if there is a space Aτ, so that

Q−(τ) = M f Pr(Aτ).

Ignoring the case in [6], the problem is solved for subcategories Q(τ) and fully in theprecedent theorem.

Theorem 2.3. The reflective subcategory Q−(τ) is (Epi,M f )-factorization of theright product K−(τ) ∗d Comp.

Proof. For Tikhonov space (X, t) let (Y, u) = β(X, t) the Stone-Cech compactification,and pX

τ : (X, tτ) −→ (X, t) and pYτ : (Y, uτ) −→ (Y, u) the K−τ -coreplique of respective

objects.

Figure 2.2

On the set Y we examine the inductive topology m, which is not mandatory beingthe Tikhonov topology, defined by applications βX and pY

τ :

G ∈ m⇐⇒ [(βX)−1(G) ∈ t and (pYτt)−1(G) ∈ uτ].

The squaregpτ(βX) = hpX

τ , (25)

is the pushout square constructed on morphisms pXτ and pτ(β(X)) in the category of

topological spaces. In the sequel we construct the pushout square on these morphismsin the category Th. Let F be the set of continued defined functions (Y,m) with valuesin the field of real numbers R:

F = Hom((Y,m),R),

and m(F) - the topology defined on the set Y of this set F:


m(F) = f −1(G)| f ∈ F and G is open in R.Let l be the canonic application. Then

(lh)pXτ = gX pτ(βX) (26)

is the pushout square constructed on morphisms pXτ and pτ(βX) in the category Th.

Let Z be the closure of set X in the space (Y,m(F)), and let the topology v be theone induced from space (Y,m(F)). Since lX is an epi in the category Th it follows thatuXiX is the Stone-Cech compactification of space (Z, v). Thus we can consider thatthe topology v is the one induced from application uXiX = βZ on set Z from spaceβ(X, t).

The closure of set X in the spaces (Y,m). A set A is closed in the space (Y,m) iffthe set h−1(A) is closed in the space (X, t) and g−1(A) is closed in the space (Y, uτ). IfX ⊂ A, then the set A is closed iff the set g−1(A) is closed in the space (Y, uτ).

The closure of set X in the spaces (Y,m) and (Y, uτ) coincides: clmX = cluτX. Thelast set we denote as T : T = cluτX.

Let us prove that the closure of set X in the space (Y, uτ) coincides with Z: cluτX =

clm(F)X.Firstly we mention that T ⊂ Z and will prove the reverse inclusion. Let be y ∈ Y\T .

Then exists a set H ⊂ Y \ T , that comprise point y so that:1. H is closed in the topology u.2. H is a Gτ-set in the topology u.3. H is closed and open in the topology uτ.4. H remains open and closed in the topology m.5. H remains open and closed in the topology m(F).The theorem is proved.

3. THE CASE OF THE SUBCATEGORY Q−(τ)Let be R and L two reflective subcategories of the category C and L ⊂ R. If

C is local and colocal small with projective limits, then there exists a class L(R) offactorization structure in the category C with the following property.

For any object X of category C let be rX : X −→ rX and lX : X −→ lX therespective replique. Then

lX = vXrX (27)

for some morphism vX . If the subcategory L is monoreflective, then in the writtenequality all morphisms are bimorphisms.

We note

U = rX |X ∈ |C| ,V = vX |X ∈ |C| .

Always U ⊥ V .


AssumeP′′(R) = P′′ = Vq; I′′(R) = I′′ = Vqx; P′(R) = P′ = Uxq; I′(R) = Ux.

Theorem 3.1. [1] Let C be local and colocal small with projective limits and thesubcategory L is monoreflective. Then:

1. (P′, I′) and (P′′, I′′) are structures of factorization in the subcategory C.2. Let (P, I) be a structure of factorization in the category C. The following affir-

mations are equivalent:a) for any object X of category C the equality (1) is (P, I)-factorization of mor-

phism lX;b) P′ ⊂ P ⊂ P′′.

Relying on this theorem we assert

Theorem 3.2. Let τ be a cardinal, τ > ℵ0.1. The reflective subcategory Q−(τ) could be obtained factorizing the right product

(the morphism vX) K−(τ) ∗d Comp following the structure of factorization (Epi,M f ).2. In the category Th there ar the structures of factorization (P, I), P

′(Q−(τ)) ⊂

P ⊂ P′′(Q−(τ)), so that the reflective subcategory Q−(τ) could be obtained doing the(P, I)- factorization of right product K−(ω) ∗d Comp.

4. PROBLEMSIn paper [4] some classes of coreflective subcategories in the category Th are ex-

amined.

Definition 4.1. Let τ be an cardinal. The topological space (X, t) is called k−τ -space,if any function definite on set X and continue on every compact K ⊂ X, | K |< τ iscontinue on space (X, t).

Let C−(τ) a full subcategory of all k−τ -spaces.

Theorem 4.1. The subcategory C−(τ) is a coreflective subcategory of category Th.

Definition 4.2. The weight w(X, t) of the topological space (X, t) is called minimal ofcardinals |B|, where B is basis of spaces (X, t).

Definition 4.3. Let τ be an cardinal. The Tikhonov space is called b−τ -space if everyfunction definite on set X and continue on every compact K ⊂ X, w(K) < τ is continueon space (X, t).

Let B−(τ) is a subcategory full of all b−τ -spaces.

Theorem 4.2. 1. The subcategory B−(τ) is a coreflective subcategory of categoryTh.2. Let be τ ≤ ω. Then B−(τ) = Disc.3. ∩B−(τ) = C - subcategory of k-functionals space.


4. Let ω ≤ α < β and α a regular cardinal and β ≥ α+. Then B−(α) ⊂ B−(β) andB−(α) , B−(β).

Problem 4.1. 1. Let us describe the subcategory R which is (Epi,M f )-factorizationof the right product C−(τ) ∗d Comp.

2. R-replique of an arbitrary object X of category Th is also the closing of set Xin the C−(τ)-coreplique of space βX (with induce topology from βX)?

3. The same thing is valid for the right product B−(τ) ∗d Comp.

References[1] Botnaru D., Structure bicategorielles complementaires, ROMAI J., 5, 2(2009), 5-27.

[2] Botnaru D., Cerbu O., Semireflexive product of two subcategories, Proc.of the Sixth Congress ofRomanian Mathematicians, Bucharest, 2007, V.1, p. 5-19.

[3] Botnaru D., Cerbu O., Some properties of semireflexivity subcategories (submitted).

[4] Botnaru D., Robu R., Some categorical aspects of Tikhonov spaces, Scientific Annals. of MSU,The series ”Physical-Mathematical Sciences”, Chisinau, 2000, 87-90 (in Romanian).

[5] Botnaru D., Turcanu A., Les produits de gauche et de droit de deux sous-categories, Acta etcoment, Chisinau, III(2003), 57-73.

[6] Cigoghidze A.C., About properties close of compactness, Uspehi mat.nauk, 35, 6 (216)(1980),177-178.

[7] Engelking R., General topology, Warszawa, 1985.

[8] Grothendieck A., Topological vector spaces, Gordon and Breach, New York, London, Paris,1973.

[9] Herrlich H., Fortsetzbarkeit stetiger abbildungen und kompactheitsgrad topologischer Raume,Math. Z., 96, 1(1967), 64-72.

CLASSIFICATION OF A CLASS OF QUADRATICDIFFERENTIAL SYSTEMS WITH DERIVATIONS

ROMAI J., 6, 2(2010), 55–67

Ilie BurdujanUniversity of Agricultural Sciences and Veterinary Medicine ”Ion Ionescu de la Brad” Iasi,Romaniaburdujan [email protected]

Abstract The classification up to a center-affinity of a class of homogeneous quadratic differentialsystems defined on R3, which have a semisimple derivation with a 2-dimensional kernelis achieved. It is proved that there exist eighteen classes of affinely nonequivalent suchsystems.

Keywords: homogeneous quadratic dynamical systems, nilpotent derivation.2000 MSC: Primary 34G20, Secondary 34L30, 15A69.

1. INTRODUCTIONThe study of a homogeneous quadratic differential system (S ) (briefly, HQDS) on

a B space A can be achieved by using suitable algebraic tools. This is possiblebecause a commutative algebra A(·) is naturally associated with any such a system(S ). In general, algebra A(·) is not associative. There exists (see [11], [1]) a 1-to-1 correspondence between the classes of affinely equivalent HQDSs on A and theclasses of isomorphic commutative algebras on A. A strong connection between thequalitative properties of HQDSs and the properties - invariant up to an isomorphism- of the corresponding commutative algebras emerges from this correspondence. Inparticular, any derivation/automorphism of (S ) is a derivation/automorphism of itsassociated algebra A(·) (see [5], [9]). As it is well known (see, for example, [1]), if aderivation of A(·) having only real eigenvalues exists, then its semisimple and nilpo-tent parts are necessarily derivations, too. The commutative algebras on R3 havinga nilpotent derivation were already classified (see [4], [6]). The problem of clas-sification of 3-dimensional real commutative algebras having at least a semisimplederivation is an intricate task and it is still unsolved.

In this paper, in order to save the space, we shall classify - up to a center-affinity- only a particular class of HQDSs on R3 which have a semisimple derivation. It isproved that this class of HQDSs is naturally partitioned in eighteen mutually disjointsubclasses consisting of nonequivalent systems, i.e., every system in one subclass isnot equivalent to any system in another subclass; moreover, two systems in the samesubclass that correspond to different values of parameters are nonequivalent, too.

55

56 Ilie Burdujan

Our results are based on the remark: the problem of classification up to an affineequivalence of HQDSs on Rn is equivalent with the problem of classification up to anisomorphism of all real n-dimensional commutative algebras. That is why we shallclassify, up to an isomorphism, the class of real 3-dimensional commutative algebrasthat correspond to the analyzed class of HQDSs. The lattices of the subalgebras ofthese algebras are used as a main tool for solving the isomorphism problems.

In a forthcoming paper we shall classify the set of all HQDSs on R3 which have asemisimple derivation.

2. REAL COMMUTATIVE ALGEBRAS ON R3

HAVING A SEMISIMPLE DERIVATIONLet us consider a nontrivial real 3-dimensional commutative algebra A(·). Then,

any basis in A allows to identify A with R3.

Now, let us suppose that A(·) has a nonzero semisimple derivation. Then Der Anecessarily contains a semisimple derivation D having its spectrum SpecD of theform SpecD = 1, λ, µ (λ, µ ∈ R). Accordingly, there exists a basis B = (e1, e2, e3)such that

D(e1) = e1, D(e2) = λe2, D(e3) = µe3.

Obviously, the case λ = µ = 1 is not possible. Since D(e21) = 2e2

1, then either e21 = 0

(if 2 < SpecD) or e21 , 0 (if 2 ∈ SpecD). More exactly, it follows

e21 = 0 i f 2 < SpecD

e21 = κe2 i f λ = 2 , µ

e21 = ωe3 i f µ = 2 , λ

e21 = κe2 + ωe3 i f λ = µ = 2.

Each of the following equations can be exploited in a similar way:

D(e1e2) = (1 + λ)e1e2, D(e1e3) = (1 + µ)e1e3, D(e22) = 2λe2

2,D(e2e3) = (λ + µ)e2e3, D(e2

3) = 2µe23.

In fact, at least one of the elements of the set 2, 1 + λ, 1 + µ, 2λ, 2µ, λ+ µ belongsto SpecD. A simple inspection of papers [4], [6], [10] provides the following casesof the spectrum of D that have to be considered:

1, 0, 0, 1, 1, 0, 1, 1, 2, 1, 2, 0, 1, 2, 2, 1, 2, 3, 1, 2, 4, 1,−1, 0.In what follows we deal with a class of algebras which have a semisimple derivationD with SpecD = 1, 0, 0, only. This is the case when dimR Ker D = 2.

3. CASE SPEC D = 1, 0, 0There exists a basis B = e1, e2, e3 such that

D(e1) = e1, D(e2) = 0, D(e3) = 0.

Classification of a class of quadratic differential systems with derivations 57

Then, from the equations

D(e21) = 2e2

1, D(e1e2) = e1e2, D(e1e3) = e1e3,

D(e22) = D(e2e3) = D(e2

3) = 0

emerges the following multiplication table of algebra A(·)

e21 = 0 e2

2 = αe2 + βe3e1e2 = κe1 e2e3 = γe2 + δe3e1e3 = ωe1 e2

3 = εe2 + θe3.

It results that A = Im D ⊕ Ker D where Re1 is an ideal and Ker D is a subalgebra ofA(·). The algebra Ker D can have (see [11]) either a basis of nilpotent elements of in-dex two or a single 1-subspace of such nilpotent elements, only. Among the algebrasof the former kind, there exists one (see [11]) which has the following multiplicationtable:

e22 = 0, e2e3 = e3, e2

3 = γe2 + e3 (γ , 0).

It suggests us to deal with algebras defined by the following multiplication table:

Table T e21 = 0 e2

2 = 0e1e2 = αe1 e2e3 = e3e1e3 = βe1 e2

3 = γe2 + e3

with α, β, γ ∈ R, γ , 0.We shall denote by A(α, β, γ) any algebra on R3 with the multiplication table T in

basis B.

Algebra A(α, β, γ)

Any algebra A = A(α, β, γ) (γ , 0) is not simple because it contains the ideal Im D.Distinct triples of real numbers (α, β, γ) (γ , 0) determine non-isomorphic algebras.Indeed, the following result is proved by a straightforward checking.

Theorem 3.1. The algebras A(α, β, γ) (γ , 0) and A(p, q, r) (r , 0) are isomorphicif and only if α = p, β = q, γ = r.

In order to find a partition for class A(α, β, γ) (γ , 0) into subclasses of noniso-morphic subalgebras we look for the family of its 1-dimensional subalgebras. To thisend we need to find its annihilator elements, nilpotent (of index two) elements andidempotent (of index two) elements.

This problem is solved in the next three propositions.

Proposition 3.1. The set Ann (A) of all annihilator elements of A = A(α, β, γ) is:

Ann (A) =

0 i f α2 + β2 , 0Im D i f α = β = 0.

58 Ilie Burdujan

Proposition 3.2. The set N(A) of all nilpotent elements (of index two) of A = A(α, β, γ)is:

N(A) =

Re1 ∪ Re2 i f α , 0SpanRe1, e2 i f α = 0.

Proposition 3.3. The set of idempotent elements (of index two) of any algebra A(α, β, γ)(γ , 0) is:

1 I(A) = ∅ if γ < −18 ,

2 I(A) = −12 e2 + 2e3 if γ = −1

8 and α − 4β + 1 , 0,3 I(A) = xe1 − 1

2 e2 + 2e3 | x ∈ R if γ = −18 and α − 4β + 1 = 0,

4 I(A) = y1e2 + z1e3, y2e2 + z2e3 | x ∈ R if γ > −18 , γ , 0,

2(αy1 + βz1) − 1 , 0 and 2(αy2 + βz2) − 1 , 0,5 I(A) = y1e2 + z1e3 ∪ xe1 + y2e2 + z2e3 | x ∈ R if γ > −1

8 , γ , 0,2(αy1 + βz1) − 1 , 0 and 2(αy2 + βz2) − 1 = 0,

6 I(A) = xe1 + y1e2 + z1e3, | x ∈ R ∪ y2e2 + z2e3 if γ > −18 , γ , 0,

2(αy1 + βz1) − 1 = 0 and 2(αy2 + βz2) − 1 , 0,7 I(A) = xe1 + y1e2 + z1e3 | x ∈ R ∪ xe1 + y2e2 + z2e3 | x ∈ R ifγ > −1

8 , γ = 0, 2(αy1 + βz1) − 1 = 0 and 2(αy2 + βz2) − 1 , 0.

Corollary 3.1. Every algebra A(α, β, γ) (γ < −18 ) is not isomorphic to any algebra

A(p, q, 18 ) or A(p, q, r) (r > −1

8 , r , 0).

Proof. It is enough to notice that any two such classes of algebras have distinct fam-ilies of idempotent elements.

Properties of algebra A(α, β, γ)

In what follows, we list the properties of these eighteen subclasses of algebras thathelp us to solve easily the isomorphism problem. Moreover, note that the lattices ofsubalgebras of each subclass of algebras were obtained.

Properties of algebra A1=A(α, β, γ) (α , 0, γ < −18 , (α − 1)2 + (2β − 1)2 , 0)

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = ∅• Subalgebras: Re1, Re2, SpanRe1, e2, SpanRe2, e3• Ideals : Re1• A2 = A• Der A = RD

• Aut (A) = T (x) | x ∈ R∗ where T (x) =

x 0 00 1 00 0 1

.


Properties of algebra A2=A(1, 12 , γ) (γ < −1

8 )

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = ∅• Subalgebras: Re1, Re2, SpanRe2, ae1 + be3 (a2 + b2 , 0)• Ideals : Re1• A2 = A

• Der A = RD ⊕ RD1 where [D1] =

0 0 10 0 00 0 0

and [D,D1] = D1

• Aut (A) = T (x, y) | x, y ∈ R, x , 0 where T (x, y) =

x 0 y0 1 00 0 1

.

Properties of algebra A3=A(0, β, γ) (β , 0, γ < −18 )

• AnnA = 0• N(A) = SpanRe1, e2• I(A) = ∅• Subalgebras: R(pe1 + qe2) (p2 + q2 , 0), SpanRe1, e2, SpanRe2, e3• Ideals : Re1• A2 = A• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A4=A(α, β,−18 ) (α , 0, α − 4β + 1 , 0)

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = −1

2 e2 + 2e3• Subalgebras: Re1, Re2, R(−1

2 e2 + 2e3), SpanRe1, e2, S panRe2, e3,SpanRe1,−1

2 e2 + 2e3• Ideals : Re1• A2 = A• Der A = RD


x 0 00 1 00 0 1

.

60 Ilie Burdujan

Properties of algebra A5=A(0, β,− 18 ) (β , 0, β , 1

4 )

• AnnA = 0• N(A) = SpanRe1, e2• I(A) = −1

2 e2 + 2e3• Subalgebras: Re1, Re2, R(pe1+qe2) (p2+q2 , 0), R(− 1

2 e2+2e3), SpanRe1, e2,SpanRe2, e3, S panRe1,−1



x 0 00 1 00 0 1

.

Properties of algebra A6=A(4β − 1, β,−18 ) (β < 14 , 1

2 )• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = xe1 − 1

2 e2 + 2e3 | x ∈ R• Subalgebras: Re1, Re2, R(xe1 − 1

2 e2 + 2e3) (x ∈ R), SpanRe1, e2,SpanRe2, e3, SpanRe1,− 1



x 0 00 1 00 0 1

.

Properties of algebra A7=A(1, 12 ,−1

8 )

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = xe1 − 1

2 e2 + 2e3 | x ∈ R• Subalgebras: Re1, Re2, R(xe1 − 1

2 e2 + 2e3) (x ∈ R), SpanRe1, e2,SpanRe2, e3, S panRe1,−1

2 e2 + 2e3, S panRe2, ae1 + be3 (a, b ∈ R),SpanRxe1 − 1

2e2 + 2e3,

cx2

e1 + be2 + ce3 (b, c ∈ R, 4b + c , 0)• Ideals : Re1• A2 = A


0 0 10 0 00 0 0

and [D,D1] = D1



x 0 y0 1 00 0 1

.

Properties of algebra A8=A(0, 14 ,−1

8 )

• AnnA = 0• N(A) = SpanRe1, e2• I(A) = xe1 − 1

2 e2 + 2e3 | x ∈ R• Subalgebras: R(pe1 + qe2) (p2 + q2 , 0), SpanRe1, e2, SpanRe2, e3,

SpanRe1,−12 e2 + 2e3, SpanRpe1 + qe2, e3(p, q ∈ R∗),

SpanR−12 e2 + 2e3, ae1 + be2 + ce3 (a, b, c ∈ R, 4b + c , 0),

SpanRxe1 − 12 e2 + 2e3, (1 − 2xq)e1 + qe2 (x, q ∈ R, xq , 0)

• Ideals : Re1• A2 = A


0 4 10 0 00 0 0

and [D,D1] = D1


x 4y y0 1 00 0 1

.

Properties of algebra A9=A(α, β, γ) (α , 0, γ , 0, γ > −18 ,

2(αy1 + βz1) − 1 , 0, 2(αy2 + βz2) − 1 , 0)

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = y1e2 + z1e3, y2e2 + z2e3• Subalgebras: Re1, Re2, R(y1e2 + z1e3), R(y2e2 + z2e3), SpanRe1, e2,

SpanRe2, e3, SpanRy1e2 + z1e3, e1, SpanRy1e2 + z1e3, e2,SpanRy2e2 + z2e3, e1, SpanRy2e2 + z2e3, e2,SpanRy1e2+z1e3, ae1+b[γ(α−z1−αz1+2βz1+1)e2+2(β+γz1+αγz1)e3] (a, b ∈ R),SpanRy2e2 +z2e3, ae1 +b[γ(α−z2−αz2 +2βz2 +1)e2 + (β+γz2 +αγz2)e3] (a, b ∈ R)• Ideals : Re1• A2 = A• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A10=A(0, β, γ) (γ , 0, γ > − 18 , 2βz1 − 1 , 0,

2βz2 − 1 , 0, β , 0)

• AnnA = 0

62 Ilie Burdujan

• N(A) = SpanRe1, e2• I(A) = y1e2 + z1e3, y2e2 + z2e3• Subalgebras: R(pe1 + qe2) (p2 + q2 , 0), R(y1e2 + z1e3), R(y2e2 + z2e3),

SpanRe1, e2, SpanRe2, e3, SpanRy1e2 + z1e3, e1, SpanRy2e2 + z2e3, e1,SpanRy1e2 + z1e3, ae1 + b[γ(1 − z1 + 2βz1)e2 + 2(β + γz1)e3] (a, b ∈ R, γ ,1), SpanRy2e2 + z2e3, ae1 + b[γ(1 − z2 + 2βz2)e2 + 2(β + γz2)e3] (a, b ∈ R, γ , 1)There exists other subalgebras if α, β, γ are connected by some appropriate equa-tions, namely:SpanRpe1 +qe2,

pq (b+βc)e1 +be2 +ce3 i f γ = 2β2−β, β , 1

4 (b, c, p, q ∈ R, q , 0)• Ideals : Re1• A2 = A• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A11=A(α, β, γ) (α , 0, γ , 0, γ > −18 ,

2(αy1 + βz1) − 1 , 0, 2(αy2 + βz2) − 1 = 0)

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = y1e2 + z1e3 ∪ xe1 + y2e2 + z2e3 | x ∈ R• Subalgebras: Re1, Re2, R(y1e2+z1e3), R(xe1+y2e2+z2e3) (x ∈ R), SpanRe1, e2,

SpanRe2, e3, SpanRy1e2 + z1e3, e1, SpanRy2e2 + z2e3, e1,If x , 0 a biparametric family of 2-dimensional algebras of type SpanRxe1 +

y2e2 + z2e3, ae1 + be2 + ce3 is obtained; they correspond to the following conditions:

4x(1 − α)b2 + 4γx(βz2 − 1)c2 + 2(z2 − 2)ab + (4γz2 + z2 − 1)ac+

+4x(γz2 + αγz2 − β + 1)bc = 0,−γxc3 + 2xb2c + xbc2 − 2(y2 − αy2 + βz2)abc + (−y2 + 2βy2 + γz2)ac2−

−2αz2ab2 = 0.

• Ideals : Re1• A2 = A• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A12 = A(0, β, γ) (γ , 0, γ > −18 , 2βz1−1 , 0, 2βz2−1 =

0, i.e. γ = 2β2 − β, β = −γz1, β , 0)

• AnnA = 0• N(A) =SpanRe1, e2• I(A) = y1e2 + z1e3 ∪ xe1 + y2e2 + z2e3 | x ∈ R


• Subalgebras: R(pe1 + qe2), R(y1e2 + z1e3), R(xe1 + y2e2 + z2e3) (x ∈ R),SpanRe1, e2, SpanRe2, e3, SpanRy1e2 + z1e3, e1, SpanRy2e2 + z2e3, e1,Spanpe1 + qe2,

pq (b + βc)e1 + be2 + ce3 (b, c, p, q ∈ R, pq , 0),

SpanRy1e2 + z1e3, ae1 + b[γ(2βz1 − z1 + 1)e2 + 2(β + γz1)e3] (a, b ∈ R, β , −1),SpanRxe1 + y2e2 + z2e3, ae1 + b( 2β−1

2 e2 + e3) (x, a, b ∈ R),

SpanRxe1 + y2e2 + z2e3, ae1 +4βa − a − 4β2xb

4βx e2 + be3 (x, a, b ∈ R, x , 0)• Ideals : Re1• A2 = A


0 1 −γz10 0 00 0 0

and [D,D1] = D1

• Aut (A) = T (x, y) | x ∈ R∗ where T (x) =

x y −γz1y0 1 00 0 1

.

Properties of algebra A13=A(α, β, γ) (α , 0, γ , 0, γ > −18 , 2(αy1 +βz1)−1 =

0 , 2(αy1 +βz2)−1 , 0, (α−1)2 +(2β−1)2 , 0, i.e. 2γ(1−α)2 +(2β−1)(α−2β) = 0)

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = xe1 + y1e2 + z1e3 | x ∈ R ∪ y2e2 + z2e3• Subalgebras: Re1, Re2, R(y1e2+z1e3),R(xe1+y2e2+z2e3) (x ∈ R), SpanRe1, e2,

SpanRe2, e3, SpanRy1e2 + z1e3, e1, SpanRy2e2 + z2e3, e1,If x , 0 a 2-parametric family of 2-dimensional algebras of type S panRxe1 +

y1e2 + z1e3, ae1 + be2 + ce3 is obtained; they correspond to the following conditions:

4x(1 − α)b2 + 4γx(βz1 − 1)c2 + 2(z1 − 2)ab + (4γz1 + z1 − 1)ac+

+4x(γz1 + αγz2 − β + 1)bc = 0,−γxc2 + 2xb2c + xbc2 − 2(y1 − αy1 + βz1)abc + (−y1 + 2βy1 + γz1)ac2−

−2αz1ab2 = 0.

• Ideals : Re1• A2 = A• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A14=A(1, 12 , γ) (γ , 0, γ > −1

8 )

• AnnA = 0• N(A) = Re1 ∪ Re2• I(A) = xe1 + y1e2 + z1e3 | x ∈ R ∪ y2e2 + z2e3

64 Ilie Burdujan

• Subalgebras: Re1, Re2, R(xe1+y1e2+z1e3) (x ∈ R), R(y2e2+z2e3), SpanRe1, e2,SpanRe2, e3, SpanRe1, y1e2 + z1e3, SpanRe1, y2e2 + z2e3,SpanRe2, ae1 + be3 (a, b ∈ R),SpanRxe1 + y1e2 + z1e3,−4xe1 + z1(e2 − 4e3),SpanRy2e2 + z2e3, ae1 + b(y1e2 + z1e3) (a, b ∈ R∗)• Ideals : Re1• A2 = A


0 0 10 0 00 0 0

and [D,D1] = D1


x 0 y0 1 00 0 1

.

Properties of algebra A15=A(0, β, γ) (γ , 0, γ > − 18 , 2βz1−1 = 0, 2βz2−1 ,

0, i.e. γ = 2β2 − β, β = −γz2)

• AnnA = 0• N(A) = S panRe1, e2• I(A) = xe1 + y1e2 + z1e3 | x ∈ R ∪ y2e2 + z2e3• Subalgebras: R(pe1 + qe2) (p, q ∈ R), R(xe1 + y1e2 + z1e3) (x ∈ R),

R(y2e2 + z2e3), SpanRe1, e2, SpanRe2, e3, SpanRy1e2 + z1e3, e1,SpanRy2e2+z2e3, e1, Spanpe1+qe2,

pq (b+βc)e1+be2+ce3 (b, c, p, q ∈ R, pq , 0),

SpanRy2e2 + z2e3, ae1 + b[γ(2βz2 − z2 + 1)e2 + 2(β + γz2)e3] (a, b ∈ R, β , −1),SpanRxe1 + y2e2 + z2e3, ae1 + b( 2β−1

2 e2 + e3) (x, a, b ∈ R),

SpanRxe1 + y1e2 + z1e3, ae1 +4βa − a − 4β2xb

4βx e2 + be3 (x, a, b ∈ R, x , 0)• Ideals : Re1• A2 = A


0 1 −γz20 0 00 0 0

and [D,D1] = D1

• Aut (A) = T (x, y) | x ∈ R∗ where T (x) =

x y −γz2y0 1 00 0 1

.

Properties of algebra A16=A(0, 0, γ) (γ < −18 )

• AnnA = Re1• N(A) = S panRe1, e2• I(A) = ∅• Subalgebras: R(pe1 + qe2) (p2 + q2 , 0), SpanRe1, e2, SpanRe2, e3• Ideals : Re1, SpanRe1, e2• A2 = A


• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A17=A(0, 0,−18 )

• AnnA = Re1• N(A) = SpanRe1, e2• I(A) = −1

2 e2 + 2e3• Subalgebras: R(pe1 + qe2) (p2 + q2 , 0), R(−1

2 e2 + 2e3), SpanRe1, e2,SpanRe2, e3, SpanRe1,−1

2 e2 + 2e3, SpanRe2,−12 e2 + 2e3

• Ideals : Re1, SpanRe1, e2• A2 = SpanR−1

8 e2 + e3, e3• Der A = RD


x 0 00 1 00 0 1

.

Properties of algebra A18=A(0, 0, γ) (γ , 0, γ > −18 )

• AnnA = Re1• N(A) = SpanRe1, e2• I(A) = y1e2 + z1e3 ∪ y2e2 + z2e3• Subalgebras: R(pe1 + qe2) (p2 + q2 , 0), R(y1e2 + z1e3),

R(y2e2 + z2e3), SpanRe1, e2, SpanRe2, e3, SpanRe1, y1e2 + z1e3,SpanRe1, y2e2 + z2e3• Ideals : Re1, SpanRe1, e2• A2 = SpanRγe2 + e3, e3• Der A = RD


x 0 00 1 00 0 1

.

These lists of properties of classes Ai allow us to assert the following result.

Theorem 3.2. Every algebra of class Ai is not isomorphic to any algebra of classA j (i , j, i, j ∈ 1, 2, ..., 18).

Accordingly, the class of homogeneous quadratic differential systems

(S )

x′ = 2αxy + 2βxzy′ = γz2

z′ = 2yz + z2

66 Ilie Burdujan

is naturally divided into eighteen classes of affinely nonequivalent systems. The lasttwo equations of (S) supply the prime integral F(y, z), namely

F(y, z) =

ln 2y2 + yz − 2γz2

y4 + 1γ√|8γ + 1|arctg 2γz − y

2γy√|8γ + 1| i f γ < −1

8

ln (4y + z)2

4y4 − 8yz + 4y i f γ = −1

8(|2y2 + yz − 2γz2|

y4

) 12√

8γ+1 2γz − (1 +√

8γ + 1)y2γz − (1 −

√8γ + 1)y

i f γ > −18 .

Consequently, any trajectory of (S) lies on a cylinder F(y, z) = const.

4. THE LATTICE OF SUBALGEBRAS OFA(α, β, γ)

It is well known that isomorphic algebras have necessarily isomorphic lattices oftheir subalgebras. Consequently, the lattices of all subalgebras of two algebras be-come a natural tool in proving whether these algebras are or are not isomorphic.

Moreover, the lattice of subalgebras of any algebra A delivers a partition of itsvector space in nonempty mutually disjoint cells. In its turn, this partition providesa partition of the set of all solutions of the associated homogeneous quadratic differ-ential system. In the previous section we have obtained the lattice of all subalgebrasof any algebra A(α, β, γ), even if, this finding is a customary task. For particularclasses of algebras this problem can be easily solved, namely for algebras A withDer A = RD (i.e. algebras with fewer symmetries). For example, the lattice ofsubalgebras for algebra A1 is

0, Re1, Re2, S panRe1, e2, S panRe2, e3, A.These subalgebras define a partition of R3 by means of their ground subspaces Ox,Oy, xOy and yOz. Accordingly, any solution of the corresponding HQDS (S) has itsorbit in one of the cells bounded by these subalgebras or it is contained in one of thesesubalgebras. Thus, some orbits lay either on the axis Ox or on Oy, other orbits arecontained in the plane xOy or yOz (but have no intersection with Ox and Oy) and theother ones are contained inside of one of the four quadrants defined in R3 by planesxOy and yOz.

The lattice of subalgebras for algebra A2 is

0, Re1, Re2, S panRe2, ae1 + be3 (a2 + b2 , 0), A.A comparison of the two lattices allows us to decide that A1 and A2 are not isomor-phic. On the other hand, each solution of the corresponding HQDS (S) is containedin a subalgebra, i.e. it is necessarily a null torsion curve. Indeed, a straightfor-ward computation confirms us that the triple scalar product (r’, r”, r”’) = 0 wherer’ = (2xy + xz)e1 + (γz2)e2 + (2yz + z2)e3. Moreover, each trajectory of a HQDSassociated to one of the algebras A16, A17, A18 has null torsion, too.


5. CONCLUSIONSThe analyzed class of commutative algebras is divided into eighteen classes of

non-isomorphic algebras. Accordingly, the corresponding class of homogeneousquadratic differential systems is partitioned into eighteen classes of affinely nonequiv-alent systems. Since A is a vector direct sum of an ideal and a subalgebra, each sucha system is decoupled into subsystems that can be solved.

The most part of classes of algebras have Der A = RD, so that the correspond-ing orbits of solutions have just a few symmetries. The algebras in the classesA2, A7, A8, A12, A14, A15 are the only algebras that have two-dimensionalderivation algebras.

References[1] I. B, Quadratic differential systems, Publ. House PIM, 2008.(Romanian)

[2] I. B, A classification of a class of homogeneous quadratic dynamical systems on R3

withderivations, Bull. I.P. Iasi, Sect. Matematica, Mecanica teoretica, Fizica, T.LIV (2008), 37–47.

[3] I. B, Homogeneous quadratic dynamical systems on R3

having derivations with complexeigenvalues, Libertas Mathematica, XXVIII (2008), 69–92.

[4] I. B, Classification of Quadratic Differential Systems on R3 having a nilpotent of Order3 Derivation, Libertas Mathematica, XXIX (2009), 47-64.

[5] I. B, Automorphisms and derivations of homogeneous quadratic differential systems, Ro-mai J., 6, (12010), 15-28.

[6] I. B, Classification of Quadratic Differential Systems on R3 having a nilpotent of Order2 Derivation (in press).

[7] T. D, Classifications and Analysis of Two-dimensional Real Homogeneous Quadratic Differ-ential Equation Systems, J. Diff. Eqs., 32(1979), 311–334.

[8] I. K, Algebras with many derivations, ”Aspects of Mathematics and its Applications”(ed. J.A. Barroso), Elsevier Science Publishers B.V., 1986, 431.

[9] K. M. K , A. A. S, Quadratic Dynamical Systems and Algebras, J. of Diff. Eqs, 117(1995), 67–127 .

[10] K. M. K , A. A. S, Automorphisms ans Derivations of Differential Equations and Al-gebras, Rocky Mountain J. of Mathematics, 24, 1,(1994), 135–153.

[11] L. M, Quadratic Differential Equations and Non-associative Algebras, in ”Contributionsto the Theory of Nonlinear Oscillations” Annals of Mathematics Studies, no. 45, Princeton Uni-versity Press, Princeton, N. Y., 1960.

[12] A.A. S, R. W, Introduction to Lie groups and Lie algebras, Academic Press, New York,1973.

[13] R. D. S, An Introduction to Nonassociative Algebras, Academic Press, New York, London,1966.

[14] N. I. V, K. S. Sı, Geometrical Classification of Quadratic differential systems (Rus-sian), Differentialnye Uravnenje, 13, 5(1977), 803–814.

ON THE 3D-FLOW OF THE HEAVY ANDVISCOUS LIQUID IN A SUCTIONPUMP CHAMBER

ROMAI J., 6, 2(2010), 69–81

Mircea Dimitrie Cazacu1, Cabiria Andreian Cazacu2

1Polytechnic University of Bucharest, Romania2”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest, [email protected], [email protected]

Abstract We present the essence of the theoretical and experimental researches, developed in thefield of the three-dimensional flow with free surface of a heavy and viscous liquid inthe suction chamber of a pump, with or without air carrying away or appearance ofthe dangerous cavitational phenomenon, problem of a special importance for a faultlessworking of a pump plant, especially of these relevant to the nuclear electric power plants,which constituted the object of our former preoccupations.

From the mathematical point of view we present the solving particularities of thetwo-dimensional boundary problems, specific to the three-dimensional domain limits ofthe pump suction chamber flow, as well as the possibility to obtain a stable numericalsolution.

From the practical point of view, we show the possibilities of laboratory modelling ofthis complex phenomenon, by simultaneously use of many similitude criterions, as wellas the manner to determine the characteristic curves, specific to each designed pumpsuction chamber.

Because the drawings of the pump station of Romanian nuclear power station Cer-navoda on Danube are classified, we shall present the theoretical and experimental re-searches performed on the model existent in the Laboratory of New Technologies ofEnergy Conversion and Magneto-Hydrodynamics, founded after the world energetic cri-sis from 1973 [1] in the POLITEHNICA University of Bucharest, Power EngineeringFaculty, Hydraulics, Hydraulic Machines and Environment Engineering Department.

Keywords: liquid three-dimensional flow with free surface, three-dimensional flow in the pump suctionchamber, complex modelling of liquid three-dimensional flow with free surface.2000 MSC: 35 G 30, 35 J 25, 35 J 40, 35 Q 30, 65 N 12, 76 D 17.

1. INTRODUCTIONThe theoretical research concerning the three-dimensional flow of the real liquid

in a suction chamber of a pump, [2], [3], is justified by the flow effect with or with-out appearance of whirls, with or without air training [4] or by the appearance of thecavitation destructive phenomenon at the liquid entry in a pump, as well as by theirnegative influence on the hydraulic and energetic pump efficiency, the quick cavita-tional erosions, the machine vibrations followed by the fast wear, eventually by the

69

70 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu

pump suction less and by their pumping cease, extremely dangerous for the pumpingstation, for instance of a nuclear electric power plant [1].

These situations may come in the sight in special work conditions linked with thelow level of the water in the river bed or in the pump suction chamber, or by aninefficient design of this installation for different cases of pumps working [5], [6].

2. NUMERICAL INTEGRATION OF THEHEAVY AND VISCOUS LIQUIDTHREE-DIMENSIONAL FLOW IN A PUMPSUCTION CHAMBER

We present the mathematical specific boundary conditions [2], [3], which inter-vene in the problem of the three-dimensional viscous liquid steady flow into the suc-tion chamber of a pump plant, represented in the Figure 1.

Thus, for the uniform flow of the viscous and heavy liquid in the entrance and go-ing out sections of the domain, we must consider the two-dimensional special bound-ary problems:

- for the fluid entrance section we must solve a mixed Dirichlet-Neumann prob-lem for a Poisson equation with an unknown a priori constant, depending of the slopeangle α of the suction chamber bottom, which can be determined by a successive cal-culus cycle of the computer program, satisfying a normalization condition, which,for the given value of mean input velocity and entrance section dimensions, ensuresthe desired flow-rate,

- for the fluid going out pipe section we must solve a Dirichlet problem for anotherPoisson equation, having the pressure drop gradient p′z in the suction pipe also asunknown a priori constant, which can be determined also by a successive calculuscycle satisfying a normalization condition and which establishes the pressure dropgradient on the pipe length for the same flow-rate, satisfying another normalizationcondition.

2.1. PARTIAL DIFFERENTIAL EQUATIONS OFA VISCOUS AND HEAVYLIQUID STEADY FLOW

The three-dimensional motion equations in Cartesian trihedron (Fig.1) are

U′T + U′XU + U′YV + U′ZW +1ρ

P′X = ν(U”X2 + U”Y2 + U”Z2) + g sinα, (1)

V ′T + V ′XU + V ′YV = V ′ZW +1ρ

P′Y = ν(V”X2 + V”Y2 + V”Z2), (2)

W′T + W′XU + W′YV + W′ZW +1ρ

= ν(W”X2 + W”Y2 + W”Z2) + g cosα, (3)

On the 3D-flow of the heavy and viscous liquid in a suction pump chamber 71

the mass conservation equation for an incompressible fluid being

U′X + V ′Y + W′Z = 0, (4)

where, as usual, (U, V, W) is the velocity of the fluid, P is its the pressure, ρ - thefluid density, and ν - its kinematical viscosity. T and X, Y, Z represent time andspatial independent variables, respectively.

2.2. THE DIMENSIONLESS FORM OF THEEQUATION SYSTEM

For more generality of the numerical solving [3], [5] - [8] we shall work withdimensionless equations, by using the characteristic physical magnitudes of flow:

- the suction chamber width B H, approximately equal with its height,- the liquid mean velocity of the entrance in suction chamber Um = Q/BH,- the air atmospheric pressure on its free surface P0,- and the period T0 of whirl appearance in the three dimensional flow.With the new dimensionless variables and functions:

x =XH, y =

YH, z =

ZH, t =

TT0, u =

UUm

, v =V

Um,w =

WUm

, p =PP0, (5)

the partial differential equations (1) to (4) become in the dimensionless form:

Sh u′t + u′xu + u′yv + u′zw + Eu p′x =1

Re ∆x,y,z

u +sinα

Fr, (1′)


Sh v′t + v′xu + v′yv + v′zw + Eu p′y =1

Re ∆x,y,z

v, (2′)

Sh w′t + w′xu + w′yv + w′zw + Eu p′z =1

Re ∆x,y,z

w +cosα

Fr, (3′)

in which one puts into evidence the following criteria of hydrodynamic flow simi-larity, numbers of: Strouhal Sh = H/T0Um, Euler Eu = P0/ρU2

m, Reynolds Re =

HUm/ν and Froude Fr = U2m/gH; the mass conservation equation is an invariant

u′x + v′y + w′z = 0. (4′)

2.3. THE NUMERICAL SOLVING METHODThe numerical integration of partial differential equation system was performed

by an iterative calculus of unknown functions u, v and w, given by the algebraic rela-tions associated to the partial differential equations, by introducing their expressionsdeduced from the finite Taylor’s series developments [8] in a cubical grid, having thesame step δx = δy = δz = χ:

f ′x,y,z =f1,2,5 − f3,4,6

2χand f ”x2,y2,z2 =

f1,2,5 − 2 f0 + f3,4,6χ2 . (6)

2.4. THE BOUNDARY CONDITIONS FOR THETHREE-DIMENSIONAL STEADY FLOW

- on the solid walls, due to the molecular adhesion condition, we consider

uSW = vSW = wSW = 0 (7)

and consequently, from equation (3’) for α ≈ 0 the hydrostatic pressure distributionbecomes

pSW(z) = 1 +γHP0

z = 1 + Ar z, (8)

in which we introduced the hydrostatic similarity number Ar = γH/P0, dedicatedto the Archimedean lift discoverer,

- on the free surface, considered to be a plane due to the liquid important weightand small flow velocities, we have the conditions

pFS(x, y, 0) = 1 and wFS(x, y, 0) = 0, (9)

excepting a null measure set of points constituted by the whirl centers on the freesurface, in whose neighbourhood the fluid has a descent motion (Fig. 1).

Neglecting the liquid friction with the air, we shall cancel both shearing stresscomponents, obtaining:

τzx |FS= u′z + w′x = u′z |FS= 0,→ u6 = u5

τxy |FS= v′x + w′y = v′z |FS= 0,→ v6 = v5.(10)


The value w6 is calculated from the mass conservation equation (4’), written in finitedifferences (that is not unstable in this local appliance [5] [6])

u1 − u3

2x+

v2 − v4

2x+

w5 − w6

2x= 0 → w6 |FS= w5 + u1 − u3 + v2 − v4, (11)

- in the entrance section, we consider the uniformly and steady flow at the normaldepth of the current parallel with the channel bottom slope i = sinα, which leads usto the condition v |ES= w |ES= 0 and to the hydrostatic pressure repartition, the dis-tribution of u(y, z) velocity component being obtained from the first motion equation(1’), which in these boundary conditions become a Poisson type equation with theconstant α a priori unknown, written also in finite differences

∆x,z

u +ReFr

sinα = 0 → u0 =14

4∑

i=1

ui +χ2

4ReFr

sinα. (12)

The numerical solving is possible by an iterative cycle on the computer, until to theobtaining of slope value i, necessary to the given Re and Fr numbers correspondingto the flow velocity Um, when the heavy force component g sinα is balanced by theinterior friction forces, acting by the liquid adhesion to the channel bottom and itssolid walls.

The boundary conditions for the Poisson equation (12) for solving in the frame ofplane the mixed problem Dirichlet-Neumann

USW = 0 on the solid walls,=zx|FS= µU′z = 0 on the free surface,

lead us to solve a mixed problem Dirichlet-Neumann, whose normalization con-dition is

B∫

0

H∫

0

Ui(Y,Z)dYdZ = Q = BHUm, (13)

or, in the iterative numerical calculus for the dimensionless case,

χ2kM−1∑

k=2

jN−1∑

j=2

uk,j +χ2

2

jN−1∑

j=2

ukM,j +χ2

8

jN−1∑

j=2

uk=2,j +χ2

4

kM−1∑

k=2

uk,j=2 = q = 1, (13′)

representing the flow-rate which is brought by this velocity repartition in the entrancesection, for the given values of Um, therefore Re and Fr numbers, as well the consid-ered grid step χ.

For the starting of the calculus it is necessary to introduce only a single velocityvalue in a point as a seed, for instance uS = 2 in the point j = 20 and k = 40 in themiddle of the free surface, in the rest of the domain the initial arbitrary values being


equal with zero, or, by admittance of a paraboloidal repartition a the initial givenarbitrary values concerning the velocity distribution in the domain, in the shape

uva(y, z) = 8(1 − z2) · (12− y) · (1

2+ y) = 8(1 − z2) · (1

4− y2),

uva( j, k) = 4us · a2( j − 1) · (k − 1) · [2(k − 1) · a][2( j − 1) · a],(14)

because we have

y = ( j − 1)a − 12

and y = 1 − (k − 1)a, (15)

- in the exit section, we shall consider that the streamlines are parallel with thevertical pipeline walls, uES = vES = 0, the velocity component w(x, y) distribution inthe uniformly and steady flow verifying also a Poisson type equation with an a prioriunknown constant p′z, but this time due to the ignorance of the pressure drop on thevertical pipeline p′z, deduced from the third motion equation (3’) in approximationcosα ≈ 1, determined above by solving the entry section problem

Re Eu p′z =ReFr

cosα + ∆x,y

w,→ w0 =14

[4∑

i=1

wi + χ2Re(1Fr− Eu p′z)]. (16)

The boundary condition to solve the Poisson equation (16) in the frame ofDirichlet problem, is given by the liquid adhesion wES = 0 on the interior solidwall of suction pipe.

The Poisson type equation solving, in which one does not know a priori the pres-sure drop p′z along the pipe, can be made by numerical way, also by an iterativecycle on computer, yielding up to the arbitrary values p′z & 0.1, which exceed a fewthe pressure distribution deduced from the hydrostatic repartition (8), until the ve-locity integration wES in the exit section (Fig. 2) shall equalize the unitary value ofdimensionless flow-rate, calculated in the case of non-symmetrical or symmetricalsupposed velocity repartitions, using the forms of [6]

qnonsym ≈ χ2 ·

20∑

i=17

10∑

j=7

wij +18·

∑

i∈17, 2

9∑

j=8

wij +18·

20∑

i=18

∑

j∈7, 1wij

= 1,

qsym ≈ χ2 · [9w17, 8 + 4(w18, 8 + w17, 7)] = 1.

(17)

2.5. CONCLUSIONS ON THE USED BOUNDARYSPECIFIC PROBLEMS IN THE 3D-FLOWS

The Dirichlet problem, proposed by the German mathematician Peter GustavLejeune Dirichlet, being generally specific to the equations of elliptic type, corre-sponding to the repartition of scalar physic magnitudes given by harmonic functions


(for example of the temperature distribution in a homogeneous medium, without con-vection i.e. U = V = 0, and without interior heat sources in virtue of Laplace equa-tion) has here a specific character.

The Neumann problem formulated by the German physicist CarlGottfried Neumann, consists in giving, on the boundary C of the domain D, thevalues of the normal derivative of the function, f ′n .

To specify the unknown function values in the domain, it is necessary also inthis case to fulfil the so-called normalization condition, giving the flux through thefrontier line

φ =

∫

Cf ′ndl. (18)

In the three-dimensional case of mixed boundary problems, by combination ofthese two classical problems, to solve the Poisson equations with unknown constants,intervene the well-known normalization conditions, even for the Dirichlet problemonly.

Initially, the computational program has been made in Cobol [4]], due to the greatnumber of the data, working successively to relax the three velocity componentsvalues in a plane, considering their values also in the two neighbouring planes, thegraphical representation of the three-dimension flow being shown by the three veloc-ity components as in Figure 2.


2.6. THE IMPORTANCE OF THE COMPLEXPHENOMENA FROM A PUMP SUCTIONCHAMBER FLOW

The faultless working of the pump plant for the nuclear electric power stations,which must work even in the case of nuclear reaction stopping, yet 10 to 12 hours tillthe complete cooling of the reactor, is the main aim to ensure the nuclear security.

In the second part of this paper we present the experimental research performed,regarding the complex modelling of the spatial flow and the specific phenomena,which can be produced in the pump suction chamber. The research has been startedin the year 1977 together with my former student and then collaborator Dr. Eng.Costel Iancu [2] for the pump station design of the nuclear power station Cernavodain Romania [1] and continued by the construction of the Model installation endowedwith the measuring apparatus for a good working research of a pump suction (Fig.1 to 3) in the Laboratory of new technologies of energy conversion and environmentprotection, realized in the year 1979 with the help of the former Eng. PhD studentDumitru Ispas [3]-[6].

3. THE COMPLEX MODELLING OF THEHEAVY AND VISCOUS LIQUID 3D- FLOW INA PUMP SUCTION CHAMBER

After doctoral Thesis defending in 1957, M.D.Cazacu published a work on the lab-oratory complex modelling of the complicated water-hammer phenomenon [9], con-sidering not only the dimensionless motion equation, but also the mass conservationand state equation of the compressible liquid, as well as the pipe elastic deformationwith influence upon the sound propagation velocity.

He decided at that time to pay attention to this very important phenomenon, takinginto consideration the high security level of a pump plant for a nuclear electric powerplant, since the research on the laboratory model was not conclusive in the consultedtechnical literature in this field.

3.1. THE EXPERIMENTAL INSTALLATIONDESCRIPTION AND MEASURINGAPPARATUS

The experimental installation is presented in Fig. 3, and it consists in 1-stainedglass chamber, 2-supporting feet, 3-tightening cap, 4-air tap, 5-flow reassure, 6- cen-trifugal pump, 7-direct current electric motor, 8-pump pressing pipeline, 9-dischargecontrol valve, 10-flow meter, 11-pump suction pipeline, 12-suction basket, 13-emptyingtap, 14-air carrying away whirl, 15-whirl on the bottom, 16-blades against the whirls.


The water level in the pump suction chamber may be changed through the tap 13,the water leaving the chamber through the pump suction pipe 12 producing two whirlswithout or with air carrying away 14, which due to the rotational flowω caused by thepump suction flow rates, smaller then the nominal flow rate, change their positions:one remaining on the free surface, but at the other chamber side the second removingon the chamber bottom, where with the solid particles uncoupled from the concreteand concentrated in the whirl axis, due to the bigger flow velocities, leads at a strongerosion of the pump suction chamber.

For the measuring of different physical magnitudes we utilized the apparatus:- a mechanic tachometer contacted with the electric motor shaft end, to measure

the pump number of rotations n,- a diaphragm to measure the pump flow rate Q,- a graduated rule from transparent plastic to measure the water level Z in the

suction chamber, fixed at the suction chamber glass,- a manual chronometer to measure the frequency of whirl appearance,- a precision metallic mano-vacuum-meter or a mercury manometer tube to mea-

sure the air pressure at the suction chamber free surface.

3.2. THE FLOW COMPLEX MODELLING USINGMORE SIMILARITY CRITERIONS

The short history concerning these inefficient foregoing modelling attempts beganany decades before, when the researchers tried to utilize the very known Froude’scriterion for the modelling of this gravitational flow in a pump suction chamber,but which gave on the Model, smaller then the Reality, built at the length scale


λ = H/H′ = D/D′ > 1, lesser velocities at which the whirls could not to appear

Fr =V2

gH= Fr′ =

V ′2

g′H′→ β =

VV ′

= λ0,5 ≥ 1 and k =QQ′

= λ2β = λ2,5, (19)

in which we noted with β the velocity ratio and with k the flow rate ratio.For this reason, in the last decade of that period, the researchers, without no justi-

fication, have enlarged the velocities on the Model till the same values as the veloc-ities of the Reality, introducing the so called equal velocity criterion, for which M.D. Cazacu gave a proof considering Weber’s criterion, thinking at the free surfacewhirls with air carrying away similarity

We =σ

ρV2 = We′ =σ′

ρ′V ′2→ β = 1 = λ0 and then k = λ2, (20)

where σ is the air-water superficial tension.As a good expert in the recommendations of the pump manufacturers and pump

station designers, that the water access to the pump suction baskets must be as possi-ble direct and because the estimation of the separation of viscous fluid flow is givenby Reynold’s number, he considered also this kind of modelling, supposing that thewater cinematic viscosity is the same

Re =VDυ

= Re′ =V ′D′

υ′→ β =

VV ′

=υ

υ′λ−1 ≈ λ−1 and k = λ, (21)

in which the velocities on the Model are greater than those from the Reality.Another very important problem was the modelling of the influence of pump work-

ing, especially at lesser flow rates than that nominal. He introduced the pump mod-elling by its rapidity criterion nq

nq =nQ1/2

H3/4 = n′q =n′Q′1/2

H′3/4, (22)

from which we can choose the pump number of rotations on the laboratory Model

n′ = nk1/2

λ3/4 = nλ1/4β1/2 (23)

and determine the convenient geometric scale in the case of composed similarityFroude and pump rapidity nq, in which case he gave in Table 1 the pressure P′0 nec-essary at the water free surface in suction chamber, to benefit also of the cavitationphenomenon modelling at pump or at the strong erosive whirl, which ends on a sidewall or on the bottom of the suction chamber built from the concrete, in the aim toeliminate his appearance in the Reality [5].


Table 1

n’[rot/min] 3.000 1.500 1.000 750 600 n = 500

λ = (n′/n)2 36 9 4 2, 25 1, 44 1

P′0 − P′v(T ′)P0 − Pv(T )

=

= 1/β2 = 1/λ = σ/σ′ 0, 0277 0, 111 0, 25 0, 444 0, 6944 1

P′0(bar) 0, 261 0, 342 0, 43 0, 5778 0, 7678 1

In this table one sees the advantage of the length scale λ = 4, corresponding tothe standardized number of rotations 1,000 rot/min of the pump model, inclusivelyunder pressure that we must assure at the model free surface, as well as that whichassures only the cavitational phenomenon similarity, for which we must have alsoaccomplished the equality relation of the cavitational criterion

σ =P0 − Pv(T )ρν2/2

= σ′ =P′0 − P′v(T ′)ρ′ν′2/2

. (24)

Finally, in the case of Strouhal’s similarity, concerning the whirl appearance fre-quency in time, we shall have the relation, from which we can deduce the frequencyof whirl appearance in reality as function of that obtained on the model

Sh =D fV

= Sh′ =D′ f ′

V ′→ f = f ′

D′VV ′D

. (25)

The symmetrical position (Fig. 1) of the two whirls from the right part can bemodified by the pump working at littler flow rates (Fig. 3), by ω rotation impressedto the chamber flow as in the left part, when one whirl arrive to us and the other beingdisposed vertically on the bottom chamber (Fig. 4), causing its accelerated wear.

3.3. CONCLUSIONS AND IMPORTANTREMARKS

With this occasion we shall observe that the greater models become too expensiveand the smaller models introduce forces of other nature, which modify completelythe development of the hydrodynamic process.

On the basis of this modelling method one sees, that we can study a complexhydro-aerodynamic similarity using 6 criterions, as: Froude, Weber, Reynolds, pumprapidity, cavitation criterion and Strouhal, not only individually but also simultane-ously.


In this way we can also determine the convenient technical-economically lengthscale of the Model, to have also a good modelling similarity. In the same time we candetermine the whirl apparition frequency in reality, that is very important becausethe erosion of the pump suction chamber walls is however a problem of time andwe can also determine the zones that can be covered with metallic plates and onecan determine for any pump plant, working at different exploitation situations itscharacteristic curves (Fig. 4), which determine the region of a good operation.

References[1] M.D.Cazacu, C.Iancu, Studiu privind aspiratia si aductiunea pompelor mari de circulatie din

termocentrale CNE si a criteriilor de similitudine pentru verificarea pe model (Study regardingthe suction and delivery of the great circulation pumps from the Electric Nuclear Power Sta-tions and the similarity criterions for the check on the model). Contr. nr.201/1977, Beneficiary −ICEMENERG − Bucharest.

[2] M.D.Cazacu, Gh.Baran, A.Ciocanea, High education and research development concerningthe recoverable, inexhaustible and new energy sources in the POLITEHNICA University ofBucharest. Internat. Conf. on Energy and Environment, 23-25 Oct.2003, Polytechnic Univ. ofBucharest, Sect. 5 - Energetic and Educational Politics, Ses.2 - Educational and Environmentalprotection Politics, 5, 120-125.

[3] M.D.Cazacu, On the boundary conditions in three-dimensional viscous flows. The 5th Congressof Romanian Mathematicians, June 22-28, 2003, University of Pitesti, Romania, p. 24.

[4] M.D.Cazacu, Boundary mathematical problems in viscous liquid three-dimensional flows. Inter-nat. Conf. on Theory and Applications of Mathematics and Informatics - ICTAMI 2007, AlbaIulia, Acta Universitatis Apulensis, Mathematics-Informatics, No. 15 / 2008, 281 - 289.

[5] C.Iancu, Contributii teoretice si experimentale la studiul curgerii fluidelor cu miscare de rotatiesi vartej legat de suprafete libere (Theoretical and experimental contributions at the study of fluidflows with rotation motion and whirl bound of free surfaces). Doctoral Thesis defended in 1970at the Polytechnic Institute of Bucharest, Romania.


[6] M.D.Cazacu, D.Ispas, Cercetari teoretice si experimentale asupra curgerii lichidului vıscos ıncamera de aspiratie a unei pompe (Theoretical and experimental research on the viscous liquidflow in a pump suction chamber), Conf. of Hydraulic Machines and Hydrodynamics, Timisoara,Romania, 18 - 19 Oct. 1985, vol. III, 67-74.

[7] M.D.Cazacu, D.A.Ispas, M.Gh.Petcu, Pipe-channel suction flow researches. Proc. of XXthIAHR Congress (Subject B.a ), Moscow SSRU, September 5 - 9, 1983, VI, 214 - 221.

[8] M.D.Cazacu, On the solution stability in the numerical integration of non-linear with partialdifferential equations. Proceedings of the Internat. Soc. of Analysis its Appl. and Computation,17-21 Sept. 2002, Yerevan, Armenia. Complex analysis, Differential equations and Related top-ics. Publishing House Gitityun, 2004, Vol. III, 99-112.

[9] D.Dumitrescu, M.D.Cazacu, Theoretische und experimentelle Betrachtungen uber die Stromungzaher Flussigkeiten um eine Platte bei kleinen und mittleren Reynoldszahlen. Zeitschrift furAngewandte Mathematik und Mechanik, 1, 50, 1970, 257- 280.

[10] M.D.Cazacu, La similitude hydrodynamique concernant le phenomene du coup de belier. Rev.Roum. de Sci. Techn. Mec. Appl., Rom. Acad. Publ. House, Tome 12, No. 2, 1967, Bucharest,317- 327.

FIXED POINT THEOREMS IN E - METRICSPACES

ROMAI J., 6, 2(2010), 83–91

Mitrofan M. Choban, Laurentiu I. CalmutchiDepartment of Mathematics, Tiraspol State University, Chisinau, Republic of [email protected]

Abstract The aim of the present article is to give some general methods in the fixed point theoryfor mappings of general topological spaces. Using the notions of the multi-metric spaceand of E-metric space, we prove the analogous of several theorems about common fixedpoints of commutative semigroups of mappings.

Keywords: fixed point, m-scale, semifield, multi-metric space, E-metric space, pseudo-metric.2000 MSC: 54H25, 54E15, 54H13, 12J17, 54E40.

1. INTRODUCTIONIn the sequel, any space is considered to be Tychonoff and non-empty. We use the

terminology from [6, 7].Let R be the space of real numbers, ω = 0, 1, 2, ... and N = 1, 2, ....In [1, 2] the metric spaces over topological semifields were introduced. The gen-

eral concept of the metrizability of spaces is contained in [14].The notion of a topological semifield may be generalized in the following way. We

say that E is a metric scale or, briefly, an m-scale if:1. E is a topological algebra over the field of reals, R;2. E is a commutative ring with the unit 1 , 0;3. E is a vector lattice and 0 ≤ xy provided 0 ≤ x and 0 ≤ y;4. For any neighborhood U of 0 in E there exists a neighborhood V of 0 in E such

that if x ∈ V and 0 ≤ y ≤ x, then y ∈ U.5. For any non-empty upper bounded set A of E there exists the suppremum ∨A.

(The set A is upper bounded if there exists b ∈ E such that x ≤ b for any x ∈ A.) Ifthe set A is not upper bounded, then we put ∨A = ∞.

From the condition 4 it follows:6. If 0 ≤ yn ≤ xn and limn→∞xn = 0, then limn→∞yn = 0 too.From the condition 5 it follows:7. If the non-empty set A ⊆ E is lower bounded, then there exists the infimum ∧A.Any topological semifield is an m-scale. A topological product of m-scales is an

m-scale.

83

84 Mitrofan M. Choban, Laurentiu I. Calmutchi

Let E be an m-scale.Denote by E−1 = x : x · y = 1 for some y ∈ E the set of all invertible elements of

E.By N(0, E) we denote some base of the space E at the point 0.We consider that 0 ≤ x 1 if 0 ≤ x < 1, 1 − x is invertible and limn→∞xn = 0.

We put E(+,1) = x ∈ E : 0 ≤ x 1. If t ∈ R and 0 ≤ t < 1, then t · 1 ∈ E(+,1). Weidentify t with t · 1 ∈ E for each t ∈ R.

A mapping d : X ×X −→ E is called a pseudo-metric over m-scale E or a pseudo-E-metric if it satisfies the following axioms:

d(x, x) = 0, (∀)x ∈ E;

d(x, y) = d(y, x), (∀)x, y ∈ E;

d(x, y) ≤ d(x, z) + d(z, y) (∀)x, y, z ∈ E.

Every pseudo-E-metric is non-negative, i.e. d(x, y) ≥ 0 for all x, y ∈ X.An E-pseudo-metric d is called an E-metric if it satisfies the following axiom:

d(x, y) = 0 if and only if x = y.

The ordered triple (X, d, E) is called a metric space over m-scale E or an E-metricspace if d is an E-metric on X.

Let d be a pseudo-E-metric on X. If A ⊆ B, B ⊆ X, x ∈ X, r ∈ E and r ≥ 0,then d(A, B) = in f d(x, y) : x ∈ A, y ∈ B and B(x, d, r) = y ∈ X : d(x, y) < r.If U ∈ N(0, E), then we put B(x, d,U) = y ∈ X : d(x, y) ∈ U for any x ∈ X.The family B(x, d,U) : x ∈ X, U ∈ N(0, E) is the base of the topology T(d) ofthe pseudo-E-metric space (X, d, E). If d is an E-metric, then the space (X,T(d)) is aTychonoff (completely regular and Hausdorff) space.

The pseudo-E-metric d generates on X the canonical equivalence relation: x ∼ y iffd(x, y) = 0. Let X/d be the quotient set with the canonical projection πd : X −→ X/ρand the metric d(u, v) = ρ(π−1

d (u), π−1d (v)).

Definition 1.1. Let A be a non-empty set and Eα : α ∈ A be a family of m-scales. Amulti-Eα-metric space is a pair (X,P), where X is a set and P = dα : X ×X −→ Eα :α ∈ A is a non-empty family of pseudo-Eα-metrics on X satisfying the condition:x = y if and only if d(x, y) = 0 for each d ∈ P.

Remark 1.1. Let A be a non-empty set and Eα : α ∈ A be a family of m-scales.Then E = ΠEα : α ∈ A is an m-scale. If (X,P) is a multi-Eα-metric space, then:

1. d(x, y) = (dα(x, y) : α ∈) is an E-metric on X and T(d) is the topology of themulti-Eα-metric space X.

2. Let ρα(x, y) = bβ ∈ Eβ : β ∈ A, where bα = dα(x, y) and bβ = 0 for all β , α.Then Q = ρα : α ∈ A is a family of pseudo-E-metrics on X and T(d) is the topologyof the multi-E-metric space (X,Q).

Fixed point theorems in E - metric spaces 85

Thus we may consider that E = Eα for any α ∈ A in the Definition 1.1. Inparticular, any multi-E-metric space is an E′-metric space.

If E = R, then the pseudo-E-metric is called a pseudo-metric and the pseudo-E-metric space is called a pseudo-metric space.

Fix a multi-E-metric space (X,P). A subset V ⊆ X is called P-open if for anyx ∈ V there exists a set U ∈ N(0, E) and a finite set A = A(x,U) ⊆ P such thatB(x, A,U) = ∩B(x, ρ,U) : ρ ∈ A ⊆ V . The family T(P) of all P-open subsets is acompletely regular Hausdorff topology on X. If T is a completely regular Hausdorfftopology on X, then T = T(P) for some family P of pseudo-E-metrics on X (see [6]).If d(x, y) = (ρ(x, y) : ρP), then d is an EP-metric on X.

Let X be a space and F be a non-empty set of mappings f : X −→ X. If Fix( f ) , ∅for each f ∈ F, then X is called a fixed point space relative to F. By Fix(F) = x ∈ X :f (x) = x for any f ∈ Fwe denote the set of all common fixed points of the mappingsf ∈ F. If ϕ : X −→ X is a mapping, then by Coin(F, ϕ) = x ∈ X : f (x) = ϕ(x) forany f ∈ F we denote the set of all coincidence points of the mappings f ∈ F andmapping ϕ.

The excellent book [7] contains the fixed point theory for metric spaces with theimportant applications in distinct domains. Several results for general topologicalspaces with interesting applications are contained in the surveys [7, 12]. Distinctgeneralizations of the Banach fixed point principle were proposed in [3, 7, 12]. Dis-tinct classes of metrizable spaces were examined in [8, 9, 14]. In [16], the followinggeneral problem arose it was arisen the following general problem: to find topolog-ical analogies of the Banach fixed point principle. Some solutions to this problemwere proposed in [5, 17]. This article is a continuation of the works [5, 16, 17, 11].

For each family γ of subsets of a set X and any x ∈ X the set S t(x, γ) = ∪U ∈ γ :x ∈ U is the star of the point x relative to γ. We put S t(A, γ) = ∪S t(x, γ) : x ∈ A.

2. COMPLETE MULTI-METRIC SPACESFix an m-scale E and an E-metric space (X, d, E).A set xµ : µ ∈ M is a net or a generalized sequence if M is a directed set.A point x ∈ X is a limit of the net xµ : µ ∈ M and we put limµ∈M xµ = x if for any

U ∈ N(0, E) there exists λ ∈ M such that d(x, xµ) ∈ U for any µ ≥ λ.A net xµ : µ ∈ M is called fundamental if for any U ∈ N(0, E) there exists λ ∈ M

such that d(xµ, xη) ∈ U for any µ, η ≥ λ. Any convergent net is fundamental. Thelimit of a fundamental sequence is unique (if the limit exists).

The space (X, d, E) is called complete if any fundamental net is convergent (see[6, 1, 2]).

Example 2.1. Let A be an uncountable set. We put Iα = I = [0, 1] for any α ∈ A.Let Y = IA = ΠIα : α ∈ A, 0A = (0α : α ∈ A) ∈ Y and X = Y \ 0A. Thespace E = RA is a topological field and an m-scale. By construction, X ⊆ IA ⊆ E.We put d(x, y) = (|xα − yα| : α ∈ A) and dα(x, y) = |xα − yα| for any pair of points


x = (xα : α ∈ A) ∈ Y and y = (yα : α ∈ A) ∈ Y. Let P = dα : α ∈ A. Then (X, d, E)is an E-metric space and (X,P) is a multi-metric space. Obviously, T(d) = T(P). Thespace Y is compact and the space X is pseudocompact. We put xn = (2−n

α : α ∈ A).Then xn : n ∈ N is a fundamental sequence. We have limPxn = 0A. Thus the spaces(X, d, E) and (X,P) are not complete.

Remark 2.1. Let (X, d, E) be an E-metric space and the space E be first countable,i.e. it is metrizable in the usual sense. Then the space X is metrizable in the usualsense too.

3. CLOSURE OPERATORS ONPSEUDO-E-METRIC SPACES

Fix an m-scale E, a topological space X and a pseudo-E-metric d on a space X.Denote by exp(X) the set of all subsets of X.

A family γ ⊆ exp(X) is centered if ∩η , ∅ for any finite subfamily η ⊆ γ.If A is a non-empty subset of X, then diam(A) = ∨d(x, y) : x, y ∈ A. Consider

that diam(∅) = 0.A closure operator on the pseudo-E-metric space (X, d, E) is a mapping C :

exp(X) −→ exp(X) satisfying the conditions:1. if A ⊆ B, then C(A) ⊆ C(B);2. A ⊆ C(A) and diam(C(A)) = diam(A);3. C(C(A)) = C(A).Let C be a closure operator on X. The set A ⊆ X is C-closed if C(A) = A. The

space X is called C-compact if ∩γ , ∅ for any centred family γ of non-empty C-closed sets.

If the space X is compact and C(A) = clXA is the closure of the set A, then C is aclosure operator on X and X is a C-compact space.

4. NIEMYTZKI FIXED POINT THEOREM FORMULTI-METRIC SPACES

The next theorem for compact metric spaces due to V. Niemytzki (see [15, 12])and L.P.Belluce and W.A.Kirk [4].

Theorem 4.1. Let f , ϕ : X −→ X be mappings of a countably compact multi-metricspace (X,P,R) into itself, ϕ(X) = X and d(ϕ(x), ϕ(y)) ≥ d(x, y) for all points x, y ∈ Xand d ∈ P. Assume that d( f (x), f (y)) < d(x, y) for any pair of distinct points x, y ∈ Xand any d ∈ P. Then:

1. The set Fix( f ) = x ∈ X : f (x) = x is non-empty and it is a singleton.2. The set Coin( f , ϕ) = x ∈ X : f (x) = ϕ(x) is non-empty and it is a singleton.

Theorem 4.2. Let f , ϕ : X −→ X be mappings of a countably compact multi-metricspace (X,P,R) into itself, ϕ(X) = X and d(ϕ(x), ϕ(y)) > d(x, y) for any pair of distinct


points x, y ∈ X and any d ∈ P. Assume that d( f (x), f (y)) ≤ d(x, y) for all pointsx, y ∈ X and d ∈ P. Then:

1. the set Fix(ϕ) = x ∈ X : ϕ(x) = x is non-empty and it is a singleton.2. the set Coin( f , ϕ) = x ∈ X : f (x) = ϕ(x) is non-empty and it is a singleton.

Proof. The mapping ϕ is one-to-one and the mappings f and ϕ−1 are uniformly con-tinuous. Moreover, if g = ϕ−1 f , i.e. g(x) = ϕ−1( f (x)) for any x ∈ X, thend(g(x), g(y)) < d(x, y) for any pair of distinct points x, y ∈ X and any d ∈ P.

Fix ρ ∈ P.Consider the projection πρ : X −→ X/ρ of (X, ρ) onto the metric space (X/ρ, ρ) and

the mapping gρ : X/ρ −→ X/ρ, where g(πρ(x)) = πρ(gρ(x)) and ρ(x, y) = ρ(g(x), g(y))for all x, y ∈ X. By construction, X/ρ is a compact metric space and ρ(g(x), g(y)) <ρ(x, y) for all x, y ∈ X/ρ. Suppose that for any n ∈ ω there exists xn, yn ∈ X such thatρ(g(xn), g(yn) ≥ (1− 2−n)ρ(x.y). Since X/ρ is a metrizable compact space, there existan infinite subset L ⊆ ω, an infinite subset M ⊆ L and two points x, y ∈ X such that:

- πρ(x) is the limit of the subsequence πρ(xn) : n ∈ L and ρ(x, xn) < 2−n for anyn ∈ L;

- πρ(y) is the limit of the subsequence πρ(yn) : n ∈ M and ρ(y, yn) < 2−n for anyn ∈ M.

There exists m ∈ ω such that 2−m+2 < ρ(x, y) − ρ(g(x), g(y)). By construction,ρ(g(x), g(y)) = lim ρ(g(xn), g(yn)) = lim ρ(xn, yn) = ρ(x, y), a contradiction. Thusthere exists a number kρ < 1 such that ρ(g(x), g(y)) ≤ k · ρ(x, y) for all x, y ∈ X.

Fix x = x0. We put xn = g(xn−1) for any n ∈ N.There exists b ∈ E such that b · (1 − k) = 1.Then d(xn, xn+1) = d(g(xn−1), g(xn)) ≤ k · d(xn−1, xn) for any n ∈ N. Hence

d(xn, xn+1) ≤ kn · d(x0, x1) for any n ∈ N. Obviously, d(xn, xm) ≤ d(xn, xn+1 +

... + d(xm−1, xm) ≤ (kn + kn+1 + ... + km−1 + km) · d(x0, x1). Therefore d(xn, xm) ≤(kn − kn+m) · b · d(x0, x1) provided n,m ∈ N and n ≤ m. Since limn→∞kn = 0, thesequence xn : n ∈ ω is fundamental in (X, ρ).

Since the space X is countably compact and ρ is an arbitrary pseudo-metric, thereexists a unique point b ∈ X such that b = lim xn. Then g(b) = lim g(xn) = lim xn+1 =

b. Thus f (b) = ϕ(b). The assertion 2 is proved. The assertion 1 follows from theassertion 2. The proof is complete.

The assertion 1 for compact metric spaces was proved by V.Niemytzki (see [15,12]) in 1936.

5. MAPPINGS OF E-METRIC SPACES WITHDIMINISHING ORBITAL DIAMETERS

Fix an m-scale E.Let (X, d, E) be a pseudo-E-metric space and let X be endowed with the topology

T(d) generated by the pseudo-E-metric d. Obviously, clA is a closure operator on X.


Consider a commutative semigroup F of mappings f : X −→ X. We can assumethat the identity mapping eX : X −→ X is an element of F.

For any point x ∈ X the set F(x) = (g(x)) : g ∈ F is the F-orbit of x. Letr(x,F, d) = ∧diam(F(g(x)) : g ∈ F. Since F is a semigroup, the family F2(x) =

F(g(x)) : g ∈ F is centered.The set L ⊆ X is F-invariant if g(L) ⊆ L for any g ∈ F.Let In(x,F, d) = ∩H ⊆ X : x ∈ X, clH = H,H is F-invariant. Obviously,

F(x) ⊆ In(x,F, d).A semigroup F is said to have diminishing orbital diameters (d.o.d.) (see [4, 10,

13]) if for every point x ∈ X we have r(x,F) < diam(F(x)) provided diam(F(x)) > 0.Our definition is more general then that in [13].

The set VCoin(F) = x ∈ X : d(x, f (x)) = 0 for any f ∈ F is the set of all d-fixedpoints of the family F. If d is a E-metric, then VCoin(F) = Coin(F).

Theorem 5.1. Let F : X −→ X be a commutative semigroup of continuous mappingswith diminishing orbital diameters of the pseudo-E-metric space (X, d, E) and thespace (X,T(d)) is compact. Then VCoin(F) , ∅. Moreover, clF(x) ∩ VCoin(F) , ∅for any point x ∈ X.

Proof. We can assume that the identity mapping eX ∈ F. Since the mappings F arecontinuous, the closure of an invariant set is invariant.

Fix a point b ∈ X.If diam(F(b)) = 0, then b ∈ F(b) ⊆ VCoin(F).Assume that diam(F(b) > 0. Since F has d.o.d. and the space (X,T(d)) is compact,

there exists a minimal invariant d-closed subset H of X such that diam(H) < ∞ andH ⊆ clF(b). We affirm that diam(H) = 0. Assume that diam(H) > 0. Fix c ∈ H. Ifdiam(F(c)) = 0), then P = clF(c) ⊆ H, diam(P) = 0 and P is a d-closed invariant set,in contradiction with the minimality of the set H. Thus 0 < diam(F(c)) ≤ diam(H).Since F has d.o.d., there exists a point y ∈ F(c) such that diam(F(y)) < diam((F(c)).Then B = clF(y) ⊆ H, diam(B) < diam(H) and B is a d-closed invariant set, acontradiction with the minimality of the set H. Therefore diam(H) = 0. Then H ⊆clF(x) ∩ VCoin(F).

Corollary 5.1. Let F : X −→ X be a commutative semigroup of mappings with di-minishing orbital diameters of the compact E-metric space (X, d, E). Then Coin(F) ,∅. Moreover, clF(x) ∩Coin(F) , ∅ for any point x ∈ X.

The Corollary 4.2 for the compact metric spaces was proved in [4, 10, 13].

Remark 5.1. If E = R and F : X −→ X is a commutative semigroup of continuousmappings with diminishing orbital diameters of the pseudo-E-metric space (X, d, E),then d( f (x), f (y)) ≤ d(x, y) for all x, y ∈ X and f ∈ F. In particular, the mappings F

are continuous.


6. MAPPINGS OF MULTI-E-METRIC SPACESWITH DIMINISHING ORBITAL DIAMETERS

Fix an m-scale E.Let (X,P) be a multi-E-metric space and let X be endowed with the topology T(P)

generated by the pseudo-E-metrics P. Obviously, clA is a closure operator on X. Forany A ⊆ X and d ∈ P the element diamd(A) = ∧d(x, y) : x, y ∈ A is the d-diameterof the set A. We put diam(A) = diamd(A) : d ∈ P. The element b = (bd : d ∈ P)is called an E-P-number if bd ∈ ∞ ∪ E for any d ∈ P. If b = (bd : d ∈ P) andc = (cd : d ∈ P) are two E-P-numbers, then b < c if and only if bd ≤ cd for all d ∈ P

and bρ < cρ for some ρ ∈ P. By construction, diam(A) is an E-P-number.Consider a commutative semigroup F of mappings f : X −→ X. We can assume

that the identity mapping eX : X −→ X is an element of F.For any point x ∈ X the set F(x) = (g(x) : g ∈ F is the F-orbit of x. Let r(x,F, d)

= ∧diamd(F(g(x)) : g ∈ F and r(x,F) = (r(x,F, d) : d ∈ P). By construction,r(x,F) is an E-P-number.

Let In(x,F) = ∩H ⊆ X : x ∈ X, clH = H,H is F-invariant. Obviously, F(x) ⊆In(x,F).

A semigroup F is said to have diminishing orbital diameters (d.o.d.) if for everypoint x ∈ X we have r(x,F) < diam(F(x)) provided diam(F(x)) > 0.

Theorem 6.1. Let F : X −→ X be a commutative semigroup of continuous mappingswith diminishing orbital diameters of the multi-E-metric space (X,P) and the space(X,T(P)) is compact. Then Coin(F) , ∅. Moreover, clF(x) ∩ Coin(F) , ∅ for anypoint x ∈ X.

Proof. The proof is similar to the proof of Therem 4.1. We can assume that theidentity mapping eX ∈ F. Since the mappings F are continuous, the closure of aninvariant set is invariant.

Fix a point b ∈ X.If diam(F(b)) = 0, then b ∈ F(b) ⊆ Coin(F).Assume that diam(F(b) > 0. Since F has d.o.d. and the space (X,T(d)) is compact,

there exists a minimal invariant closed subset H of X such that diam(H) < diam(F(b))and H ⊆ clF(b). We affirm that diam(H) = 0. Assume that diam(H) > 0. Fixc ∈ H. If diam(F(c)) = 0), then P = clF(c) ⊆ H, diam(P) = 0 and P is a closedinvariant set, a contradiction with the minimality of the set H. Thus 0 < diam(F(c)) ≤diam(H). Since F has d.o.d., there exists a point y ∈ F(c) such that diam(F(y)) <diam((F(c)). Then B = clF(y) ⊆ H, diam(B) < diam(H) and B is a closed invariantset, in contradiction with the minimality of the set H. Therefore diam(H) = 0. ThenH ⊆ clF(x) ∩Coin(F).

By the virtue of the following example, the method of multi-metrics is more effec-tive than the method of metrics.


Example 5.2. Let Xn = [0, 1] and ρn(x, y) = (n + 1) · |x − y| for all x, y ∈ Xnand n ∈ ω. We put X = ΠXn : n ∈ ω and d(x, y) = (n + 1) · |xn − yn| for allx = (xn : n ∈ ω) ∈ X, y = (yn : n ∈ ω) ∈ X and n ∈ ω. If P = dn : n ∈ ω,then (X,P) is a metrizable multi-metric space. Let f (x) = (2−1xn : n ∈ ω) andg(x) = (2−n−1xn : n ∈ ω) for any x = (xn : n ∈ ω) ∈ X. Denote by F the semigroupwith identity generated by the mappings f and g. Since f g = g f , the semigroup F iscommutative. If t ∈ R, then t = (xn = t : n ∈ ω) ∈ Rω. If 0 < x = (xn : n ∈ ω) ∈ X,then diam(F(g(x)) < diam(F(x) and diam(F(g(x)) < 1. Moreover, r(x,F) = 0 forany x ∈ X. Thus the semigroup F has diminishing orbital diameters. By construction,diam(F(t)) = (2−1 ·(n+1)·t : n ∈ ω) for any t ∈ [0, 1]. If t > 0, then supdiamdn(F(t) :n ∈ ω = ∞.

7. ASYMPTOTICAL PROPERTIES OFMAPPINGS

Fix an m-scale E. Let E(−,1) = E(+,1) ∩ E−1. If ε ∈ E(−,1) and 0 < t < 1, thent · ε, t · 1 ∈ E(−,1).

Consider an E-metric space (X, d, E) and a non-empty commutative semigroupF : X −→ X of continuous mappings of X into itself.

A semigroup F : X −→ X is called asymptotically non-expansive (see [10, 13]) iffor any x, y ∈ X there exists g ∈ F such that d( f g(x), f g(y)) ≤ d(x, y) for all f ∈ F.

If A ⊆ X, then the set AF = z ∈ X : there exists x ∈ A such that for every f ∈ F

and ε ∈ E(−,1) there exists g ∈ F with d( f g(x), z) < ε is called the F-closure of theset A in X.

Proposition 7.1. Let F : X −→ X be a non-empty commutative semigroup of con-tinuous asymptotically non-expansive mappings of an E-metric space (X, d, E) intoitself. If z ∈ XF, then:

1. For all f ∈ F and ε ∈ E(−,1) there exists g ∈ F such that d( f g(z), z) < ε.2. For any f ∈ F the mapping f |F(z) is an isometrical embedding of F(z) into

F(z).3. diam(F(z)) = diam(F(g(z))) for all g ∈ F.

Proof. Is similar to the proof of Propositions 1 and 2 from [10].

Corollary 7.1. ([13], Proposition 1, for metric space). Let F : X −→ X be a non-empty commutative semigroup of continuous asymptotically non-expansive mappingsof an E-metric space (X, d, E) into itself. If the semigroup F has diminishing orbitaldiameters and z ∈ XF, then F(z) = z and z ∈ Coin(F).

References

[1] M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, An outline of the theory of topologicalsemifields, Russian Math. Surveys, 21:4 (1966), 163-192.


[2] M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, Topological semifields and their ap-plications to general topology, American Math. Society. Translations, Series 2, 106, 1977.

[3] V. Berinde, M. Choban, Remarks on some completeness conditions involved in several commonfixed point theorems, Creative Mathematics and Informatics, 19:1(2010), 3-12.

[4] L. P. Belluce, W. A. Kirk, Fixed point theorems for certain classes of nonexpansive mappings,Proceed. Amer. Math. Soc., 20 (1969), 141-146.

[5] L. Calmutchi, M. Choban, On mappings with fixed points, Buletin Stiintific. Universitatea dinPitesti, Matematica si Informatica, 3 (1999), 91-96.

[6] R. Engelking, General Topology, PWN. Warszawa, 1977.

[7] A. Granas, J. Dugundji, Fixed point theory, Springer, New York, 2003.

[8] G. Gruenhage, Generalized Metric spaces, In. Handbook of Set-Theoretic Topology, K. Kunenand J. E. Vaughan, eds., Elsevier, Amsterdam, 1984.

[9] G. Gruenhage, Metrizable spaces and generalizations, In: Recent Progress in General TopologyII, M. Husek and J. van Mill, eds., Elsevier, Amsterdam, 2002, 201-225.

[10] R.D.Molmes, P.P.Narayanaswami, On asymptotically nonexpansive semigroups of mappings,Canadian Math. Bull., 13 (1970), 209-214.

[11] K. Iseki, On Banach theorem on contractive mappings, Proceed. Japan Academy, 41 :2 (1965),145-146.

[12] A. A. Ivanov, Fixed points of mappings of metric spaces, Journal of Soviet Mathematics 12:1(1979), 1-64.

[13] Mo Tak Kiang, Semigroups with diminishing orbital diameters, Pacific J. Math. 41:1 (1972),143-151.

[14] S. I. Nedev, M.M.Choban, General conception of the metrizability of topological spaces, Annalesof the Sophia University, Mathematics, 65 (1973), 111-165.

[15] V. Niemytzki, The method of fixed points in analysis, Uspekhi Matem. Nauk, 1 (1936), 141-174.

[16] J. Stepfans, S. Watson, W. Just, A topological fixed point theorem for compact Hausdorff spaces,York University, Preprint, 1991.

[17] I. V. Yashchenko, Fixed points of mappings and convergence of the iterations of the dual map-pings, In: Obshchaya Topologia: Otobrajenia, Proizvedenia i Razmernosti Prostranstv, Moskva:Moskov. Gosud. Uni-t, 1995, 131-142.

DISLOCATIONS AND DISCLINATIONS INFINITE ELASTO-PLASTICITY

ROMAI J., 6, 2(2010), 93–105

Sanda Cleja-TigoiuFaculty of Mathematics and Computer Science, University of Bucharest, [email protected]

Abstract The paper deals with analytical description of the dislocation and disclination, whichare lattice defects of the crystalline materials. The evolution equations for disclinations,having as source the screw dislocations are derived within the constitutive framework ofsecond order plasticity developed by the author in the papers appeared in Int. J. Fracture(2007), (2010).

Keywords: dislocation, disclination, elastic and plastic distortions, micro and macro forces, balanceequations, connections, evolution and constitutive equations.2000 MSC: 74C99, 74A20.

1. INTRODUCTIONThe plastic deformability of metals, which are crystalline materials, is produced

because of the existence of lattice defects inside the microstructure. The lattice de-fects, among which the dislocations, disclinations and point defects are mathemati-cally modeled by the differential geometry concepts, as torsion, curvature and metricproperty of certain connection, see Kroner [10], [11], de Wit [7], but without anyelasto-plastic constitutive equations. The elastic models for crystal defects can befound in Teodosiu [14]. We are not dealing with curved space but with curved geom-etry in flat space as stated de Wit [7].

1. The nature of the geometry is determined by the linear connection Γ, fixed byits coefficients, the curvature tensor R, the Cartan torsion or torsion tensor S;

2. the metric tensor C, to measure the distance;3. the non-metricity measure Q, of the connection relative to the metric tensor, i.e.

in terms of Γ and C.Geometry for which R,S,Q are non-vanishing is non-metric, non-Riemannian.

We restrict ourself to the case Q = 0, which means that the geometry is metric. IfR = 0 the geometry is called flat. If S = 0 the geometry is called symmetric. If S = 0the geometry is called Riemannian. If R = 0, S = 0 geometry is called Euclidean.

In this paper the dislocations and disclinations are lattice defects of interest. Thedislocations are characterized by the Cartan torsion, or by the non-zero curl of plasticconnection, which means that the plastic distorsion can not be derived from a certainpotential. The disclinations are characterized by a non-vanishing curvature R.

93

94 Sanda Cleja-Tigoiu

The mathematical description of the continuously distributed dislocations is givenby Noll [13], and the differential geometry support within the context of continuummechanics can be found also in [12].

In this paper a peculiar mathematical problem is analyzed: find the disclinations,which are solutions of the appropriate evolution equation in such a way that the microbalance equation are satisfied, when the distribution of the dislocations is given.

To give the mathematical description of the problem, we precise the general con-stitutive framework which is able to capture the dislocations and disclinations. Wemention here the direction developed by Clayton et al. [3] within a micropolar elasto-plastic model, in order to emphasize the translational (dislocation) and rotational(disclinaton) defects.

The behaviour of elasto-plastic body is described within the constitutive frame-work of second order plasticity, see Cleja-Tigoiu [5], [6], based on the decompositionrule of the motion connection into the elastic and plastic second order deformations,see Cleja-Tigoiu [4], and on the existence of configurations with torsion. The socalled configuration with torsion is denoted by K, and it is described through a pair,composed by Fp, an invertible second order tensor which is called plastic distorsion

and(p)Γ , a third order tensor field which is called plastic connection.

The pair (Fp,(p)Γ ) defines the second order plastic deformation with respect to the

reference configuration of the body B. The actual configuration of the body is asso-ciated with t he motion function, which is defined at every time t by χ(·, t) : B −→ E,E being the Euclidean (flat space) with a three dimensional vector space V. The sec-ond order elastic deformation, as a consequence of the decomposition rule, is definedwith respect to the so called configuration with torsion K, which is time dependent.Two type of forces, macro and micro forces, are considered in the model. They areviewed as pairs of stress (a second order tensorial field) and stress momentum (athird order tensorial field), and they obey their own balance laws. The micro forcessatisfy the viscoplastic type constitutive equations, in Kt. The evolution equationsfor Kt, which means the plastic distortion and plastic connection, have to be given.They have been derived to be compatible with an appropriate dissipation postulate.The energetic arguments, like a virtual power principle, macro and micro balanceequations, and especially a dissipation postulate (see energy imbalance principle, forinstance in Gurtin [9]) in order to prove thermomechanics restrictions, see Gurtin [9],Cleja-Tigoiu [5].

In our adopted formalism the measure of the dislocations is characterized by thenon-vanishing plastic curl (see Bilby [1], Noll [13]), while the disclinations has beenrelated to a certain second order tensor Λ which enters the expression of the plas-tic connection and generates a non-zero curvature, apart from de Wit [7], where ameasure of disclination is considered to be a second order curvature tensor.

The following notations, definitions and relationships will be used in the paper:

Dislocations and disclinations in finite elasto-plasticity 95

u · v,u × v, u ⊗ v denote scalar, cross and tensorial products of vectors;a ⊗ b and a ⊗ b ⊗ c are a second order tensor and a third order tensor defined by(a ⊗ b)u = a(b · u), (a ⊗ b ⊗ c)u = (a ⊗ b)(c · u), for all vectors u.For A ∈ Lin (Lin- the space of second order tensors), we introduce:the tensorial product A ⊗ a for a ∈ V, is a third order tensor, with the property(A ⊗ a)v = A(a · v),∀v ∈ V.I is the identity tensor in Lin, AT denotes the transpose of A ∈ Lin,∇A is the derivative (or the gradient) of the field A in a coordinate system xa(with respect to the reference configuration), ∇A =

∂Ai j

∂xk ei ⊗ e j ⊗ ek.

Definition 1.1. The curl operator is defined for any smooth second order field, sayA, through

(curlA)(u × v) = (∇A)u)v − ((∇A)v)u, ∀ u, v, z ∈ V. (1)

Lin(V, Lin) = N : V −→ Lin, linear− defines the space of all third ordertensors and it is given by N = Ni jkii ⊗ i j ⊗ ik.

The scalar product of two second order tensor A,B is A · B := tr(ABT ) = Ai jBi j,and the scalar product of third order tensors is given by N ·M = Ni jkMi jk, in a Carte-sian coordinate system. AB denotes the composition of A,B ∈ Lin. The compositionsof A ∈ Lin and N, a third order tensor, defines the appropriate third order tensorialfields, via the formulae NAu = Ni jkAkpupii ⊗ i j, and ANu = Ai jN jpkukii ⊗ ip, whichare written in a Cartesian basis, for any vector u.

The affine connection is defined in a coordinate system by its coordinate represen-tation

Γ = Γimkei ⊗ em ⊗ ek. (2)

We introduce the third order tensor field Γ[F1,F2],which is generated by a third orderfield Γ together with the second order tensors F1,F2 through the formula

(Γ[F1,F2]u)v = (Γ(F1u)) F2v, ∀u, v ∈ V. (3)

For any Λ1, Λ2 ∈ Lin we define a third order tensor associated with them, denotedΛ1 × Λ2, by

((Λ1 × Λ2)u)v = (Λ1u) × (Λ2v), ∀u, v ∈ V. (4)

For A, a third order tensor, we define the vector field tr(2)A through the relationshipwritten for all vectors

(tr(2)A) · u = tr(Au). (5)

Three types of second order tensors, A B, A r B and A l B will be associatedwith any pair A,B of third order tensors, following the rules written for all L ∈ Lin

(A B) · L = A[I,L] ·B = AiskLsnBink(A r B) · L = A · (LB) = Ai jkLinBn jk(A l B) · L = A · (BL) = Ai jkBi jnLkn.

(6)


2. GEOMETRIC RELATIONSHIPSLet F(X, t) = ∇χ(X, t) be the deformation gradient at time t, X ∈ B, and Γ =

F−1∇F be the motion connection or the material connection.∇χF is a gradient in the actual configuration, while the gradient in the configuration

with torsion K, ∇KF, is calculated by

∇KF := (∇F)(Fp)−1. (7)

Ax.1 The decomposition of the second order deformation, (F,Γ), associated with

the motion of the body B, into elastic, (Fe,(e)ΓK), and plastic, (Fp,

(p)Γ ), second order

deformations is given by

F = FeFp,

Γ =(p)Γ +(Fp)−1

(e)ΓK [Fp,Fp], Γ = F−1∇F.

(8)

Here the plastic connection with respect to the configuration with torsion,(p)ΓK, is

related with the plastic connection,(p)Γ , previously defined with respect to the reference

configuration, by the following relationships

(p)ΓK= −Fp

(p)Γ [(Fp)−1, (Fp)−1]. (9)

The plastic metric tensor Cp and strain gradient C are defined with respect to thereference configuration, while the elastic metric tensor Ce is defined in configurationwith torsion by

Cp := (Fp)T Fp, Ce = (Fp)−T C(Fp)−1, C = FT F. (10)

Definition 2.1. The Bilby’s type plastic connection is defined in a coordinate systemthrough

(p)A:= (Fp)−1∇Fp. (11)

Remark 2.1. The Cartan torsion that belongs to the Bilby’s connection is given by(Su)v = (Fp)−1(((∇Fp)u)v − ((∇Fp)u)v

), while the curvature tensor is vanishing.

Moreover, if the second order torsion tensor N is defined by N(u×v) = (Su)v, then ithas the representation N = (Fp)−1curlFp. Consequently, we can say that the torsiontensor S is equivalent to curlFp.

Let us introduce the expression for the plastic connection with respect to the ref-erence configuration built by Cleja-Tigoiu in [5], which has metric property withrespect to Cp, and that allows a represented under the form

(p)Γ=

(p)A +(Cp)−1(Λ × I), (12)


where the third order tensor Λ × I, generated by the second order (covariant) tensorΛ is defined by (4), namely

((Λ × I)u)v = Λu × v, ∀u, v ∈ V. (13)

Λ is called the disclination tensor.

3. SCREW DISLOCATIONFirst the skew dislocation will be defined in connection with the definition of the

Burgers vector. The Burgers vector can be defined in terms of the plastic distortionFp, by considering a closed curve (circuit) C0 in the reference configuration of thebody and A0 a surface with normal N bounded by C0

b =

∫

C0

Fp dX =

∫

A0

(curl(Fp))NdA =

∫

AK

αKnKdAK, (14)

where αK is Noll’s dislocation density in [13]

αK ≡ 1detFp (curl(Fp))(Fp)T . (15)

The expression of the Burgers vector can be aproximate by the formula

b ' curl(Fp)N area(A0). (16)

In crystal plasticity the presence of the defects inside the crystals is measured by non-vanishing Burgers vector. The integral representation (14) shows that non-zero curlof plastic distortion, supposed to be continuum and non-zero in a certain materialneighborhood, leads to a non-vanishing Burgers vector.

Definition 3.1. We say that the plastic distortion Fp characterizes a screw dislocationif the generated Burgers vector through a circuit with the appropriate normal N iscollinear with the normal, i.e. b ‖ N, in contrast with the edge dislocation whenb ⊥ N.

Let us introduce a Cartesian basis (e1, e2, e3) and a plastic distortion Fp whichdefines a screw dislocation that correspond to a Burgers vector directed to e3, givenby

Fp = I + e3 ⊗ τ, τ ⊥ e3, with

τ : D ⊂ R2 −→ V, τ := F p31e1 + F p

32e2, Jp := det(Fp) = 1.(17)

The mathematical description of the problem related to the screw dislocation withinthe finite elasticity is performed by Cermelli and Gurtin [2].

If we consider that τ = τ(x1, x2), the curl of the plastic distortion, see (1), is givenby

curl Fp =(∂F p

31

∂x2 −∂F p

32

∂x1

)e3 ⊗ e3, (18)


and Bilby’s type plastic connection

(p)A:= (Fp)−1∇Fp = e3 ⊗ ∇τ, ∇τ =

∂τβ

∂xδeβ ⊗ eδ, β, δ ∈ 1, 2. (19)

The representation given in (18) justifies the name attributed to the plastic distor-tion introduced in (17), taking into account the expression for the Burgers vector, fora plane curve with the normal e3.

Let us remark its appropriate trace tr(2)(p)A, defined by the formula (5)

tr(2)(p)A ·u := tr(

(p)A u) = tr((e3 ⊗ ∇τ)u) = e3∇τ(u) =

∂τ3

∂xk uk = 0, (20)

is zero, since τ3 : τ · e3 = 0, τ being a vector function orthogonal on e3.The plastic metric tensor Cp has the representation

Cp = I + τ ⊗ e3 + e3 ⊗ τ + τ ⊗ τ. (21)

Let us introduce the disclination tensor Λ represented in terms of Frank vector ω,like in Cleja-Tigoiu [5]

Λ := ηω ⊗ ζ, (22)

ζ the tangent vector line for the disclination field, and the scalar valued function ηhave to be defined. It follows that

∇Λ := ω ⊗ ζ ⊗ ∇η, Λ := ηω ⊗ ζ, (23)

if we take constant value for ζ.Hypothesis. We assume that Frank and Burgers vectors are orthogonal, ω · b = 0,

here b = e3. This hypothesis corresponds to the physical meaning attributed to thesetypes of lattice defects. See for instance [3].

Remark 3.1. In Fig.1 the Burgers vector produced by the screw dislocation and therotation produced by the disclinations have been plotted, following the commentsfrom [3].4. MACRO AND MICRO FORCES

Within the constitutive framework developed in [6], we consider that the free en-ergy in the reference configuration can be introduce in terms of the deformation fieldslisted below

ψ = ψ(Ce,Fp,(p)A,Λ,∇Λ) ≡ ψev(C,Fp,

(p)A) + ψd(Λ,∇Λ), (24)

but in contrast with [6] we have no influence of the elastic connection.Concerning the free energy function we assume that the defect energy is given by

ψd(∇Λ) =κ2

2β2

2∇Λ · ∇Λ, (25)


Fig. 1. Lattice defects: Screw dislocaton with Burgers vector b, disclination with Frank vector ω.

with β2 a length scale parameter.Ax.2 The macro forces are T− the non-symmetric Cauchy stress and macro stress

momentum, µ, which is described by a third order tensor field, satisfy the macrobalance equation

div T + ρb = 0, −2Ta = div µ, (26)

Ta is the skew symmetric part of the stress tensor. The equations (26) are similar tothose proposed in Fleck et al. [8], see also Cleja-Tigoiu [4], [6]

We denote by ρ, ρ, ρ0 the mass densities in the actual configuration, in the config-uration with torsion and in the reference one.

When we pull back the macro stress µ to the configuration with torsion, the appro-priate expression for the macro stress momentum µK is derived

1ρ0µK := (Fe)T 1

ρµ[(Fe)−T , (Fe)−T ], where Fe = F(Fp)−1. (27)

Ax.3 The micro stress denoted by Υλ and micro momentum respectively, denotedby µλ, which are associated with the disclinations satisfy in K their own balanceequation (see [6]):

Υλ = divK µλ + ρBλ ⇐⇒ Jp Υλ = div(Jp µλ(Fp)−T )

+ ρBλ. (28)

Here ρBλ is mass density of the couple body force, Jp =| detFp | .Ax.4 The micro stress Υp and micro stress momentum µp are associated with the

plastic mechanism, and they satisfy their own balance equation in the configuration


with torsion K :

Υp = divK (µp − µK) + ρBλ ⇐⇒

Jp Υp = div(Jp (µp − µK)(Fp)−T )

+ ρBp.(29)

Here ρBp is appropriate mass density of the couple body force.The equation (29)could be found in a modified form in [5].

The constitutive restrictions imposed by the free energy imbalance, that have beenobtained by Cleja-Tigoiu in [6], are adapted to the proposed here model as it follows.

The elastic type constitutive functions are derived from the free energy function,viewed like a potential for Cauchy stress

1ρ

T = 2F(∂Cψ)FT , µ = ∂Γψ, (30)

but here µ = 0, as a consequence of the supposition made in (24) that the free energyfunction is not dependent on Γ.

We postulate here an energetic (non-dissipative) constitutive equation for the microstress momentum associated with plastic behaviour and disclinations, respectively

1ρ0µ0

p = ∂(p)Aψ

1ρ0µλ0 = ∂∇Λψ

d(∇Λ) = κ2β22 ω ⊗ ζ ⊗ ∇η.

(31)

The last equality is a consequence of (25) together with (23).The expressions for the micro stress plastic momentum µp and the macro stress

momentum µK both of them being considered with respect to K, pulled back to thereference configuration cab be expressed under the form

1ρ0µp

0 := (Fp)T 1ρµp[(Fp)−T , (Fp)−T ],

1ρ0µ0 := (Fp)T 1

ρµK[(Fp)−T , (Fp)−T ],

(32)

as well as the micro stress momentum associated with the disclination in K can beexpressed in terms of µλ0 by

1ρµλ := (Fp)−T 1

ρ0µλ0[(Fp)T , (Fp)T ]. (33)

The viscoplastic type, dissipative evolution equations for plastic distorsion Fp and Λ

have been postulated in [6] to be compatible with the appropriate dissipation inequal-ity. The viscoplastic evolution equation for plastic distortion, giving rise to the rate


of plastic distortion with respect to the reference configuration is given by

lp = −(Fp)−1Fp

1ρ0

(Σ0 − Σ0p) + (Fp)T∂Fpψ = ξ1 lp.

(34)

Definition 4.1. The Mandel type stress measures in the reference configuration areassociated with the appropriate stresses as it follows

1ρ0

Σp0 =

1ρ

(Fp)T Υp(Fp)−T ,1ρ0

Σ0 =1ρ

FT TF−T (35)

and1ρ0

Σλ0 =

1ρ

(Fp)T Υλ(Fp)−T . (36)

The viscoplastic type, dissipative evolution equations are postulated for disclina-tion Λ

( 1ρ0

Σ0λ − ∂Λ ψ

)+

( (p)A 1

ρ0µλ0

)−

−( 1ρ0µλ0 r

(p)A

) − 1ρ0µλ0

(tr(2)(

(p)A)

)= ξ3 Λ.

(37)

Ax.5 The scalar constitutive functions ξ1, ξ3 are defined in such a way to be compat-ible with the dissipation inequality

ξ1 lp · lp + ξ3 Λ · Λ ≥ 0. (38)

Note that the dissipation inequality is reduced to the expression written in the lefthand side of (39) if the viscoplastic type equations (34) and (38) have been accepted.The Mandel type stress measure associated with the disclination mechanism, Σ0

λ, isrelated to the micro stress Υλ via (37), while µλ0 is expressed in (31).

5. DISCLINATIONS GENERATED BY APLASTIC DISTORTION

We suppose that the second order disclination tensor Λ is described in terms ofFrank vector ω, with the scalar intensity function η and the disclination line ζ, sayconstant during the deformation process, have to be found.

We are now able to solve the problem: Find the disclination tensor Λ, having theexpression (22), to be solution of the evolution equation (38) with the micro stressesrelated by (37), and which is compatible with the micro balance equation associatedwith the disclination, (28)2.

We take into account the physically motivated hypothesis that the Frank and Burg-ers vectors are (fixed) orthogonal, as well as that the plastic distorsion (17) character-izes the screw dislocation, then

ω · e3 = 0, τ · e3 = 0, ∇(τ)u · e3 = 0, ∀ u ∈ V. (39)


First of all we eliminate the micro stress from (38), via (28) together with (37).By definitions and the hypotheses, from (33) together with (31)2 we get

µλ(Fp)−T = ρ0(Fp)−T∂∇Λψ[I, (Fp)T ]. (40)

As a direct consequence of (24) together with (25) and (23) we can prove the follow-ing relationship

1ρ0µλ(Fp)−T = κ2β

22 ω ⊗ ζ ⊗ ∇η + (ζ · e3)(ω ⊗ ζ ⊗ ∇η). (41)

Consequently, we apply the divergence operator to (42) and then

div(µλ(Fp)−T )

= κ2β22ρ0 ∆ηω ⊗ ζ + (ζ · e3) ω ⊗ τ+

+κ2β22ρ0(ζ · e3)

(ω ⊗ ((∇ τ)∇η)

(42)

holds. Here ∆η denotes the Laplacean of the scalar function η.

Proposition 5.1. In the evolution equation for Λ, written in (38) the term

( (p)A 1

ρ0µλ0

)= κ2β

22(e3 ⊗ ∇τ) (

ω ⊗ ζ ⊗ ∇η), (43)

is vanishing, since its components are given by

( (p)A 1

ρ0µλ0

)sn = κ2β

22 (es · ζ)

(en · (∇τ)∇η)(e3 · ω) = 0, (44)

while the following expression

( 1ρ0µλ0 r

(p)A

)= κ2β

22(ζ · (∇τ)∇η) e3 ⊗ ω. (45)

has to be introduced inside.

In order to prove the above relationships we calculate the components of the sec-ond order tensor field, starting from the definitions introduced in (6). The scalarproduct of the tensor written in the left hand side of (44) with (en ⊗ es) gives rise tothe components sn of the tensor. We perform the tensorial operations and we arriveat

κ2(β2)2((e3 ⊗ ∇τ) (ω ⊗ ζ ⊗ ∇η)) · (en ⊗ es) =

= κ2(β2)2 (e3 ⊗ ∇τ)[I, en ⊗ es] · (ω ⊗ ζ ⊗ ∇η) =

= κ2(β2)2 (e3 ⊗ es ⊗ (∇τ)T en) · (ω ⊗ ζ ⊗ ∇η) =

= κ2(β2)2 (e3 ⊗ ω)(es · ζ)en · ((∇τ)∇η).

(46)


Note the formula proved below via (3) has been used in the previously written ex-pression, namely

(e3 ⊗ ∇τ)[I, en ⊗ es]u = (e3 ⊗ (∇τ)u)(en ⊗ es) =

= (e3 ⊗ es)(∇τ)u · en) = (e3 ⊗ es ⊗ (∇τ)en)u,(47)

hold for all u ∈ V, since ∇τ ∈ Lin.We proceed similarly to the calculus of the term written in the left hand side of (46).We take the scalar product with (en ⊗ es) and we use the definitions introduced in (6)

( 1ρ0µλ0 r

(p)A

) · (en ⊗ es) =

= κ2β22(ζ · (∇τ)∇η) · (en ⊗ es)

(p)A,

(48)

as well as the formula

(en ⊗ es) (e3 ⊗ ∇τ) = δ3sen ⊗ ∇τ. (49)

We use the balance equation for micro forces associated with the dislocation (28)composed on the left with (Fp)T = I + τ ⊗ e3 and on right with (Fp)−T = I − τ ⊗ e3in order to express the Mandel’s type stress tensor (37)

1ρ0

Σλ0 = κ2β

22 ∆η

(ω ⊗ ζ + (ζ · e3)ω ⊗ τ)+

+(ζ · e3)ω ⊗ ((∇τ)∇η) − κ2β22 ∆η(ζ · τ)(ω ⊗ e3)−

−κ2β22 ∆η(ζ · e3)∆η | τ |2 +((∇τ)∇η) · τω ⊗ e3.

(50)

Proposition 5.2. The evolution equation for Λ written in (38) becomes

1ρ0

Σ0λ − κ2β

22(ζ · (∇τ)∇η) e3 ⊗ ω = ξ3 ηω ⊗ ζ, (51)

where the first term is given by (51), since tr(2)(p)A= 0.

We investigate now the consequences that follows from (52) and (51).If ζ · e3 = 0, the projection on ζ of the equation (52) together with (51) leads to

ξ3η = κ2β22∆η. (52)

When we return to the equation (52), we conclude that the following conditions

(ζ · τ)∆η = 0, ζ · ∇τ)∇η = 0 (53)


necessarily hold.If ζ · e3 , 0, as ω · e3 = 0, τ · e3 = 0, the projection of the evolution equation on

the Burgers vector, here on e3, is reduced to

+κ2β22( − ∆η (ζ · τ)ω + (ζ · e3)(1− | τ |2)∆η−

−((∇τ)∇η) · τω)= ξ3 η(ζ · e3)ω.

(54)

If we restrict to the condition ζ = e3, the equation (55) becomes

κ2β22 ∆η(1− | τ |2) − ((∇τ)∇η) · (τ) = ξ3 η. (55)

When we return to the equation (51) together with (50) the compatibility condition

∆η τ + (∇τ)∇η = 0 (56)

follows. We consider the scalar product of (57) with τ and we arrive at the equality

∆η | τ |2 +τ · ((∇τ)∇η) = 0. (57)

Note that the equation (56) together with (58) becomes (53). Thus we proved thefollowing result.

Theorem 5.1. Let us assume that e3 · ω = 0.

1 If the disclination line ζ is orthogonal to the Burgers vector, namely e3 · ζ = 0the evolution equation for the density of disclination is given by

ξ3η = κ2β22∆η, (58)

and the compatibility condition is reduced to ζ · τ = 0.

2 If the disclination line is collinear with Burgers vector, i.e. ζ ‖ e3, the evolutionequation for the disclination intensity is still given by (59), while the compati-bility condition, namely between the plastic distortion expressed in terms of τand η, is derived under the form

τ∆η + (∇τ)∇η = 0. (59)

6. CONCLUSIONSIn Theorem 5.1 under the assumption that the plastic distortion is described by

a screw dislocation we derived the non-local evolution for scalar disclination den-sity. We assumed that the disclination is described in terms of Frank vector anddisclination line. If the Frank and Burgers vectors are orthogonal, and moreover thedislocation and the disclination lines are either collinear, or perpendicular one to theanother, then the evolution of scalar disclination density η is not influenced by the


evolution of plastic distorsion. A non-local evolution equation for η is derived. Theresult is similar with those obtained in [6], for the case of plastic shear distortion, butthere the directions of the Frank vector and the Burgers vector could be arbitrarilygiven. The same type of the analysis can be conducted for the edge dislocation.

Acknowledgments. The author acknowledges support from the Romanian Ministryof Education and Research through PN-II, IDEI program (Contract no. 576/2008).

References[1] B.A. Bilby, Continuous distribution of dislocations, In: Sneddon IN, Hill R (eds) Progress in

Solid Mechanics. North-Holland, Amsterdam, 1960.

[2] P. Cermelli, M.E. Gurtin, The Motion of Screw Dislocations in Crystalline Materials UndergoingAntiplane Shear: Glide, Cross-Slip, Fine Cross-Slips, Arch. Rat. Mech. Anal., 148(1999), 3-52.

[3] J.D. Clayton, D.L. McDowell, D.J. Bammann, Modeling dislocations and disclinations with finitemicropolar elastoplasticity, Int. J. Plasticity, 22, 2(2006), 210-256.

[4] S. Cleja-Tigoiu, Couple stresses and non-Riemannian plastic connection in finite elasto-plasticity, ZAMP, 39(2002), 996-1013.

[5] S. Cleja-Tigoiu, Material forces in finite elasto-plasticity with continuously distributed disloca-tions, Int J Fracture, 147, 1-4(2007), 67-81.

[6] S. Cleja-Tigoiu, Elasto-plastic materials with lattice defects modeled by second order deforma-tions with non-zero curvature, Int. J. Fracture, 166, 1-2(2010), 61-75.

[7] R. de Wit, A view of the relation between the continuum theory of lattice defects and non-Euclidean geometry in the linear approximation, Int. J. Engng. Sci., 19(12)(1981), 1475-1506.

[8] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory andexperiment, Acta Metall. Mater., 42, 2(1994), 475-487.

[9] M.E. Gurtin, On the plasticity of single crystal: free energy, microforces, plastic-strain gradients,J. Mech. Phys. Solids, 48, 5(2000), 989-1036.

[10] E. Kroner, The Differential geometry of Elementary Point and Line Defects in Bravais Crystals,Int. J. Theoretical Physics, 29, 11(1990), 1219-1237.

[11] E. Kroner, The internal mechanical state of solids with defects, Int. J. Solids Structures, 29, 14-15(1992), 1849-1857.

[12] J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity, (1983) Prentice-Hall, Inc.,Englewood Cliffs, New Jersey, 1983.

[13] W. Noll, Materially Uniform Simple Bodies with Inhomogeneities, Arch. Rat. Mech. Anal., 1967,and in The Foundations of Mechanics and Thermodynamics, Selected papers (1974) Springer-Verlag/ Berlin Heidelberg New York, 1974.

[14] C. Teodosiu, Elastic Models of Crystal Defects, Ed. Academiei, Springer-Verlag Berlin Heidel-berg New York, 1982.

ANALYSIS AND NUMERICAL APPROACH TOUNIDIMENSIONAL ELASTO-PLASTIC PROBLEMWITH MIXED HARDENING

ROMAI J., 6, 2(2010), 107–120

Sanda Cleja-Tigoiu1, Nadia Stoicuta2, Olimpiu Stoicuta2

1Faculty of Mathematics and Computer Science, University of Bucharest, Romania2University of Petrosani, [email protected]

Abstract In this paper is determined by the finite element method combined with the Euler methodand Newton method, the solution of the problem of elasto-plastic deformation of a one-dimensional bar using weak formulation of the problem with initial and boundary data.

Keywords: elasto-plastic model, constitutive equation, weak solution, finite element method, numericalsimulation.2000 MSC: 97U99.

1. INTRODUCTIONThe constitutive framework which describes the set of the constitutive and evo-

lution equation in the mechanical problem can be found in the references [1], [5],and [11]. The initial and boundary value problem, which describes the dynamic be-haviour of one-dimensional elasto-plastic material with isotropic and mixed harden-ing has been solved by J.C. Simo and T. J. Hughes in [10]. In order to solve the one-dimensional problem, the finite element method is applied together with the Newtonmethod and the Return Mapping Algorithm. From the continuum model which de-scribes the constitutive equation in the elasto-plastic model, through an elastic typeconstitutive equation together with the evolution equation for plastic strain and in-ternal variables, the discrete algorithmic equations are obtained in [12] by applyingan Euler type difference scheme. The Return Mapping Algorithm proposed in [12]involve the computation of the elastic trial stress together with the test for plasticloading. To solve the one-dimensional, initial and boundary value problem, we startfrom the quasi-static rate boundary value problem formulated at a generic momentof time and associated with the equilibrium equation. In order to determine the weaksolution of the rate boundary value problem, which represents the time derivative ofthe displacement field u, i.e. the velocity at time t in the one-dimensional body, weapply the finite element method, those main results are resumed from [2], [8] and[11]. The method proposed here works for loading elsto-plastic process only. To up-date the current values of σ, εp, α, k, of the unknowns the Euler method and Newton

107

108 Sanda Cleja-Tigoiu, Nadia Stoicuta, Olimpiu Stoicuta

method are applied in order to integrate the differential equation system. The numeri-cal algorithms combine the finite element method with the numerical methods relatedto the differential type equations, i.e. the Euler and Newton method, and the coupledproblems are simultaneously solved.

2. THE UNIDIMENSIONAL MATHEMATICALMODEL OF ELASTO-PLASTIC PROBLEMWITH MIXED HARDENING

Let x ∈ Ω be a material point of the body Ω ⊂ R. Ω is an one dimensional body,which deforms in time, say on the interval I = [0,T ). We denote by u : Ω × I →R - the displacement field, by σ : Ω × I → R - the Cauchy stress field and with

ε : Ω × I → R, with ε(x, t) =∂u∂x

(x, t) ≡ ε(u)(x, t) -the strain. We introduce alsothe hardening variables: α : Ω × I → R the kinematic hardening variable andk : Ω× I → R the isotropic hardening variable, with k > 0, which play different rolesin describing the deformability of the surface of the plasticity in the stress space,during the irreversible deformation process. Following Cleja-Tigoiu and Cristescu[1985] (see also Chabauche [1989], Paraschiv-Munteanu and Cleja-Tigoiu [2004],we introduce the basic assumptions within the constitutive framework of the elasto-plastic model with mixed hardening. [see 2004]:1. the rate of the strain tensor ε can be decomposed into the rate of elastic part andthat of plastic part, denoted by εe and εp, respectively

ε = εe + εp, (1)

2. the elastic type constitutive equation is given by

σ = Eεe, (2)

3. the irreversible properties of the material are described in terms of the yield func-tion F(σ, α, k), which dependent on the stress and hardening variables; the rate of theplastic strain tensor is described by the associated flow rule through

εp = λ∂F(σ, α, k)

∂σ, (3)

where λ, the so-called plastic factor is a function of the state of the material, and it isdefined through the Kuhn-Tucker and consistency conditions

λ ≥ 0, F ≤ 0, λF = 0 (4)

λF = 0. (5)

4. the variation of the kinematic hardening variable α is given by Armstrong andFrederik [1996] law

α = Cεp − γαk, (6)

Analysis and numerical approach to unidimensional elasto-plastic problem... 109

C and γ are material parameters, while the isotropic hardening variable k is given by

k =

√23εp · εp (7)

5. we add the initial condition

σ(0) = 0, ε(0) = 0, εp(0) = 0, α(0) = 0, k(0) = 0 (8)

which correspond to a physical body undeformed and unstressed at time t = 0.

In the considered one-dimensional model the yield function is given by

F(σ, α, k) := |σ − α| − F(k) (9)

where F is a strictly increasing function, defined through

F(k) = R(k) + κ, R(0) = 0, R′(k) > 0, k > 0 (10)

κ = σY represents the initial yield condition in the uniaxial test. The isotropic chang-ing in the dimension of the yield surface is represented by Chaboche [1989]

R(k) = Q(1 − e−bk), Q, b > 0 (11)

Q and b are material constants.Taking into account the expression (10), the yield function is derived under the

formF(σ, α, k) = |σ − α| −

[Q(1 − e−bk) + σY

](12)

The plastic factor is calculated from (5)

λ =〈β〉h

H(F) (13)

where the complementary plastic factor β and the hardening variable h are definedthrough

β =Eε(u)(σ − α)

Q(1 − e−bk) + σY(14)

h = E + C −√

23γα

σ − αQ(1 − e−bk) + σY

+

√23

Qbe−bk (15)

We introduce the supposition that h > 0 on F(σ, α, k) = 0. If h becomes zero inthe process, then the solution cannot be defined since εp → ∞. We can say that thematerial is damaged.In relation (14)

ε(u(x, t)) =∂u(x, t)∂x

≡ du(x, t)dx

, (15)


with the last notation introduced for the sake of simplicity.The Heaviside function, from the relationship (13) is

H(F) =

0, if F < 01, if F ≥ 0 (17)

By eliminating the elastic part of the strain we arrive at the following result.

Proposition 2.1. The one-dimensional elasto-plastic model with mixed hardening isdescribed by the following differential system

εp =〈β〉h

σ − αQ(1 − e−bk) + σY

H(F)

α =〈β〉h

Cσ − α

Q(1 − e−bk) + σY−

√23γα

H(F)

σ = Eε(u) − E〈β〉h

σ − αQ(1 − e−bk) + σY

H(F)

k =

√23〈β〉h

H(F)

(17)

where β as a function on u and h are given by the relations (14) together with (16).

The equilibrium equation in terms of one-dimensional Cauchy stress σ(x, t) hasthe form:

divσ(x, t) + b1(x, t) = 0 in Ω × I, with divσ :=∂σ

∂x(19)

where b1 ∈ R is the body force.The boundary conditions are formulated on the boundary ∂Ω = Γ of the body

which is divided into two parts Γu and Γσ, such that Γu∪Γσ = Γ and Γu∩Γσ = ∅.They are given by

σ11(x, t)n = f1(x, t) in Γσ × I ; u(x, t) = g1(x, t) in Γu × I. (19)

Here n is the outward unit normal field on ∂Ω and n = 1 in the one-dimensionalcase. The functions f1(x, t) and g1(x, t) are given.

Problem P: Given the functions b1(x, t), f1(x, t) and g1(x, t), find the real valuedfunctions u, σ, ε, εp, α, k that are defined on Ω × [0,T ) and satisfy (19) together with(20) and the differential type constitutive equations, listed in (18) together with (14)and (16).

To solve it we start from the variational formulation of the equilibrium problemfor elasto-plastic one-dimensional bar.


3. THE VARIATIONAL FORMULATIONIn what follows, we recall the main ideas from Johnson [1987] and Hughes [1987],

which are appropriate to our description. Unlike Johnson and Hughes, which used a

local momentum equation of the form∂

∂xσ + ρb = ρ

∂v∂t

in Ω × I where v =∂u∂t

, inthis paper the equilibrium equation is given by relation (19).

We determine the weak solutions of the rate quasi-static boundary value problemassociated at a generic moment t, which is derived by taking the time derivative ofthe equilibrium equation (19) together with the boundary condition (20)

div[∂σ(x, t)∂t

]+∂b1(x, t)

∂t= 0 in Ω × I

∂σ(x, t)∂t

=∂ f1(x, t)∂t

in Γσ × I

∂u(x, t)∂t

=∂g1(x, t)

∂tin Γu × I

(20)

For the problem P, at a generic stage of the process the current values, i.e. at thetime t, the current plastic domain is the form:

Ωpt = x ∈ Ω|F(σ(x, t), α(x, t), k(x, t)) = 0 . (22)

The set of kinematically admissible velocity field is denoted by:

Vad =

w |w : Ω→ R ;

∂w∂x∈ L2(Ω), w

/Γu = g1

⊂ H1(Ω) (23)

where

L2(Ω) =

w

∣∣∣∣∣∣∣∣∣w : Ω→ R;

∫

Ω

w2dx = ‖w‖2L2(Ω) < ∞

(24)

is the space of the square integrable functions on Ω, and

H1(Ω) =

w ∈ L2(Ω)

∣∣∣ ∂w∂x∈ L2(Ω)

(25)

is the Sobolev space.

Theorem 3.1. At every time t the rate of the displacement field, u, satisfy the follow-ing relationships:

∫

Ω

Eep (x, t)du(x, t)

dxdw(x, t)

dxdx =

∫

Ω

db1(x, t)dt

w(x, t)dx+

+

[d f1(x, t)

dtw(x, t)

]∣∣∣∣∣∣Γσ

+

[dσ(x, t)

dtdg1(x, t)

dt

]∣∣∣∣∣∣Γu

(25)


which hold for every admissible vector field w ∈ Vad.

Proof. By multiplying the first equation of equation (21) with an admissible displace-ment w, integrating on a domain Ω and applying Green’s formula we get

∫

Ω

dσ(x, t)dt

dw(x, t)dx

dx =

[dσ(x, t)

dtw(x, t)

]|Γ +

∫

Ω

db1(x, t)dt

w(x, t)dx. (27)

Further, we perform some transformation in the left hand side of the relation (27)by introducing the third relation from (18) and we have

∫Ω

E du(x,t)dx

dw(x,t)dx dx −

∫

Ω

Eλσ(x, t) − α(x, t)

Q(1 − e−bk(x,t)) + σY

dw(x, t)dx

HFdx =

=∫Ω

db1(x,t)dt w(x, t)dx +

[d f1(x,t)

dt w(x, t)]∣∣∣∣

Γσ+

[dσ(x,t)

dtdg1(x,t)

dt

]∣∣∣∣Γu

(28)

where the rate of strain ε(u(x, t)) is replaced from (16).The expression of β and h are replaced by the relations (14) and (16), but the

positive part of the expression of β enters (28).We remark that only under the assumption that

Eε(u(x, t))(σ − α) > 0 (29)

along the process, i.e. when no unloading is produced, (28) becomes∫Ω

[E

(1 − E

h(x,t)

((σ(x,t)−α(x,t))

Q(1−e−bk(x,t))+σY

)2H(F)

)]du(x,t)

dxdw(x,t)

dx dx =

∫Ω

db1(x,t)dt w(x, t)dx +

[d f1(x,t)

dt w(x, t)]∣∣∣∣

Γσ+

[dσ(x,t)

dtdg1(x,t)

dt

]∣∣∣∣Γu.

(29)

In (30) following notation has been introduced, but only under the hypothesis for-mulated in (29),

Eep(x, t) =

E if H(F) = 0,

E(1 − E

h(x,t)

(σ(x,t)−α(x,t)


)2)

if H(F) = 1, (31)

where Eep is the so-called elasto-plastic modulus.We introduced the supposition that h > 0 on F(σ, α, k) = 0. Consequently the

equality (30) together with (31) becomes:∫

Ω

Eep (x, t)du(x, t)

dxdw(x, t)

dxdx =

∫

Ω

db1(x, t)dt

w(x, t)dx+

+

[d f1(x, t)

dtw(x, t)

]∣∣∣∣∣∣Γσ

+

[dσ(x, t)

dtdg1(x, t)

dt

]∣∣∣∣∣∣Γu

.

(32)


The relation (32) is in fact the variational representation of the solution.

Remark. In case when the displacement u(x, t) = 0 on the boundary Γu, i.e.g1(x, t) = 0, the variational formulation given above becomes:

∫

Ω

Eep (x, t)du(x, t)

dxdw(x, t)

dxdx =

∫

Ω

b1(x, t)w(x, t)dx +[f1(x, t)w(x, t)

]∣∣∣∣Γσ, (33)

written for all w ∈ Vad, which are vanishing on Γu.As a consequence of the above theorem, the following statement holds.

Theorem 3.2. Find a displacement field u (·, t), solution of the variational formula-tion

a(u,w) = 〈L,w〉 ∀u ∈ Vad,w ∈ Vad (34)

where a (·, ·) : Vad × Vad → R is the bilinear and symmetric form defined by

a(u,w) =

∫

Ω

Eep(x, t)ε(u(x, t))ε (w(x, t)) dx. (35)

Here L is a linear functional

〈L,w〉 =

∫

Ω

b1(x, t)w(x, t)dx +[f1(x, t)w(x, t)

]∣∣∣∣Γσ

; ∀t ∈ [0,T ] (36)

where

Eep(x, t) =

E if H(F) = 0,

E(1 − E

h(x,t)

(σ(x,t)−α(x,t)


)2)

if H(F) = 1, (37)

but under the hypothesis written in (29).

Hypothesis. We assume that the material properties are given in such a way toensure that a(,) be a bilinear, symmetric, continuous and coercive form.

Next,we replace the material particle x ∈ Ω with ξ ∈ Ω, to avoid misunderstand-ings. The problem to be solved is presented below

Problem P1. Consider the following differential system

ddt

x(t) = f (x(t), u (ξ, t)) ; x(t0) = x0 (38)

where the vector x(t) has the components

x(t) = (x1(t) x2(t) x3(t) x4(t))T (39)

x1 = εp, x2 = α, x3 = σ, x4 = k (40)


while the vector valued function which defines the system (38) has the form:

f (x(t), u (ξ, t)) = ( f1 (x, u) f2 (x, u) f3 (x, u) f4 (x, u))T (41)

f1 (x, u) = λ (x, u)x3 − x2

Q(1 − e−bx4) + σY

f2 (x, u) = λ (x, u)

Cx3 − x2

Q(1 − e−bx4) + σY−

√23γx2

f3 (x, u) = E (ε (u) − f1 (x, u))

f4 (x, u) =

√23λ (x, u)

(42)

λ (x(t), u (ξ, t)) =〈β (x(t), u (ξ, t))〉

h(x(t))H (F (x(t))) (43)

β (x(t), u (ξ, t)) =Eε (u (ξ, t)) (x3(t) − x2(t))

Q(1 − e−bx4(t)) + σY(44)

h(x(t)) = E + C −√

23γx2(t)

x3(t) − x2(t)Q(1 − e−bx4(t)) + σY

+

√23

Qbe−bx4(t) (45)

H(F(x(t))) =

0, if F(x(t)) < 01, if F(x(t)) ≥ 0 (46)

F(x(t)) = |x3(t) − x2(t)| −[Q(1 − e−bx4(t)) + σY

]. (47)

Determine the displacement u ∈ Vad field, such that u(·, t) ∈ Vad, the plastic defor-mation εp, the internal variables α, k and the stress σ which satisfy at every time t thevariational formulation

∫ L

0Eep (x(t)) ε(u(ξ, t))ε(w(ξ, t))dξ =

=

∫ L

0b1(ξ, t)w(ξ, t)dξ +

[f1(ξ, t)w(ξ, t)

]∣∣∣∣ξ=L

(48)

Eep (x(t)) =

E if H(F) = 0

E(1 − E

h(x(t))

(x3(t)−x2(t)

Q(1−e−bx4(t))+σY

)2)

if H(F) = 1 (49)

and having the time evolution for any fixed particle given by the above differentialsystem.


4. DISCRETIZATION BY FINITE ELEMENTMETHOD

Finite element method is generally used to solve a variational problem, or thediscretization form of certain variational formulation of the problem. Here we usethe finite element method to find the weak solutions of the problem, which satisfy thevariational equation (48) coupled with the system of the differential equations (38).

The problem P1 is solved using the finite element method, and hence we brieflypresents this method, following the papers and books given by Ferreira [8], Fish [9],Johnson [10].

We consider one-dimensional body, which is identified with an interval of the real

axis, i.e. Ω = [0, L]. The time interval [0, T ] is discretized by [0,T ] =N⋃

n=1[tn, tn+1].

[0,L] is at its turn discretized in ne network elements, where a network element hasthe form Ωe =

[ξe

1, ξe2, ξ

e3

], e = 1, ne. Let nN the total number of nodes used in the

discretization.We apply the finite element method to the elasto-plastic problem formulated for

the one-dimensional bar. Thus, we divide the bar into ne elements with nN nodes,each element having three nodes.

Since we consider the one-dimensional case, the number of degrees of freedomngl is equal to one for each node in the network.

Concerning the boundary conditions: ξ = 0 (Γu) is considered to be the fixed endof the bar thus the displacement is zero, while at a traction boundary condition isapplied at ξ = L (Γσ).

Further we proceed to the effective implementation of finite element method pro-posed by Fish [9], Johnson [10]. The global approximation of the trial solution u(ξ, t)is

u(ξ, t) =

ne∑

e=1

Ne(ξ)ue(t) ≡ N(ξ)u(t). (50)

The vectors N(ξ) and u(t) which enter the relation (50) have the form

N(ξ) =[N1(ξ) N2(ξ) N3(ξ) ... NnN (ξ)

](51)

u(t) =[u1(t) u2(t) u3(t) ... unN (t)

]T . (52)

In the same way, the weight function w(ξ, t) is calculated using the relationship

w(ξ, t) =

ne∑

e=1

Ne(ξ)we(t) ≡ N(ξ)w(t) (53)

where the vector w(t) is given under the form

w(t) =[w1(t) w2(t) w3(t) ... wnN (t)

]T . (54)


Also, the displacements ue(ξ, t) and the weight function we(ξ, t) of the three nodesof each element can be calculated using the following relations:

ue(ξ, t) = Ne(ξ)ue(t) ≡ (Ne(ξ)Le) u(t) (55)

we(ξ, t) = Ne(ξ)we(t) ≡ (Ne(ξ)Le) w(t) (56)

where the vectors ue(t) and we(t) are calculated using the relationships:

ue(t) = Leu(t) (57)

we(t) = Lew(t) (58)

Here Le, called the selection matrix, is composed by a number of lines equal tothe number of degrees of freedom per element and a number of columns equal to thetotal number of degrees of freedom in the network and is build with the help of therelationships

Lei j = δIne(i,e), j =

1, Ine(i, e) = j0, Ine(i, e) , j. (59)

In the relation (59), Ine is the matrix of connection, that is a matrix that has a linefor each item. The line e of the matrix Ine contains the nodes that make up the item.

In the relations (55) and (56), the vector of the interpolation function of an elemente, if this element has three nodes, is of the form

Nei (ξ) =

[Ne

1(ξ) Ne2(ξ) Ne

3(ξ)]

(60)

and the displacements vector for an element in the three nodes are

ue(t) =[ue

1(t) ue2(t) ue

3(t)]

(61)

we(t) =[we

1(t) we2(t) we

3(t)]. (62)

For the unidimensional case when the network element contains three nodes, theshape functions Ne

i (ξ) given by the relation (60) are built as is follows:

Ne1(ξ) =

(ξ − ξe

2

) (ξ − ξe

3

)(ξe

1 − ξe2

) (ξe

1 − ξe3

) (63)

Ne2(ξ) =

(ξ − ξe

1

) (ξ − ξe

3

)(ξe

2 − ξe1

) (ξe

2 − ξe3

) (64)

Ne3(ξ) =

(ξ − ξe

1

) (ξ − ξe

2

)(ξe

3 − ξe1

) (ξe

3 − ξe2

) . (65)


When we take the time derivate of the displacement written in (55), we have:

due(ξ, t)dt

≡ ue(ξ, t) =(Ne(ξ)Le) u(t) (66)

Thus the rate of the strain tensor is calculated using the following relationship:

ε(ue(ξ, t)) =due(ξ, t)

dξ=

dNe(ξ)dξ

Leu(t) ≡ (Be(ξ)Le) u(t) (67)

where Be(ξ) =dNe(ξ)

dξ, and Be(ξ) =

[Be

1(ξ) Be2(ξ) Be

3(ξ)].

On the other hand, the represented of the strain measure is:

ε(we(ξ, t)

)=

(Be(ξ)Le) w(t). (68)

In the weak form (48), the integral over (0, L) is viewed as a sum of integrals overindividual element domain, Ωe. Using the notations (67) and (68) in the variationalformulation (48) we have:

ne∑e=1

∫Ωe

[Be(ξ)Lew(t)

]T (Eep)e (x(t))[Be(ξ)Leu(t)

]dξ =

ne∑e=1

∫Ωe

[Ne(ξ)Lew(t)

]T b1(ξ, t)dξ +ne∑

e=1

[[Ne(ξ)Lew(t)

]T f1(ξ, t)]∣∣∣∣

Γeσ

.(69)

Moreover (69) can be written under the form

w(t)T

ne∑

e=1LeT

∫

Ωe

Be(ξ)T (Eep)e (x(t)) (Be(ξ)) dΩe

Le

u(t) =

= w(t)T

ne∑

e=1LeT

∫Ωe

Ne(ξ)T b1(ξ, t)dΩe +ne∑

e=1LeT

[Ne(ξ)T f1(ξ, t)

]Γeσ

.(70)

We introduce the following notation relations

Ke(x(t)) =∫

Ωe

Be(ξ)T (Eep (x(t)))e (Be(ξ)) dΩe

Reb(t) =

∫Ωe

Ne(ξ)T b1(ξ, t)dΩe

Ref (t) =

[Ne(ξ)T f1(ξ, t)

]Γeσ

(71)

which allow us to rewrite (70) in the form:

w(t)T

ne∑

e=1

LeT KeLe

u(t) −

ne∑

e=1

LeT Rbe +

ne∑

e=1

LeT R fe

= 0. (72)


The matrices introduced in (71) have the meaning, Ke is the element stiffnessmatrix, Rb

e is the matrix of the external forces and Ref is the matrix of the internal

forces. If

K(x(t)) =

ne∑

e=1

LeT Ke(x(t))Le (73)

R(t) = Rb(t) + R f (t)→

Rb(t) =ne∑

e=1LeT Re

b(t)

R f (t) =ne∑

e=1LeT Re

f (t)(74)

then the relation (72) can be written in a shorter form

w(t)T [K(x(t))u(t) − R(t)] = 0 ; ∀ w(t) (75)

As w(t) is arbitrarily given, relationship (75) becomes

K(x(t))u(t) = R(t). (76)

Thus, in view of the above, in the following we apply the Gauss quadrature formulato the integral from relationship (70), i.e.:

Kemn

(x(t)) = h2

n−1∑

i=0

AiBem(h1 + h2τi)T (

Eep (x(t)))e Be

n(h1 + h2τi) (77)

where the matrix Ke is symmetric, and m = 1, nN and n = 1, nN .The integral Re

b(t) given by the relation from (71) can be similarly computed byusing the Gauss quadrature formula, as follows:

(Re

b

)m

(t) =

h2

n−1∑

i=0

Nem(h1 + h2τi)T b(h1 + h2τi, t)

, m = 1, nN (78)

where we used the notations h1 =b + a

2, h2 =

b − a2

. We obtain the components

Fm(x(t), u(t)) =

nN∑

n=1

[Kmn(x(t))un(t)] − Rm(t) (79)

of the relation (76), that can be written in following form

F(x(t), u(t)) := K(x(t))u(t) − R(t) = 0. (80)

To solve the system of equations (80) relative to the unknown u(t), we apply New-ton’s method. Thus we have:

u (tn+1) = u (tn) −[∂F (x(tn), u (tn))

∂u(tn)

]−1

F (x(tn), u (tn)) , n = 0, 1, ... . (81)


Simultaneously we apply the Euler method to the non-linear system of differentialequations given by (38). The iterative formulae can be derived under the form:

x (tn+1) = x (tn) + ∆t f (x (tn) ,N(ξ)u(tn)) , x (t0) = 0, n = 0, 1, 2, ... (82)

as a consequence of (50), where ∆t = tn+1 − tn is the step of the method.We present now the main steps of the algorithm applied to the formulated problem:

u(t0) = 0; x (t0) = 0∣∣∣∣∣∣∣∣∣∣∣∣∣∣

For n = 0 to N∣∣∣∣∣∣∣∣∣

x (tn+1) = x (tn) + ∆t f (x (tn) ,N(ξ)u(tn))

u (tn+1) = u (tn) −[∂F (x(tn), u (tn))

∂u(tn)

]−1

F (x(tn), u (tn))

end

(83)

To solve the problem P1, the algorithm presented in the relationship (83) is run inevery point of the network.

5. NUMERICAL APPLICATIONNumerical application presented here, aims to highlight the numerical algorithm

of solving the problem P1.

Fig. 1. The graph of the stress depending on the strain in the point 30


In this sense, the material chosen is steel DP 600, whose parameters are givenbelow (Broggiato [2008]):

E = 182.000 [MPa] ;σY = 349, 4 [MPa] ; Q = 50, 1 [MPa]C = 17.400 [MPa] ; b = 27, 5 [−] ; γ = 125, 9 [−]

. (84)

As an example we consider an uni-dimensional bar, whit Ω = [0, 30]. The barhas length L = 30 [cm]. The bar is fixed in the node ξ = 0, and the traction forcef1(ξ, t) = 450 sin t is located in the node ξ = 6. The body force is considered byform b1 = 0, 5 t2. We specify also that for the meshing, the bar was divided intoten elements, each element having three nodes, total number of nodes in the networkbeing 21. Since, the bar is fixed in the node ξ = 0, the displacement in this node iszero. In the network, the elements have been divided into equal intervals. In theseconditions, after running the program, which implements the numerical algorithm ofthe problem P1, we obtain the following information presented in the figure 1.

Acknowledgement: Sanda Cleja-Tigoiu acknowledges the support from the Ministery of EducationResearch and Inovation under CNSIS PN2 Programm Idei, PCCE, Contract No. 100/2009.

References[1] P. J. Armstrong, C. O. Frederick, A mathematical representation of the multiaxial Bauschinger

effect, G.E.G.B. Report RD-B-N 731, 1996.[2] J. Bathe, Finite element procedure, Vols. I, II, III, Prentice Hall, Englewood Cliffs, New Jersey,

1996.[3] T. Belytschko, K. L. Wing, B. Moran, Nonlinear Finite Elements for Continua and Structures,

British Library Cataloguing in Publication Data, Toronto, 2006.[4] G. B.Broggiato , F. Campana, L. Cortese The Chaboche nonlinear kinematic hardening model:

calibration methodology and validation , Meccanica, 43, 2(2008), 115-124.[5] J. L. Chaboche, Constitutive equations for cyclic plasticity and cyclic visco-pasticity , Int. J.

Plasticity, 1989.[6] S. Cleja-Tigoiu, N. Cristescu, Teoria plasticitatii cu aplicatii la prelucrarea materialelor, Ed.

Univ. Bucuresti, 1985.[7] S. Cleja-Tigoiu, E. Soos, Elastoplastic model with relaxed configurations and internal state vari-

ables, Appl. Mech. Rev., 43, 7(1990), 131-151.[8] A. J. M. Ferreira, Matlab Codes for Finite Element Analysis Solid and Structures , Springer,

Berlin, 2009.[9] J. Fish, T. Belytschko, A First Course in Finite Elements , John Wiley and Sons, Ltd., USA, 2007.

[10] C. Johnson, Numerical solution of partial diferential equation by the finite element method ,Cambridge University Press, Cambridge, 2002.

[11] I. Paraschiv-Munteanu, S. Cleja-Tigoiu, E. Soos, Plasticitate cu aplicatii ın geomecanica , Ed.Universitatii din Bucuresti, 2004.

[12] J. C. Simo, T. J. R. Hughes, Computational Inelasticity , Springer, New-York,1998.

SOME NATURAL DIAGONAL STRUCTURESON THE TANGENT BUNDLES AND ON THETANGENT SPHERE BUNDLES

ROMAI J., 6, 2(2010), 121–130

Simona-Luiza Druta-Romaniuc1, Vasile Oproiu2

1Department of Sciences, ”Al. I. Cuza” University of Iasi, Romania2Faculty of Mathematics, ”Al. I. Cuza” University of Iasi, [email protected], [email protected]

Abstract We study the properties of some geometric structures defined on the tangent and tan-gent sphere bundles of a Riemannian manifold by using the natural lifts. These liftsare obtained from the Riemannian metric g of the base manifold. We get some almostHermitian structures defined on the tangent bundle and find the conditions under whichthey are Kahlerian. Then we study some specific properties of such structures (to haveconstant holomorphic sectional curvature, to be Einstein, etc). Similar problems are con-sidered for the tangent sphere bundles Tr M endowed with the the induced almost contactstructures (the quality of Tr M to be a Sasaki space form, or an η-Einstein manifold).

Keywords: tangent bundle, natural lift, Einstein structure, tangent sphere bundle, almost contact struc-ture, Sasaki space form, η-Einstein manifold.2000 MSC: 53C05, 53C15, 53C55.

1. INTRODUCTIONSome new interesting geometric structures on the tangent bundle of a Riemannian

manifold (M, g) may be obtained by lifting the metric g from the base manifold.In 1998, in a joint work with N. Papaghiuc ([16]), the second author introduced

on the tangent bundle of a Riemannian manifold, a Riemannian metric defined by aregular Lagrangian depending on the energy density only.

The same author has studied some properties of a natural lift G, of diagonal type,of the Riemannian metric g (so that it is no longer obtained from a Lagrangian) anda natural almost complex structure J of diagonal type on T M (see [12]-[15]). Thecondition for (T M,G, J) to be a Kahler Einstein manifold leads to the conditionsfor (M, g) to have constant sectional curvature, and for (T M,G, J) to have constantsectional holomorphic curvature or to be a locally symmetric space. In [12] and[15], the second author excluded some important cases which appeared, in a certainsense, as singular cases. The case where the Riemannian metric G is defined by theLagrangian L (excluded in [12]) was studied in [16]. Other singular cases excludedin [12] and [13] were studied by N. Papaghiuc in [18] and [19]. Other geometricstructures obtained by considering natural lifts of the metric g from the base manifoldto its tangent bundle have been studied in (see [1], [8], [17]).

121

122 Simona-Luiza Druta-Romaniuc, Vasile Oproiu

In this paper we present a survey on the geometric structures defined by the naturaldiagonal lifts of the metric from the base manifold to the tangent bundle, as well asthe structures defined by the restrictions of these lifts to the tangent sphere bundlesof constant radius r, seen as hypersurfaces of the tangent bundles, consisting of thetangent vectors of norm equal to r only.

It is known that every almost Hermitian structure from the tangent bundle inducesan almost contact structure on the tangent sphere bundle of constant radius r. Impor-tant results in this direction may be found in the recent surveys [1] and [9], but alsoin the papers [2]-[4], [11], [20]. The most part of the authors who worked in this fieldconsidered unitary tangent sphere bundles, endowed with the induced Sasaki metric(see [21]), but O. Kowalski and M. Sekizawa showed that the geometry of the tangentsphere bundles depends on the radius. They determined in [9] the g-natural metrics ofthe mentioned type on the tangent sphere bundles of constant radii, having constantscalar curvature, and in [2] M. T. K. Abassi and O. Kowalski obtained the conditionsunder which the g-natural metrics on the unit tangent sphere bundle are Einstein.

In the paper [5] we considered on the tangent bundle T M of a Riemannian mani-fold M a natural diagonal metric, obtained by the second author in [15], and denotedin the section 4 by G. We proved that the tangent sphere bundle Tr M endowed withthe Riemannian metric induced from G is never a space form, then we found theconditions under which (Tr M,G) is an Einstein manifold.

In [6] we obtained some almost contact metric structures (ϕ, ξ, η,G) on the tan-gent sphere bundles, induced by some almost Hermitian structures (G, J) of naturaldiagonal lift type on the tangent bundle of a Riemannian manifold (M, g). The abovealmost contact metric structures are not automatically contact metric structures. Inorder to get such structures we made some rescalings of the metric, of the funda-mental vector field, and of the 1-form. Then we gave the characterization of theSasakian structures of natural diagonal lift type on the tangent sphere bundles. In thiscase, the base manifold must be a space form. We studied in [7] the holomorphicφ-sectional curvature of the obtained Sasakian manifolds, and we proved that thereare no natural diagonal tangent sphere bundles of constant holomorphic φ-sectionalcurvature. On the other hand, the study of the conditions under which the obtainedSasakian tangent sphere bundles (Tr M, ϕ, ξ, η,G) are η-Eistein manifolds, namely thecorresponding Ricci tensor field has the form

Ric = ρG + ση ⊗ η,

where ρ and σ are smooth real functions, led in [6] to two cases, which will bepresented in the final section of this paper.

The manifolds, tensor fields and other geometric objects we consider in this paperare assumed to be differentiable of class C∞ (i.e. smooth). The well known sum-mation convention is used throughout this paper, the range of the indices h, i, j, k, l, rbeing always 1, . . . , n.

Some natural diagonal structures... 123

2. PRELIMINARY RESULTSConsider a smooth n-dimensional Riemannian manifold (M, g) and its tangent

bundle τ : T M → M. The total space T M has a structure of a 2n-dimensionalsmooth manifold, induced from the smooth manifold structure of M. This structureis obtained by using local charts on T M induced from usual local charts on M. If(U, ϕ) = (U, x1, . . . , xn) is a local chart on M, then the corresponding induced localchart on T M is (τ−1(U),Φ) = (τ−1(U), x1, . . . , xn, y1, . . . , yn), where the local coor-dinates xi, y j, i, j = 1, . . . , n, are defined as follows. The first n local coordinates ofa tangent vector y ∈ τ−1(U) are the local coordinates in the local chart (U, ϕ) of itsbase point, i.e. xi = xi τ, by an abuse of notation. The last n local coordinatesy j, j = 1, . . . , n, of y ∈ τ−1(U) are the vector space coordinates of y with respectto the natural basis in Tτ(y)M defined by the local chart (U, ϕ). Due to this specialstructure of differentiable manifold for T M, it is possible to introduce the concept ofM-tensor field on it (see [10]).

Denote by ∇ the Levi Civita connection of the Riemannian metric g on M. Thenwe have the direct sum decomposition

TT M = VT M ⊕ HT M (1)

of the tangent bundle to T M into the vertical distribution VT M = Ker τ∗ and thehorizontal distribution HT M defined by ∇ (see [22]). The vertical and horizontal liftsof a vector field X on M will be denoted by XV and XH respectively. The set of vectorfields ∂

∂y1 , . . . ,∂∂yn on τ−1(U) defines a local frame field for VT M, and for HT M we

have the local frame field δδx1 , . . . ,

δδxn , where δ

δxi = ∂∂xi − Γh

0i∂∂yh , Γh

0i = ykΓhki, and

Γhki(x) are the Christoffel symbols of g.

The set ∂∂yi ,

δδx j i, j=1,n, denoted also by ∂i, δ ji, j=1,n, defines a local frame on T M,

adapted to the direct sum decomposition (1).Consider the energy density of the tangent vector y with respect to the Riemannian

metric g

t =12‖y‖2 =

12

gτ(y)(y, y) =12

gik(x)yiyk, y ∈ τ−1(U). (2)

Obviously, we have t ∈ [0,∞) for every y ∈ T M.

3. NATURAL DIAGONAL LIFTEDSTRUCTURES ON THE TANGENT BUNDLE

The second author constructed in [15] an (1,1)-tensor field J on the tangent bundleT M, obtained as natural 1-st order lift of the metric g from the base manifold to thetangent bundle T M:

Jδi = a1(t)∂i + b1(t)g0iC, J∂i = −a2(t)δi − b2(t)g0iC, (3)

where a1, a2, b1, b2 are smooth functions of the energy density, C = yh∂h is the Liou-ville vector field and C = yhδh is the geodesic spray.


The invariant expressions of the above (1,1)-tensor field are

JXHy = a1(t)XV

y + b1(t)gτ(y)(X, y)yVy ,

JXVy = −a2(t)XH

y − b2(t)gτ(y)(X, y)yHy ,∀X ∈ T1

0(M), ∀y ∈ T M. (4)

Studying the conditions under which J2 = −I, it was easy to prove the followingresult.

Proposition 3.1. ([15]) The (1, 1)-tensor field J given by the relations (3) or (4) de-fines an almost complex structure on the tangent bundle if and only if a2 = 1/a1, b2 =

−b1/[a1(a1 + 2tb1)].

Then it was considered on T M a natural Riemannian metric of diagonal type G onT M, induced by g. The expression in local adapted frame are defined by the M-tensorfields

G(1)i j = G(∂i, ∂ j) = c1gi j + d1g0ig0 j,

G(2)i j = G(δi, δ j) = c2gi j + d2g0ig0 j,

G(∂i, δ j) = G(δ j, ∂i) = 0.(5)

where the coefficients c1, c2, d1, d2 are smooth functions depending on the energydensity t ∈ [0.∞) and the conditions for G to be positive definite are given by c1 >0, c2 > 0, c1 + 2td1 > 0, c2 + 2td2 > 0, for every t ≥ 0.

The invariant expressions of the above metric are

G(XHy ,Y

Hy ) = c1(t)gτ(y)(X,Y) + d1(t)gτ(y)(X, y)gτ(y)(Y, y),

G(XVy ,Y

Vy ) = c2(t)gτ(y)(X,Y) + d2(t)gτ(y)(X, y)gτ(y)(Y, y),

G(XVy ,Y

Hy ) = G(XH

y , XVy ) = 0,

(6)

∀X, Y ∈ T10(T M), ∀y ∈ T M.

Theorem 3.1. ([14]) Let J be a natural diagonal almost complex structure on T Mdefined by g and the functions a1, a2, b1, b2. Then the family of natural diagonalRiemannian metrics G, given by (5) or (6) is almost Hermitian with respect to J, i.e.G(JX, JY) = G(X, Y), if and only if

c1

a1=

c2

a2= λ,

c1 + 2td1

a1 + 2tb1=

c2 + 2td2

a2 + 2tb2= λ + 2tµ, (7)

where λ > 0, µ > 0 are functions of t.

In [14], the second author obtained some classes of natural almost Hermitian struc-tures (G, J) on T M, these classes are obtained from the well known classification ofthe almost Hermitian structures in sixteen classes. The results concerning this classi-fication are given by Theorems 4, 5, 6 and 7 in [14].

Some very important particular cases of the almost Hermitian structure obtainedin Theorem 3.1 have been studied further. So, if we consider the case when c1(t) =


a1(t) = u(t), b1(t) = d1(t) = v(t), b2(t) = d2(t) = w(t), a2(t) = c2(t) = 1u(t) , λ(t) = 1

and µ(t) = 0, where w(t) = − vu(u+2tv) , then all the conditions from Proposition 3.1 and

Theorem 3.1 are satisfied, and we obtain the almost Hermitian structure defined andstudied by the second author in [15].

Theorem 3.2. ([15]) Under the above conditions, the almost Hermitian manifold(T M,G, J) is an almost Kahler manifold. Moreover, the almost complex structureJ on T M is integrable (i.e. (T M,G, J) is a Kahler manifold) if (M, g) has constantsectional curvature c and the function v is given by

v =c − uu′

2tu′ − u.

In [15] the second author obtained a Kahler-Einstein structure even in the casewhere (M, g) has positive constant sectional curvature, but only on a tube around thezero section in T M ([15, Theorem 4.2]). He also obtained on T M a structure ofKahler manifold with constant holomorphic sectional curvature ([15, Theorem 5.1]).

The particular case of the above situation, when the function u(t) = 1, for allt ∈ [0,∞), has been studied by the same author in [12]. In this case he stated

Theorem 3.3. ([12]) (T M,G, J) is an almost Kahlerian manifold and the almostcomplex structure J is integrable if and only if (M, g) has constant sectional curvaturec and v = −c. If c < 0 is obtained a Kahler structure on whole T M. In the case wherethe constant c is positive is obtained a Kahler structure in the tube around the zerosection in T M defined by t < 1

2c .

Moreover, in the same paper it was proved

Theorem 3.4. If (M, g) has negative constant sectional curvature then its tangentbundle has a structure of Kahler Einstein manifold. If (M, g) has positive constantsectional curvature then a tube around the zero section in T M has a structure ofKahler Einstein manifold.

Another important case which appear as a singular case is when v(t) = 0 for allt ∈ [0,∞) which implies w(t) = 0. This case has been studied by N. Papaghiuc in[18], [19], and he obtained

Theorem 3.5. ([18]) Assume that the function u(t) satisfies the condition u(0)u′(0) ,0. Then the almost complex structure J on T M or on a tube around the zero sectionin T M is integrable if and only if the base manifold (M, g) has constant sectionalcurvature c and the function u(t) satisfies the ordinary differential equation uu′ = c,i.e. u(t) =

√2ct + A, where A is an arbitrary real constant.


Another result from [18] (see also [19]) is given by

Theorem 3.6. If n , 2, then the Kahlerian manifold (T M,G, J) cannot be an Einsteinmanifold and cannot have constant holomorphic sectional curvature. If n = 2, thenthe Kahlerian manifold (T M,G, J) is Ricci flat.

Remark that the singular case when v(t) = u′(t) was studied in [16].

4. NATURAL DIAGONAL STRUCTURES ONTHE TANGENT SPHERE BUNDLES OFCONSTANT RADIUS R

Let us recall some results from [5], concerning tangent sphere bundles endowedwith a natural diagonal lifted metric.

We denote by Tr M = y ∈ T M : gτ(y)(y, y) = r2, with r ∈ (0,∞), and theprojection τ : Tr M → M, τ = τ i, where i is the inclusion map.

The horizontal lift of any vector field on M is tangent to Tr M, but the vertical liftis not always tangent to Tr M. The tangential lift of a vector X to (p, y) ∈ Tr M, usedin some recent papers like [3], [9], [11], [20], is defined by

XTy = XV

y −1r2 gτ(y)(X, y)yV

y .

A generator system for the tangent bundle to Tr M is given by δi and ∂Tj = ∂ j −

1r2 g0 jyk∂k, i, j, k = 1, n. Remark that the vector fields ∂T

j j=1,n satisfy the relationy j∂T

j = 0, so they are not independent. In any point of Tr M the vectors ∂Ti i=1,...n

span an (n − 1)-dimensional subspace of TTr M.The vector field N = yi∂i is normal to Tr M in T M.From now on we shall denote by G the metric G defined by the relations (5) or

(6), and by G′ the metric on Tr M induced by the metric of T M. Remark that thefunctions c1, c2, d1, d2 become constant, since in the case of the tangent spherebundle of constant radius r, the energy density t becomes a constat equal to r2

2 . Itfollows

G′(XHy ,Y

Hy ) = c1gτ(y)(X,Y) + d1gτ(y)(X, y)gτ(y)(Y, y),

G′(XTy ,Y

Ty ) = c2[gτ(y)(X, Y) − 1

r2 gτ(y)(X, y)gτ(y)(Y, y)],G′(XH

y , YTy ) = G′(YT

y , XHy ) = 0,

(8)

∀X, Y ∈ T10(M), ∀y ∈ Tr M, where c1, d1, c2 are constants. The conditions for G′ to

be positive are c1 > 0, c2 > 0, c1 + r2d1 > 0.The Levi-Civita connection ∇, associated to the Riemannian metric G′ on the tan-

gent sphere bundle Tr M has the form∇∂T

i∂T

j = Ahi j∂

Th , ∇δi∂

Tj = Γh

i j∂Th + Bh

jiδh,

∇∂Tiδ j = Bh

i jδh, ∇δiδ j = Γhi jδh + Ch

i j∂Th ,


the expressions of the involved M-tensor fields being given in [5].In [5] we obtained the horizontal and tangential components of the curvature ten-

sor field K, denoted by sequences of H and T , to indicate horizontal, or tangentialargument on a certain position. For example, we may write

K(δi, δ j)∂Tk = HHT Hh

ki jδh + HHTT hki j∂

Th .

Then we proved the following result

Theorem 4.1. ([5]) The tangent sphere bundle Tr M, with the Riemannian metricinduced by the natural metric G of diagonal lift type on the tangent bundle T M, hasnever constant sectional curvature.

We computed the Ricci tensor field of the manifold (Tr M,G′), and we obtainedthe components Ric(δ j, δk) = RicHH jk and Ric(∂T

j , ∂Tk ) = RicTT jk:

RicHH jk = Ric jk − d1(2c1+r2d1)2c2(c1+r2d1) g jk +

d1(2c1+r2d1)[c1n+r2d1(n−1)]2r2c1c2(c1+r2d1) g0 jg0k

− c22c1

Rh0klR

ljh0 + c2d1

2c1(c1+r2d1) Rh0k0Rh0 j0

(9)

RicTT jk =r4d2

1−2c1(c1+r2d1)(n−2)2c1r2(c1+d1r2) ( 1

r2 g0 jg0k − g jk) +c2

24c2

1Rhik0Rhi

j0

− c22d1

2c21(c1+r2d1)

Rh0 j0Rh0k0.

(10)

Analyzing the vanishing conditions of the difference Ric(X, Y) − ρG′(X,Y), forevery X, Y ∈ T1

0(Tr M) we stated

Theorem 4.2. ([5]) The tangent sphere bundle Tr M of an n-dimensional Riemannian(M, g) of constant sectional curvature c is Einstein with respect to the metric G′induced by the natural diagonal lifted metric G defined on T M, i.e. it exists a realconstant ρ such that Ric(X, Y) = ρG(X, Y), for every X,Y ∈ T1

0(Tr M), if and only if

c1 =r2d1nn − 2

, c2 =d1n

c(n − 2), ρ =

c(n − 1)2(n − 2)r2d1n2 .

In the paper [6] we studied the almost contact metric structures (ϕ, ξ, η, G) on thetangent sphere bundles, induced by the natural diagonal almost Hermitian structures(G, J) on T M, characterized in Theorem 3.1, and we proved

Theorem 4.3. The almost contact metric structure (ϕ, ξ, η,G) on Tr M, with ϕ, ξ, η,and G given respectively by

ϕXH = a1XT , ϕXT = −a2XH +a2r2 g(X, y)yH ,

ξ = 12λr2α

yH , η(XT ) = 0, η(XH) = 2αλg(X, y), G = αG′,

for every tangent vector fields X,Y ∈ T10(M), and every tangent vector y ∈ Tr M,

where α = c1+r2d14r2λ2 and the metric G′ is given by (8), is a contact metric structure,


and it is Sasakian if and only if the base manifold has constant sectional curvature

c =a2

1r2 .

The contact metric manifold (Tr M, ϕ, ξ, η,G) is η−Einstein if the correspondingRicci tensor may be written as

Ric = ρG + ση ⊗ η, (11)

where ρ and σ are smooth real functions.Using the expressions (9) and (10), we wrote the relation (11) with respect to the

generator system δi, ∂Tj i, j=1,n, and taking into account Theorem 4.3 we proved

Theorem 4.4. ([6]) The Sasakian manifold (Tr M, ϕ, ξ, η,G), characterized in Theo-rem 4.3, is η−Einstein if and only if

Case I) c1 = cc2r2, ρ = cc2(2n − 3) − d1

2cc22r2

, σ = λ2 d1n − cc2(n − 2)2cc2

2(cc2 + d1)r2;

CaseII) d1 =cc2r2 − c1(n − 2)

(n − 1)r2 , ρ =n(n − 2)(c1 + cc2r2)

2c1c2(n − 1)r2 ,

σ = λ2 2r2cc1c2n[n(n − 4) + 6] − 2 − n2(n − 2)(c21 + c2c2

2r4)

2c1c2(c1 + cc2r2)2 ,

where c =a2

1r2 is the constant sectional curvature of the base manifold M.

In [7] we studied the condition under which the Sasaki manifold (Tr M, ϕ, ξ, η,G)characterized in Theorem 4.3 is a Sasaki space form, i.e. it has constant holomorphicφ-sectional curvature. The tensor field corresponding to the curvature of the Sasakianspace form (Tr M, ϕ, ξ, η,G) is given by:

K0(X,Y)Z = k+34 [G(Y,Z)X −G(X,Z)Y] + k−1

4 [η(X)η(Z)Y − η(Y)η(Z)X+G(X,Z)η(Y)ξ −G(Y,Z)η(X)ξ + G(Z, ϕY)ϕX −G(Z, ϕX)ϕY + 2G(X, ϕY)ϕZ],

where k is a constant.With respect to the generator system δi, ∂

Tj i, j=1,...,n, we have six components of

K0. We exemplify by the expressions of two of them:

K0(∂Ti , ∂

Tj )∂T

k = TTTT0hki j∂

Th , K0(∂T

i , ∂Tj )δk = TT HH0

hki jδh,

where the M-tensor fields involved as coefficients have the expressions:

TTTT0hki j = k+3

4 α(G′(2)k j δ

hi −G′(2)

ki δhj ),

TT HH0hki j = a2

2k−1

4 αG′(1)kl

1r2

[g0i(δh

jyl − δl

jyh) − g0 j(δh

i yl − δliy

h)]

+ δljδ

hi − δl

iδhj

,

Studying the vanishing conditions of the difference between the curvature tensorfield K of the Riemannian manifold (Tr M,G) and the curvature tensor field K0 cor-responding to the Sasaki manifold (Tr M, ϕ, ξ, η,G), we obtained that some compo-nents of this difference vanish under certain very restrictive conditions, but TTTT h

ki j−


TTTT0hki j takes the form

1r2

[δh

i (g jk − 1r2 g0 jg0k) − δh

j(gik − 1r2 g0ig0k)

], 0,

which is never equal to zero, so we proved the final theorem from [7].

Theorem 4.5. The Sasakian manifold (Tr M, ϕ, ξ, η,G), given by Theorem 4.3, isnever a Sasaki space form.

Acknowledgements. The first author was supported by the Program POSDRU/89/1.5/S/49944,”AL. I. Cuza” University of Iasi, Romania.

References[1] M. T. K. Abbassi, g-natural metrics: New horizons in the geometry of the tangent bundles of

Riemannian manifolds, Note di Matematica, 28(2009), suppl. 1, 6-35.[2] M. T. K. Abbassi, O. Kowalski, On Einstein Riemannian g-Natural Metrics on Unit Tangent

Sphere Bundles, Ann. Global An. Geom, 38, 1(2010), 11-20.[3] E. Boeckx, L. Vanhecke, Unit tangent sphere bundles with constant scalar curvature, Czechoslo-

vak. Math. J., 51(2001), 523-544.[4] Y.D. Chai, S.H. Chun, J. H. Park, K. Sekigawa, Remarks on η-Einstein unit tangent bundles,

Monaths. Math., 155, 1(2008), 31-42.[5] S. L. Druta, V. Oproiu, Tangent sphere bundles of natural diagonal lift type, Balkan J. Geom.

Appl., 15(2010), 53-67.[6] S. L. Druta-Romaniuc, V. Oproiu, Tangent sphere bundles which are η-Einstein, submitted.[7] S. L. Druta-Romaniuc, V. Oproiu, The holomorphic φ-sectional curvature of Sasakian tangent

sphere bundles, submitted.[8] O. Kowalski, M. Sekizawa, Natural transformations of Riemannian metrics on manifolds to met-

rics on tangent bundles - a classification, Bull. Tokyo Gakugei Univ. 40, 4(1988), 1-29.[9] O. Kowalski, M. Sekizawa, On Riemannian geometry of tangent sphere bundles with arbitrary

constant radius, Archivum Mathematicum, 44, 5(2008), 391-401.[10] K.P. Mok, E.M. Patterson, Y.C. Wong, Structure of symmetric tensors of type (0,2) and tensors of

type (1,1) on the tangent bundle, Trans. Amer. Math. Soc., 234(1977), 253-278.[11] M.I. Munteanu, Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bun-

dles of a Riemannian Manifold, Mediterranean Journal of Mathematics, 5, 1(2008), 43-59.[12] V. Oproiu, A Kaehler Einstein structure on the tangent bundle of a space form, Int. J. Math. Math.

Sci., 25, 3(2001), 183-195.[13] V. Oproiu, A locally symmetric Kaehler Einstein structure on the tangent bundle of a space form,

Beitrage Algebra Geom/Contributions to Algebra and Geometry, 40(1999), 363-372.[14] V. Oproiu, Some classes of natural almost Hermitian structures on the tangent bundle, Publ. Math.

Debrecen, 62(2003) 561-576.[15] V. Oproiu, Some new geometric structures on the tangent bundles, Publ. Math. Debrecen, 55,

3-4(1999), 261-281.[16] V. Oproiu, N. Papaghiuc, A Kaehler structure on the nonzero tangent bundle of a space form, Diff.

Geom. Appl. 11(1999), 1-14.[17] V. Oproiu, N. Papaghiuc, General natural Einstein Kahler structures on tangent bundles, Diff.

Geom. Appl., 27(2009), 384-392.


[18] N. Papaghiuc, Another Kaehler structure on the tangen bundle of a space form, DemonstratioMathematica, 31, 4(1998), 855-866.

[19] N. Papaghiuc, On an Einstein structure on the tangent bundle of a space form, Publ. Math. De-brecen, 55(1999), 349-361.

[20] J. H. Park, K. Sekigawa, When are the tangent sphere bundles of a Riemannian manifold η-Einstein?, Ann. Global An. Geom, 36, 3(2009), 275-284.

[21] S. Sasaki,On the differential geometry of the tangent bundle of Riemannian manifolds, TohokuMath. J., 10(1958), 238-354.

[22] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, M. Dekker Inc., New York, 1973.

A LINEARIZATION PRINCIPLE FORTHE STABILITY OF THE CHEMICALEQUILIBRIUM OF A BINARY MIXTURE

ROMAI J., 6, 2(2010), 131–138

Adelina Georgescu1, Lidia Palese2

1Academy of Romanian Scientists, Bucharest, Romania2Dipartimento di Matematica, Universita di Bari, [email protected]

Abstract We consider the stability of a chemical equilibrium of a thermally conducting two com-ponent reactive viscous mixture which is situated in a horizontal layer heated from belowand experiencing a catalyzed chemical reaction at the bottom plate. We provide a methodfor an easy derivation of the nonlinear stability bound, derived explicitly in terms of theinvolved physical parameters. It enables us to derive a linearization principle in a largesense, i.e. to prove that the linear and nonlinear stability bounds are equal.

Keywords: hydrodynamic stability, horizontal thermal convection, energy method.2000 MSC: 76E15, 76E30.

1. INTRODUCTIONThe convective instability and the nonlinear stability of a homogeneous fluid in a

gravitational field heated from below, the classical Benard problem, is a well knowninteresting problem in several fields of fluid mechanics [1], [2], [3], [4], [5].

The stability problem for mechanical equilibria of binary mixture in the case ofcompeting effects (e.g., temperature, concentration, magnetic field) is of a big im-portance in astrophysics, geophysics, oceanography, meteorology. That is why it re-ceived a large attention, mostly in the Oberbeck-Boussinesq approximation [5], [6],[7].

The point of loss of linear stability is usually also a bifurcation point at which theconvective motions set in [4], [8]. However, in certain situations the linear and nonlinear stability bounds are not equal. In particular, subcritical instability may occurexplaining unusual phenomena. Whence a special interest for the study of the rela-tionship between linear and nonlinear stability bounds and, thus, of the linearizationprinciple [9], [10], [11], [12].

A very strong approach to deal with linearization principle (in the sense of the co-incidence of linear and nonlinear stability bounds) in convection problem was settledin [2], where an energy, defined in terms of linear combinations of the concentrationand temperature fields, was used.

131

132 Adelina Georgescu, Lidia Palese

In addition, a parametric differentiation was employed to get the best non linearbound.

For reactive fluids of technological interest, chemical reactions such as the disso-ciation of nitrogen, oxygen or hydrogen gas near the gas-solid interface of a spacevehicle when returning to the earth’s atmosphere [13], [14], [15], [16], can give tem-perature and concentration gradients which influence the transport process and canalter hydrodynamic stabilities.

In [17] a nonlinear stability analysis of the conduction-diffusion Benard problem,with the upper surface stress free and the lower one experiencing a catalyzed chemicalreaction is performed, obtaining a condition of nonlinear stability of the equilibriumsolution, and showing that, in a limiting case, there is the possibility of subcriticalbifurcation in a well-determined region of the parameters’ space.

In the present paper we consider a fluid mixture in a horizontal layer heated frombelow, the bottom plate being catalytic. We evaluate the effects of heterogeneoussurface catalyzed reactions on the hidrodynamic stability of the chemical equilibrium.

The model adopted is that of Bdzil and Frisch.We consider a Newtonian fluid model and, by introducing some linear combina-

tions of temperature and concentration, we derive a system equivalent to the perturba-tion evolution equations, generalizing the Joseph’s method of parametric differentia-tion [18], [19], [20], changing the given problem governing the evolution equationsin order to obtain an optimum stability bound.

The determination of the nonlinear stability bound is reduced to the solution of analgebraic system. With symmetrization arguments for the involved linear operatorswe can determine the nonlinear stability bound in terms of the physical parameters,in the case in which the Prandtl and Schmidt numbers are equal, in a region of theparameters’ space.

In the last section the equality between linear and nonlinear stability bounds wasproved, at least in the class of normal modes perturbations.

2. THE INITIAL/BOUNDARY VALUEPROBLEM FOR PERTURBATION

We consider the mixture composed of the dimer A2 and the monomer A, (A2, A)described by a Newtonian model, in the layer bounded by the surfaces z = 0 andz = 1, being the lower surface catalytic, that is, the interconversion (A2 A) occursvia the surface z = 0 [15]. However, the conditions that must be satisfied at thecatalyzed boundary z = 0, are [15]: ~J · ~k = 0, ~Q · ~k = 0 where ~J is the mass flux, ~kis the unit vector in the vertical upward direction, and ~Q is the heat flux.

The chemical equilibrium S 0 is characterized by the temperature (T ) and degreeof dissociation (fraction of pure monomers) (C) fields [13], [15]:

T (z) = T1 + β(1 − z), C(z) = C1 + γ(1 − z), (1)

A linearization principle for the stability of the chemical equilibrium of a binary mixture 133

where C1 and T1 are the values of C and T at z = 1 and the constants β and γ aregiven in [13], [15].

Let us now perturb S 0 ≡ (~0, P, T , C) up to a cellular motion (we denote with Va periodicity cell) characterized by a velocity ~u = ~0 + ~u, a pressure p = P + p′ atemperature T = T + θ and a concentration C = C + γ fields, where ~u, p′, θ, γ are thecorresponding perturbation fields.

In the Boussinesq approximation the evolution of the perturbation fields is gov-erned by the following equations, written in nondimensional coordinates [17],

∂

∂t~u + (~u · ∇)~u = −∇p′ + ∆~u + (Rθ + Cγ)~k, , (2)

Pr(∂

∂tθ + ~u · ∇θ) = ∆θ − Rw, (t, ~x) ∈ (0,∞) × V (3)

S c(∂

∂tγ + ~u · ∇γ) = ∆γ + Cw, (4)

in a subset of L2(V), namely ,

N =(~u, p, θ, γ) ∈ L2(V) | div~u = 0; ∂u

∂z = ∂v∂z = w = 0 on ∂V2,

~u = 0 on ∂V1, θ = γ = 0 on ∂V2,∂θ∂z = −sγ, ∂γ

∂z = rγ on ∂V1.

(5)

where ~u = (u, v,w), ∂V is the boundary of V , ∂V1 = ∂V∩z = 0, ∂V2 = ∂V∩z = 1.The perturbation fields depend on the time t and space ~x = (x, y, z) and R2, C2, arethe thermal and concentrational numbers of Rayleigh, while Pr and S c are Prandtland Schmidt numbers, respectively. In addition, r, s > 0 are dimensionless surfacereactions numbers.

The basic state S 0 corresponding to the zero solution of the initial-boundary valueproblem for (2)- (5) in the class N is called non linearly stable if a Lyapunov functionE(t), the energy, remains bounded when t → ∞ [2]. The stability or instability of S 0depends on six physical parameters occurring in (2) - (5): Pr, S c, R, C, r and s.

In the following section we apply the Joseph’s generalized method from [18] - [20]to derive the evolution equation for the energy E.

3. THE EVOLUTION EQUATIONS FOR THEPERTURBATION FIELDS

In order to derive the stability boundaries we perform the operations a(3) + (4)g3and b(4) + (3)g2, we obtain

aPr(∂θ∂t

+ ~u · ∇θ) + ag3S c(∂γ∂t

+ ~u · ∇γ) = a(Cg3 − R)w + a∆θ + ag3∆γ, (6)

bS c(∂γ∂t

+ ~u · ∇γ) + bg2Pr(∂θ∂t

+ ~u · ∇θ) = b(C − g2R)w + bg2∆θ + b∆γ, (7)


where a, b, g2, g3 are some positive constants.By multiplying (2) by ~u, (6) by θ , and (7) by γ, integrating them over V in N and

adding the results, if we require that the coefficients of θ∂γ

∂tand γ

∂θ

∂tbe equal, i.e.

bg2 = τag3, (8)

where τ = S c/Pr, we obtain

12

ddt< |~u|2 + aPrθ

2 + bS cγ2 + 2ag3S cθγ >= − < |∇~u|2 + a|∇θ|2 + b|∇γ|2+

+(ag3 + bg2)|∇θ · ∇γ| > +aR(1a

+ αg3 − 1) < θw > +bR(α

b+ α − g2) < γw > +

+

∫

∂V1

[θγ(s − rg3)a + γ2(sg2 − r)b

]dxdy, (9)

where < f >=∫

V f dxdydz, α = C/R.By choosing

g3 =sr, g2 =

rs, (10)

in the case τ = 1, if we require that

ag3 + bg2 = 2√

ab, (11)

then the terms in θ and γ in < · > from the right-hand side of (9) form a perfectsquare. From (11) it follows that

b = a( sr

)2. (12)

Let us introduce

φ1 =√

aθ +√

bγ ≡√

ar

(rθ + sγ). (13)

Moreover, if we require that the coefficients of < θw > and < γw > satisfy the relation

1√a

+√

a(αsr− 1) =

sr

( α√a

(rs)2 + (α − r

s)√

a), (14)

it followsα =

sr. (15)

Then (9) is in the form of the evolution equation for energy E(t)

dEdt

= − < |∇~u|2 + |∇φ1|2 > +R( 1√

a+√

a(αsr− 1)

)< φ1w >, (16)

whereE(t) =< |~u|2 + Prφ

21 > /2. (17)


4. THE NONLINEAR STABILITY BOUND ANDITS EQUALITY WITH THE LINEARSTABILITY BOUND

In order to find the nonlinear stability bound we follow the Joseph’s generalizedmethod of parametric differentiation [18]-[20]. Denoting

2A = R| 1√a

+√

a(αsr− 1)| (18)

relation (16) impliesdEdt≤ −ξ2

(1 − A/

√Ra∗

)E(t) (19)

where [4], [5], [21], [22], [23]

ξ2 = minu,φ1

2 < |∇u|2 + |∇φ1|2 >< |u|2 + Prφ

21 >

,1√Ra∗

= maxu,φ1

2 < φ1w >

< |∇u|2 + |∇φ1|2 >. (20)

IfA <

√Ra∗, (21)

the basic state S 0 is (nonlinearly) stable because (16) implies

dEdt≤ −ξ2(1 − A/

√Ra∗)E. (22)

Relations (18) and (21) show that S 0 is stable if

R < RE ≡ 2√

Ra∗/| 1√

a+√

a(αsr− 1)|. (23)

Therefore, RE is a non linear stability bound.But RE is maximum if 1/

√a +√

a(αs/r−1) is minimum. Thus we shall determinea with the aid of the Joseph’s parametric differentiation method [2], i.e. imposing

dda| 1√

a+√

a(αsr− 1)| = 0. (24)

If srα > 1, the solution

a =1

srα − 1

(25)

of (24) gives, in terms of the physical quantities, the non linear stability bound [23]

RE =√

Ra∗(√

(sr

)2 − 1)−1, (26)

whence the following


Theorem 4.1. For physical parameters τ = 1, R, C = αR, s/r = α, (s/r)2 > 1, thezero solution of (2)-(5), corresponding to the basic conduction state, is non linearlyasymptotically stable if R < RE , where RE is given by (26), or, equivalently, if

C2 − R2 < Ra∗

where Ra∗ is given by (20).

Let us consider the steady problem, obtained by linearizing (2) - (4) about thetrivial solution,

∆u + (Rθ + Cγ)k = ∇p, (27)

−Ru · k + ∆θ = 0, (28)

Cu · k + ∆γ = 0. (29)

Elimination of u · k between (28) multiplied by r and (29) multiplied by s implies

∆φ1 + R(√

(sr

)2 − 1)w = 0, (30)

moreover (27), taking into account (15), is equivalent to

∆u + R(√

(sr

)2 − 1)φ1k = ∇p, (31)

Then from (31)-(30) it follows

∆u + µφ1k = ∇p,∆φ1 + µw = 0, (32)

where µ = R(√

( sr )2 − 1

).

We proved that, at least in the class of normal mode perturbations, the smallesteigenvalues of the problems (27)-(29) and (32) are the same.

The system (32) represents the Euler-Lagrange equations for the minimum of thefunctional < |∇u|2 + |∇φ1|2 > /(2 < φ1w >), namely this minimum is µ.

Taking into account (20)2 it follows that µ =√

Ra∗.As a consequence the chemical equilibrium has the linear stability bound RL which

satisfies the relation

µ2 ≡ R2L

((sr

)2 − 1), (33)

this implies, taking into account (26), RL = RE , whence the following result:

Theorem 4.2. For physical parameters τ = 1, R, C = αR, s/r = α, (s/r)2 > 1, thezero solution of (2)-(5), corresponding to the basic conduction state is non linearlyasymptotically stable if

R < RE


and, in the class of normal mode perturbations,

RL = RE ,

whence the linear and non linear stability bounds coincide.

5. CONCLUSIONSWe applied a generalized Joseph’s approach to some convection problems.In the Joseph’s approach it is used a linear combination of temperature and con-

centration to define the perturbation energy.Our idea was to incorporate the Joseph’s relation for the rate of change of the prod-

uct of temperature and concentration directly into the evolution equations obtainingequations with better symmetries. In this way, the term in velocity from the tempera-ture equation contributed to the symmetric part of the obtained equations. Otherwise,if the initial evolution equations were used this contribution is null and, correspond-ingly, the stability criterion, weaker. This is due to the sign minus in the quoted termin the initial equation for temperature.

In [23] we derive the evolution equation for the perturbation energy, and, in thesame region of the parameters’ space, we determine a nonlinear stability bound interms of the involved physical parameters. This limit is the same as in the presentpaper, because the evolution equation for the perturbation energy from [23] can bederived considering some more general evolution equations with better symmetries,which incorporate the given equations, such as (6), (7).

Indeed in this paper the given problem governing the perturbation evolution waschanged in order to obtain an optimum energy inequation. The initial equations (3)-(4) were replaced by (6)-(7) in which the initial evolution equations were presentthrough the terms in g2 and g3.

These terms drastically changed the linear part of the initial equations and allowsus a much more advantageous symmetrization and an equivalent formulation of thelinear stability problem that was nothing else but the Euler system associated to themaximum problem of the nonlinear stability, whence a linearization principle in thesense of the coincidence of both linear and nonlinear stability bounds.

References[1] S.Chandrasekhar, Hydrodynamic and Hydromagnetic stability, Oxford, Clarendon Press, 1968.[2] D.D. Joseph, Global Stability of the Conduction-Diffusion Solution, Arch. Rat. Mech. Anal., 36,

4(1970), 285-292.[3] D.D. Joseph, Stability of fluid motions I-II, Springer, Berlin, 1976.[4] A. Georgescu, Hydrodynamic stability theory, Kluwer, Dordrecht, 1985.[5] B. Straughan, The Energy Method, Stability and Nonlinear Convection, Springer, New York,

2003.[6] S. Rionero G. Mulone, On the nonlinear stability of a thermodiffusive fluid mixture in a mixed

problem, J. Math. Anal. Appl., 124, (1987), 165-188.


[7] S. Rionero G. Mulone, A nonlinear stability analysis of the magnetic Be nard problem throughthe Lyapunov direct method,, Arch. Rational Mech. Anal., 103, 4 (1988), 347-368.

[8] A. Georgescu, I. Oprea, Bifurcation theory from the applications point of view, MonografiiMatematice. Universitatea de vest din Timisoara, 51, 1994.

[9] G. Prodi, Teoremi di tipo locale per il sistema di Navier-Stokes e stabilita delle soluzionistazionarie, REND. SEM. MAT. Padova, 32, (1962), 374-397.

[10] V. I. Yudovich, On the stability of stationary flow of viscous incompressible fluids, Dokl. Akad.Nauk. SSSR, 161, 5 (1965), 1037-1040. (Russian) (translated as Soviet Physics Dokl., 10, 4(1965), 293-295.)

[11] D. H. Sattinger, The mathematical problem of hydrodynamic stability, J. Math. Mech., 19, 9(1970), 197-217.

[12] O. A. Ladyzhenskaya, V. A. Solonnikov, The linearization principle and invariant manifolds forproblems of magnetohydrodynamics, J. SOV. MATH., 4, 8 (1977) 384-422.

[13] J. Bdzil, H.L. Frisch, Chemical Instabilities. II. Chemical Surface Reactions and HydrodynamicInstability Phys. Fluids, 14, 3 (1971), 475-481.

[14] J. Bdzil, H.L. Frisch, Chemical Instabilities. IV. Nonisothermal Chemical Surface Reactions andHydrodynamic Instability, Phys. Fluids, 14, 6 (1971), 1077-1086.

[15] C. L. Mc Taggart, B. Straughan, Chemical Surface Reactions and Nonlinear Stability by theMethod of Energy, SIAM J. Math. Anal., 17, 2 (1986), 342-351.

[16] D.E. Loper, P.H. Roberts, On The Motion of an Airon-Alloy Core Containing a Slurry, Geophys.Astrophys. Fluid Dynamics, 9, 1978, 289-331.

[17] B. Straughan, Nonlinear Convective Stability by the Method of Energy: Recents Results, Attidelle Giornate di Lavoro su Onde e Stabilita nei mezzi continui, Cosenza (Italy), 1983, 323-338.

[18] A. Georgescu, L. Palese, Extension of a Joseph’s criterion to the non linear stability of mechani-cal equilibria in the presence of thermodiffusive conductivity, Theor. Comp. Fl. Mech., 8, 1996,403-413.

[19] A. Georgescu, L. Palese, A. Redaelli, On a New Method inHydrodynamic Stability Theory, Math.Sci. Res. Hot Line, 4, 7 (2000), 1-16.

[20] A. Georgescu, L. Palese, A. Redaelli, The Complete Form for the Joseph Extended Criterion,Annali Universita di Ferrera , Sez. VII, Sc. Mat. 48, (2001), 9-22.

[21] A. Georgescu, L. Palese, Stability Criteria for Fluid Flows, Advances in Math. for Appl. Sc.,81, World Scient. Singapore, 2010.

[22] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon andBreach, New York, 1969.

[23] A. Georgescu, L. Palese, On the nonlinear stability of a binary mixture with chemical surfacereactions, Preprint.

ASYMPTOTIC WAVES AS SOLUTIONS OFNONLINEAR PDES DEDUCED BY DOUBLE-SCALE METHOD IN VISCOANELASTIC MEDIAWITH MEMORY

ROMAI J., 6, 2(2010), 139–155

Adelina Georgescu1, Liliana Restuccia2

1Academy of Romanian Scientists, Bucharest, Romania2Department of Mathematics, University of Messina, Italy,[email protected]

Abstract In this paper the double scale method is applied to investigate non-linear dissipativewaves in isotropic viscoanelastic media with shape and volumetric memory, that werestudied by one of the authors(L. R.) in more classical way. The system of PDEs describing these media include termscontaining second order derivatives multiplied by a small parameter. The solutions arelooked for in the form of asymptotic expansions with respect to an asymptotic sequenceof powers of the small parameter, related to the thickness of internal layers across whichthe solutions or/and some of their derivatives vary steeply. An one-dimensional applica-tion is worked out containing original results.

Keywords: Partial differential equations, double-scale method, non-linear dissipative waves, asymp-totic methods, rheological media.2000 MSC: 34E05, 34E10, 34E13, 73B20, 73B99.

1. INTRODUCTIONIn previous papers we sketched out the general use of the double-scale method

to nonlinear hyperbolic partial differential equations (PDEs) in order to study theasymptotic waves [1] and as examples the models governing the motion of anelasticmedia (Maxwell media) [2] and viscoanelastic media (Jeffreys media) [3], both ofthem without shape and volumetric memory, were studied. In this paper the dou-ble scale method is applied to investigate asymptotic waves in viscoanelastic mediawith shape and bulk memory, in which a viscous flow phenomenon occurs, that werestudied by one of the authors (L. R.) in a more classical way [4].

Since the closed-form solutions of nonlinear PDEs are rare, usually the solutionis looked for in the form of an asymptotic series of powers of a small parameter,which is related to the thickness of internal layers, across which the solution or/andsome of its derivatives varies steeply. This asymptotic expansion is called asymptoticsolution of the system of PDEs. Correspondingly, a new independent (fast) variableξ is defined that models just this fast variation across the internal layers situated near

139

140 Adelina Georgescu, Liliana Restuccia

some surfaces S and the slow variation along S. The asymptotic method involving ξis known as the double-scale method [1].

The multiple-scale method, and, in particular, the double-scale approach, is ap-propriate to phenomena which possess qualitatively distinct aspects at various scales.For instance, at some well-determined times or space coordinates, the characteristicsof the motion vary steeply, while at larger scale the characteristics are slow and de-scribe another type of motion. Furthermore, the scales are defined by some smallparameters.

The mathematical aspects involved into the study of asymptotic waves belong tosingular perturbation theory, namely the double-scale method [9] - [16]. This ap-proach was initiated in the papers of Poincare, Krylov and Bogolinskow [9], [10].The interest in nonlinear waves was manifest as early as the years ’50 and ’60 of thelast century. Subsequently, a lot of applications to various equations from elasticity,fluid mechanics and thermodynamics were worked out [17]- [8].

The use of nonlinear asymptotic waves received a particular attention soon afterthe seminal work [6], especially from the thermodynamicists. In the contest of rheo-logical media studies were carried out, some of them belonging to one of the authors(L. R.) (see [4], [29] and [30].) In this paper previous treatments performed in [4],using a method of G. Boillat [6], generalized by D. Fusco [31], are seen in the modernlight of double-scale method, as it is presented in a monograph on asymptotic treat-ments of the other author A. G. [16]. In Section 2 we give a physical interpretation ofthe new (fast) variable ξ, related to the surface across which the solution or/and someof its derivatives vary steeply. In Section 3 the equations governing the motion ofviscoanelastic media with memory in the framework of classical irreversible thermo-dynamics (TIP) with internal variables [32]-[38] are introduced and the mechanicalrelaxation equation valid for these media is described. In Section 4 we present thedouble-scale method, obtaining the first and second asymptotic approximation. Theapplication of the double-scale method is standard: the solution U(xα, ξ) is written asan asymptotic power series of the small parameter, the coefficients Ui, (i = 1, 2, ...)being functions of xα and ξ. Introducing the series in the PDEs, after the matchingof the series in the right and left-hand sides, the equations for Ui, with i ≥ 1 are ob-tained. They are called equations of order i and Ui are the asymptotic approximationsof order i. A special meaning is given for U0. It is taken as the initial, unperturbedstate, where no small parameter occurs. In Sections 5 and 6 the derivation of thefirst approximation of the wave front and of the first approximation of the solutionU is discussed in details. In Section 7 an one-dimensional application, containingoriginal results and revealing the influence of the internal variable on the mechanicalrelaxation of the medium, is considered.

Asymptotic waves as solutions of nonlinear PDEs ... 141

2. APPLICATION OF DOUBLE-SCALE METHODTO NONLINEAR PDES

We deal with those smooth solutions U(xα), called asymptotic waves, which evolveas progressive waves, i.e. there exists a family of hypersurfaces S (defined by theequation ϕ(xα) = 0) moving in the Euclidean space E3+1, consisting of points ofcoordinates xα (α = 0,1,2,3) or, equivalently, of the time t = x0 and the space coordi-nates xi (i=1,2,3),

ϕ(t, xi) = ξ = const, (1)

such that the solutions U or/and some of their derivatives vary steeply across S, whilealong S their variation is slow [1]. This means that around S there exist (asymptotic)internal layers, such that the order of magnitude (i.e. the scale) of the solutions Uor/and some of their derivatives inside these layers and far away from them differvery much. Therefore, it is natural to introduce a new independent variable ξ, relatedto the hypersurfaces S,

ξ = ωξ = ωϕ(t, xi), (2)

where ξ is asymptotically fixed , i.e. ξ = Ord(1) as ω−1 → 0. ω is a large parameter(ω 1) and usually its dimension is a frequency. Asymptotically, this means thatboth ”slow” (old) and ”fast” (new) variables are necessary to characterize completelythe behaviour of the solution as some parameter tends to zero [16]. Each variable(slow or fast) has a characteristic scale. That is why, an approximate solution islooked for as depending on the old as well as on the new variables, where the newvariables are thought of as independent of the old variables. When no interior layersare present, all characteristic quantities have the same scale. If only one new variableof a different scale is imposed, we say that the problem has a double scale. In thiscase, the appropriate method, used to derive solutions of asymptotic approximationis called the double-scale method. Usually, the new variable is a space variable xi

or the time t multiplied by a function of the small parameter. Then, we have U =

U(xα, ξ). Furthermore, taking into account that U is sufficiently smooth, hence it hassufficiently many bounded derivatives, it follows that, except for the terms containingω, all other terms are asymptotically fixed and the computation can proceed formally.

In this way, if xα = xα(s) are the parametric equations of a curve C in E3+1, wehave

dUds

= ω∂U∂ξ

∂ϕ

∂s+∂U∂xα

dxα

ds,

where the dummy index convention is used. This relation shows that, indeed, alongC, U does not vary too much if C belongs to the hypersurface S (in this case dϕ

ds = 0)but it has a large variation if C is not situated on S. For these reasons, ξ is referred toas the fast variable. Once introduced the fast variable ξ, in order to apply the double-scale method we must define the equations to which it applies. Thus, let E3+1 be anEuclidean space, let P ∈ E3+1 be a current point, let U = U(P) be the unknown vector


function solution of a system of PDEs written in the following matrix form

Aα(U)Uα + ω−1[Hk ∂

2U∂t∂xk + Hik ∂2U

∂xi∂xk

]= B(U), (3)

(α = 0, 1, 2, 3; i, k = 1, 2, 3),where x0 = t (time), x1, x2, x3 are the space coordinates, U is the vector of the un-known functions (which depend on xα), Uα = ∂U

∂xα , Aα, Hk, Hik are appropriatematrices, and

Aα(U)Uα = B(U), (4)is the associated system of nonlinear hyperbolic PDEs.

In [4] it was shown that the motion of isotropic viscoanelastic media with memo-ry, in the isothermal case, is described by a system of nonlinear PDEs having thematrix form (3) and including terms containing second order derivatives, multipliedby a very small parameter, that have a balancing effect on the non-linear steepeningof waves. In [4] non-linear dissipative waves were worked out by one of the author(L. R.) using a asymptotic method developed by G. Boillat [6] and generalized by D.Fusco [31]. In this paper we study the asymptotic waves deduced in [4] from the pointof view of double scale-method. Following A. Jeffrey in [24], the solution hypersur-faces of systems of PDEs are referred to as waves because they may be interpreted asrepresenting propagating wavefronts. When physical problems are associated withsuch interpretation the solution on the side of the wavefront towards which propa-gation takes place may then be regarded as being the undisturbed solution ahead ofthe wavefront, whilst the solution on the other side may be regarded as a propagatingdisturbance wave which is entering a region occupied by the undisturbed solution.

We conclude this Section reminding that the wavefront ϕ is an unknown function.In order to determine it, we recall its equation is ϕ(t, x1, x2, x3) = 0. This implies thatalong the wavefront we have dϕ

dt = 0, implying ∂ϕ∂t + v · gradϕ = 0 or, equivalently,

∂ϕ∂t

|gradϕ| + v · gradϕ|gradϕ| = 0. Obviously,

gradϕ|gradϕ| = n, (5)

such that the previous equality reads∂ϕ∂t

|gradϕ| + v · n = 0. (6)

Introducing the notation

λ = −∂ϕ∂t

|gradϕ| , (7)

we haveλ(U, n) = v · n, (8)

where λ is called the velocity normal to the progressive wave.


3. GOVERNING EQUATIONS FORVISCOANELASTIC MEDIA WITH MEMORY

In [32] a general thermodynamic theory for mechanical phenomena in continuousmedia was developed by G. Kluitenberg, using the methods of classical irreversiblethermodynamics with internal variables [33]-[36]. It was assumed that several micro-scopic phenomena occur which give rise to inelastic strains due to effects of latticedefects, like slip, dislocations, atom vacancies, disclinations. From the theory it fol-lows that several types of macroscopic stress fields may occur in a medium: a stressfield τ(eq)

αβ (α, β = 1, 2, 3) of thermoelastic nature, a stress field τ(vi)αβ analogous to the

viscous stress in ordinary fluids and n stress fields τ(k)(m)αβ(k = 1, 2, ..., n), in arbitrary

number, connected with microscopic stress fields surrounding imperfections in themedium. The stress field τ(eq)

αβ + τ(vi)αβ is the mechanical stress field which occurs in

the equations of motion and in the first law of thermodynamics. The stress fieldsτ

(eq)αβ +τ(k)

(m)αβ play the role of thermodynamic affinities in the phenomenological equa-tions which are generalizations of Levy’s law. Temperature effects were fully takeninto account. Explicit mechanical relaxation equations, with constant phenomeno-logical coefficients, were deduced, linearizing equations of state. For distortionalphenomena in isotropic media the following linear relation among the deviators ofthe mechanical stress tensor, the first n derivatives with respect to time of this tensor,the tensor of total strain (the sum of the elastic and inelastic strains) and the first n+1derivatives with respect to time of this last tensor was deduced, where n is the numberof phenomena that give rise to inelastic deformations

R(τ)(d)0τik +

n−1∑

m=1

R(τ)(d)m

dm

dtm τik +dn

dtn τik = R(ε)(d)0εik +

n+1∑

m=1

R(ε)(d)m

dm

dtm εik (i, k = 1, 2, 3). (9)

In (9) ddt is the material time derivative, the quantities R(τ)

(d)m (m = 0, 1, ..., n − 1),

R(ε)(d)m (m = 0, 1, ..., n + 1) are algebraic functions of the coefficients occurring in the

phenomenological equations and in the equations of state. In [37] and [38] it wasshown that the relation between τ, the scalar part of the stress tensor, and ε, the scalarpart of the strain tensor, is given by

R(τ)(v)0τ

′ +n−1∑

m=1

R(τ)(v)m

dmτ′

dtm +dnτ′

dtn = R(ε)(v)0ε +

n+1∑

m=1

R(ε)(v)m

dmε

dtm

+R(T )(v)0(T − T 0) +

n∑

m=1

R(T )(v)m

dm

dtm T, (10)

τ′ = τ − τ0,

where τ0 and T 0 are the scalar part of the stress tensor and the temperature, respec-tively, of the medium in a state of thermodynamic equilibrium. This equilibrium


state plays the role of a reference state. The quantities R(τ)(v)m (m = 0, 1, ..., n − 1),

R(ε)(v)m (m = 0, 1, ..., n + 1) and R(T )

(v)m (m = 0, 1, ..., n) are algebraic functions of the co-efficients occurring in the phenomenological equations and in the equations of state.The well-known Burgers equation is a special case of this relation for n = 2, i.e.when only two internal variables of mechanical origin are taken into consideration.The rheological relations for ordinary viscous fluids, for thermoelastic media and forMaxwell, Kelvin (Voigt), Jeffreys, Poyting-Thomson, Prandtl-Reuss, Bingham, SaintVenant and Hooke media are special cases of these more general mentioned aboverelations too (see also [37]- [41]). The total strain is given by elastic and inelasticdeformations εαβ = ε(el)

αβ + ε(in)αβ (α, β = 1, 2, 3). The inelastic strain is due to lattice

defects and related phenomena: ε(in)αβ =

∑nk=1 ε

(k)αβ, where ε(k)

αβ is the k-th contribution tothe inelastic strain of the k-th microscopic phenomenon and it is called partial inelas-tic strain tensor. In this theory n − 1 partial inelastic strain tensors ε(k)

αβ are introducedas internal variables in the thermodynamic state vector. Furthermore, it is assumedthat the gradient of the displacement field is small. This implies that the deformationsare supposed to be small from a kinematical (or geometrical point of view). Howeverthe translations and the velocity of the medium may be large [36]. Then, the straintensor εik is assumed to be small, i.e. εik = 1

2

(∂∂xk ui + ∂

∂xi uk)

(i, k = 1, 2, 3), where ui

is the i-th component of the displacement field u and xi is the i-th component of theposition vector x in Eulerian coordinates in a Cartesian reference frame. It shouldbe emphasized, however, that the same physical ideas which are developed in thistheory can also be reformulated for the case where the deformations are large from akinematical point of view [36].

Then, assume that only one microscopic phenomenon gives rise to inelastic strain.In the isothermal case, the mechanical relaxation equations describing the behaviourof isotropic viscoanelastic media of order one (n = 1), i.e. when only one inter-nal variable of mechanical origin is taken into consideration, with shape and bulkmemory, can be written in the following form [4]

R(τ)(d)0Pik +

ddt

Pik + R(ε)(d)0εik + R(ε)

(d)1ddtεik + R(ε)

(d)2d2

dt2 εik = 0 (i, k = 1, 2, 3), (11)

R(τ)(v)0P′ +

ddt

P′ + R(ε)(v)0ε + R(ε)

(v)1ddtε + R(ε)

(v)2d2

dt2 ε = 0, (12)

where Pik and εik are the deviators of the mechanical pressure tensor Pik and of thestrain tensor εik, respectively, and dεik

dt = 12

(∂vi∂xk

+∂vk∂xi

). We define Pik in terms of the

symmetric Cauchy tensor Pik = −τik, P′ = P − P0 = −(τ − τ0) and, hence, we have

Pik = Pik − 13

Pssδik, P =13

Pss, Pss = trP,

Pik = Pik + Pδik, Pss = 0,


where the scalar part P of the tensor Pik is the hydrostatic pressure. Analogous defi-nitions are valid for εik and ε.

Moreover, in eqns. (11) and (12) the coefficients satisfy the following relations

R(τ)(d)0 = a(1,1)η(1,1)

s ≥ 0, (13)

R(ε)(d)0 = a(0,0)(a(1,1) − a(0,0))η(1,1)

s ≥ 0, (14)

R(ε)(d)1 =

a(0,0)

(1 + 2η(0,1)

s

)+ a(1,1)

[η(0,0)

s η(1,1)s +

(η(0,1)

s

)2]≥ 0, (15)

R(ε)(d)2 = η(0,0)

s ≥ 0, R(τ)(v)0 = b(1,1)η(1,1)

(v) ≥ 0, (16)

R(ε)(v)0 = b(0,0)(b(1,1) − b(0,0))η(1,1)

(v) ≥ 0, R(ε)(v)2 = η(0,0)

v ≥ 0, (17)

R(ε)(v)1 =

b(0,0)

(1 + 2η(0,1)

v

)+ b(1,1)

[η(0,0)

v η(1,1)v +

(η(0,1)

v

)2]≥ 0, (18)

R(ε)(d)1 − R(τ)

(d)0R(ε)(d)2 ≥ 0, R(ε)

(d)1R(τ)(d)0 − R(ε)

(d)0 ≥ 0, R(ε)(v)1 − R(τ)

(v)0R(ε)(v)2 ≥ 0

and R(ε)(v)1R(τ)

(v)0 − R(ε)(v)0 ≥ 0.

In eqns. (13) - (18) a(0,0), a(1,1), b(0,0) and b(1,1) are scalar constants which occurin the equations of state, while the coefficients η(0,0)

s , η(0,1)s , η(1,1)

s , η(0,0)v , η(0,1)

v and η(1,1)v

are called phenomenological coefficients.Indicating by vi the i-th component of the velocity field v, we have

vi =dui

dt=∂ui

∂t+ v j

∂ui

∂x j , (19)

where the definition of material derivative is taken into consideration. The balanceequations for the mass density and momentum in the case of isotropic viscoanelasticmedia with shape an bulk memory read

∂ρ

∂t+

∂

∂xi(ρvi) = 0 (i = 1, 2, 3), (20)

ρ

(∂

∂tvi + vk

∂

∂xk vi

)+

∂

∂xk Pik +∂

∂xiP = 0. (21)

4. EQUATIONS OF FIRST AND SECONDASYMPTOTIC APPROXIMATION

Let us now apply the double-scale method to the system (3), written in matrixform, for viscoanelastic media with memory. To this aim all the quantities, depen-ding on xα, are considered as depending on xα and ξ. Consequently, the derivativeswith respect to xα must be replaced by ∂

∂xα = ∂∂xα + ∂

∂ξ∂ξ∂xα = ∂

∂xα + ω ∂∂ξ

∂ϕ∂xα . Then, let


us choose the solution of the equations as an asymptotic series of powers of the smallparameter, ε, namely with respect to the asymptotic sequence

1, εa+1, εa+2, ...,

or

1, ε1p , ε

2p , ...,

, as ε → 0. In [4], [29] and [30] it is considered p = 1, and ε = ω−1,

such that U(xα, ξ) is written as an asymptotic series with respect to the asymptoticsequence 1, ω−1, ω−2, ..., as ω−1 → 0 , the Ui

(i = 1, 2, ...) being functions of xα and ξ,

U(xα, ξ) ∼ U0(xα, ξ) + ω−1U1(xα, ξ) + O(ω−2), as ω−1 → 0, (22)

where U0(xα, ξ) is a known solution of

Aα(U0)Uα(U0) = B(U0) (23)

and it is taken constant and as the initial, unperturbed state, where no small parame-ters occur [31]. We recall that ξ = ωϕ(xα), ω 1 is a real parameter and ϕ(xα) is theunknown wavefront which is to be determined.

Then, the derivative Uα = ∂U∂xα has the following form

∂U∂xα ∼ ω−1

(∂U1

∂xα + ω∂U1

∂ξ∂ϕ∂xα

)+ ω−1 ∂2U2

∂ξ∂ϕ∂xα + O(ω−2), as ω−1 → 0

and the following asymptotic expansions are deduced for Aα, Hk, Hik and B:

Aα(U) ∼ Aα(U0) +1ω∇Aα(U0)U1 + O

(1ω2

), as ω−1 → 0, (24)

Hk(U) ∼ Hk(U0) +1ω∇Hk(U0)U1 + O

(1ω2

), as ω−1 → 0 (k = 1, 2, 3), (25)

Hik(U) ∼ Hik(U0) +1ω∇Hik(U0)U1 + O

(1ω2

), as ω−1 → 0 (i, k = 1, 2, 3), (26)

B(U) ∼ B(U0) +1ω∇B(U0)U1 + O

(1ω2

), as ω−1 → 0, (27)

where ∇ = ∂∂U .

Then, we introduce the asymptotic expansions (24) - (27) into the equations ofthe system (3) in order to obtain the sets of equations of various order of asymptoticapproximation. Each such set has as a solution one approximation Ui(i = 1, 2, ...) ofU. For the first and second asymptotic approximation we have

(Aα)0Φα∂U1

∂ξ= 0 (α = 0, 1, 2, 3), (28)

(Aα)0(Φα

∂U2

∂ξ

)= −

[(Aα)0

∂U1

∂xα + (∇Aα)0U1(Φα

∂U1

∂ξ

)

+ (Hk)0Φ0Φk∂2U1

∂ξ2 + (Hik)0ΦiΦk∂2U1

∂ξ2 − (∇B)0U1],

(29)


where Φα =∂ϕ∂xα , Φk =

∂ϕ

∂xk (k = 1, 2, 3) and the symbol (...)0 indicates that thequantities are computed in U0. Equation (28) is linear in U1, while (29) is affine inU2.

Now, by introducing in eq. (28) the quantities (5) and (7), we obtain

(A0n − λI)∂U1

∂ξ= 0, (30)

where (An)0 = A0n and An(U) = Aini. In the case where the eigenvalues are real andthe eigenvectors of the matrix An are linearly independent, the system of PDEs (4)is hyperbolic (see [1] for the definition of hyperbolicity). Furthermore, in [6] it wasshown that only for the waves propagating with a velocity λ, that does not satisfy theLax - Boillat exceptionality condition, i.e. ∇λ · r , 0 (with r the right eigenvector of(An)0 corresponding to the eigenvalue λ), our results are valid. Eq. (30) shows that∂U1

∂ξ can be taken as equal to the right-eigenvector r of A0n, corresponding to someeigenvalue λ. By integration, it follows that U1(xα, ξ) has the form

U1(xα, ξ) = u(xα, ξ)r(U0,n) + ν1(xα), (31)

where u is a scalar function to be determined and ν1 is an arbitrary vector of integra-tion which can be taken as zero, without loss of generality. Consequently, in order todetermine U1 we must determine u, which gives rise to a phenomenon of distortion ofthe signals and governs the first-order perturbation obeying a non-linear partial differ-ential equation, that will be studied in Section 6. Of course, equations of asymptoticapproximation of higher order can be written and they are affine, but their solution isvery difficult.

5. FIRST APPROXIMATION OF WAVE FRONTIn this section we show how the wave front ϕ(t, x1, x2, x3) = 0 can be determined

(see [4]). Following the general theory [5], [6] we introduce the quantity

Ψ(U,Φα) = ϕt + |gradϕ|λ(U,n). (32)

The characteristic equations for (32) are

dxα

dσ=

∂Ψ

∂Φα,

dΦα

dσ= − ∂Ψ

∂xα(α = 0, 1, 2, 3), (33)

where σ is the time along the rays.The i−th component of the radial velocity Λ is defined by

Λi(U,n) ≡ dxi

dσ=∂Ψ

∂φi= λni +

∂λ

∂ni−

(n · ∂λ

∂n

)ni = λni + vi − (nkvk)ni (34)


(i = 1, 2, 3).

Hence,Λ(U,n) = v − (vn − λ)n. (35)

Since we are considering the propagation into an uniform unperturbed state, it isknown that the wave front ϕ satisfies the partial differential equation

Ψ(U0,Φα) = ϕt + |gradϕ|λ(U0,n0) = Ψ0 = 0, (36)

where n0 is the constant value of n and represents the normal vector at the point (xi)0

defined by

n0 =

(gradϕ|gradϕ|

)

t=0=

grad0ϕ0

|grad0ϕ0| , with (grad0)i ≡ ∂

∂(xi)0 (i = 1, 2, 3). (37)

The characteristic equations for (36) are

dxα

dσ=∂Ψ0

∂Φα(α = 0, 1, 2, 3), (38)

dΦα

dσ= −∂Ψ0

∂xα(α = 0, 1, 2, 3), (39)

where σ is the time along the rays. By integration of eq.(38) one obtains

x0 = t = σ, xi = (xi)0 + Λ0i (U0,n0)t, with (xi)0 = (xi)t=0 (i = 1, 2, 3). (40)

If we denote by ϕ0 the given initial surface ϕ0[(xi)0

]= (ϕ)t=0 and if the Jacobian of

the transformation x→ x|t=0

J = det

∣∣∣∣∣∣∣δik +∂Λ0

k

∂(xi)0 t

∣∣∣∣∣∣∣ , 0 (i, k = 1, 2, 3)

is non vanishing x0i can be deduced from (40) and ϕ takes the form

ϕ(t, xi) = ϕ0(xi − Λ0

i (U0,n0)t). (41)

6. FIRST APPROXIMATION OF UIn [4] it is shown that, by utilizing (28) and (29) (see [6] and [31]), the following

equation for u(xα, ξ) can be obtained:

∂u∂σ

+ (∇Ψ · r)0 u∂u∂ξ

+1√J

∂√

J∂σ

u + µ0 ∂2u∂ξ2 = ν0u, (42)

where(∇Ψ · r)0 = (|gradϕ|)0 (∇λ · r)0 , (43)


µ0 =

[l ·

(Hk ∂ϕ

∂t∂ϕ

∂xk + Hik ∂ϕ∂xi

∂ϕ

∂xk

)r]0

(l · r)0, (44)

ν0 =(l · ∇B r)0

(l · r)0, (45)

with l the left eigenvector and r the right eigenvector corresponding to the eigenvalueλ, that does not satisfy the Lax-Boillat condition. By using the transformation ofvariables (see [31])

u =z√J

ew, κ =

∫ σ

0

ew√

J(∇Ψ · r)0 dσ, with w =

∫ σ

0ν0dσ, (46)

equation (42) can be reduced to an equation of the type

∂z∂κ

+ z∂z∂ξ

+ µ0 ∂2z∂ξ2 = 0, where µ0 =

µ0√

Je−w

(∇Ψ · r)0, (47)

which is similar to Burger’s equation and is valid along the characteristic rays. Equa-tion (47)1 can be reduced to the semilinear heat equation [42]

∂h∂κ

= µ0 ∂2h∂ξ− hlog

hµ0

dµ0

dκ, (48)

that has been extensively studied and for which the solution is known, using thefollowing Hopf transformation

z(ξ, κ) = µ0∂

∂ξlogh(ξ, κ). (49)

7. ONE-DIMENSIONAL CASEIn this Section the one dimensional case is studied, containing original results.

Consider the system of equations (11), (12), (19), (20) and (21). Assume that v2 =

v3 = 0, u2 = u3 = 0, x2 = x3 = 0 and that the involved physical quantities dependonly on x1, denoted by x. Denote v1(x, t) by v, u1(x, t) by u, P′ by P and the compo-nents of the deviator of the mechanical pressure tensor Pik by Dik, being Dik = Dki.Then, the system (11), (12), (19), (20) and (21) becomes

v =∂u∂t

+ v∂u∂x, (50)

∂ρ

∂t+ v

∂ρ

∂x+ ρ

∂v∂x

= 0, (51)

∂v∂t

+ v∂v∂x

+1ρ

∂D11

∂x+

1ρ

∂P∂x

= 0, (52)


∂D21

∂x= 0, (53)

∂D31

∂x= 0, (54)

∂D11

∂t+ v

∂D11

∂x+

23

R(ε)(d)0

∂u∂x

+23

R(ε)(d)1

∂v∂x

+23

R(ε)(d)2

∂2v∂t∂x

+23

R(ε)(d)2

∂2v∂x2 v + R(τ)

(d)0D11 = 0, (55)

∂D12

∂t+ v

∂D12

∂x+ R(τ)

(d)0D12 = 0, (56)

∂D13

∂t+ v

∂D13

∂x+ R(τ)

(d)0D13 = 0, (57)

∂D22

∂t+ v

∂D22

∂x− 1

3R(ε)

(d)0∂u∂x− 1

3R(ε)

(d)1∂v∂x

−13

R(ε)(d)2

∂2v∂t∂x

− 13

R(ε)(d)2

∂2v∂x2 v + R(τ)

(d)0D22 = 0, (58)

∂D23

∂t+ v

∂D23

∂x+ R(τ)

(d)0D23 = 0, (59)

∂P∂t

+v∂P∂x

+13

R(ε)(v)0

∂u∂x

+13

R(ε)(v)1

∂v∂x

+13

R(ε)(v)2

∂2v∂t∂x

+13

R(ε)(v)2

∂2v∂x2 v + R(τ)

(v)0P = 0. (60)

Eqns. (53) and (54) show that D21 = f (t), D31 = f1(t), where f and f1 arefunctions of t. Therefore, from eqns. (56) and (57) we obtain the following results

D12 = e−R(τ)(d)0t

+ D012, D13 = e−R(τ)

(d)0t+ D0

13,

revealing that the medium presents a relaxation time for its mechanical properties,due to the presence of a tensorial internal variable. Then, the remained system ofequ.s (50) - (52), (55) and (58) - (60) takes the following matrix form (3)

Ut + AUx + ω−1[H1 ∂

2U∂t∂x

+ H11 ∂2U∂x2

]= B(U), (61)

having the following associated system of nonlinear hyperbolic PDEs

Ut + AUx = B(U), (62)

where A0(U) = I, U = (u, ρ, v,D11,D22,D23, P)T ,

B = (v, 0, 0,−R(τ)(d)0D11,−R(τ)

(d)0D22,−R(τ)(d)0D23,−R(τ)

(v)0P)T ,


A =

v 0 0 0 0 0 00 v ρ 0 0 0 00 0 v 1

ρ 0 0 1ρ

23 R(ε)

(d)0 0 23 R(ε)

(d)1 v 0 0 0−1

3 R(ε)(d)0 0 − 1

3 R(ε)(d)1 0 v 0 0

0 0 0 0 0 v 013 R(ε)

(v)0 0 13 R(ε)

(v)1 0 0 0 v

, (63)

H1 =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 2

3 R′(ε)(d)2 0 0 0 00 0 − 1

3 R′(ε)(d)2 0 0 0 00 0 0 0 0 0 00 0 1

3 R′(ε)(v)2 0 0 0 0

, H11 =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 2

3 R′(ε)(d)2v 0 0 0 0 00 −1

3 R′(ε)(d)2v 0 0 0 0 00 0 0 0 0 0 00 1

3 R′(ε)(v)2v 0 0 0 0 0

,

(64)with R(ε)

(d)2 = ω−1R′(ε)(d)2 and R(ε)(v)2 = ω−1R′(ε)(v)2. The symbol (...)T means that U and B are

column vectors.We obtain the following expressions for the eigenvalues of the matrix A:• λ1 = v (of multiplicity equal to 5);

• λ(±)2 = v ± γ, with γ =

√2R(ε)

(d)1+R(ε)(v)1

3ρ .

The eigenvalues λ(±)2 are simple eigenvalues. The right and left eigenvectors corre-

sponding to λ(±)2 can be taken, respectively, as

r(±)2 =

0, ρ,−(v − λ(±)2 ),

23

R(ε)(d)1,−

R(ε)(d)1

3, 0, (v − λ(±)

2 )2 − 23

R(ε)(d)1

T

, (65)

l(±)2 =

2R(ε)

(d)0 + R(ε)(v)0√

3ρ(2R(ε)

(d)1 + R(ε)(v)1

) , 0, −(v − λ(±)

2

),

1ρ, 0, 0,

1ρ

. (66)

Only the eigenvalues λ(±)2 do not satisfy the Lax - Boillat exceptionality condition

because ∇λ(±)2 · r(±)

2 , 0. Then, our results are valid for them.Now, let us consider only the longitudinal wave traveling in the right direction and

the case where the propagation occurs into an uniform unperturbed state U0, i. e.

λ(+)2 = v + γ and U0 = (0, ρ0, 0, 0, 0, 0, 0),


where ρ0 is constant. The characteristic rays, given by eq. (40), are

x0 = σ = t, x = (x)0 + λ(+)2 (U0)σ = (x)0 + γ0t, (67)

whence the wave front isϕ(t, x) = ϕ0

[x(t) − γ0t

], (68)

implying ϕx = 1. Here, γ0 =

√2R(ε)

(d)1+R(ε)(v)1

3ρ0 .Now, we compute the terms in (42). We have

∇Ψ · r(+)2 = ϕx(∇λ(+)

2 · r(+)2 ),

with ∇ =

(∂

∂u,∂

∂ρ,∂

∂v,

∂

∂D11,

∂

∂D22,

∂

∂D23,∂

∂P

). Hence, we have

(∇λ(+)

2 · r(+)2

)0

=γ0

2, being ∇λ(+)

2 =

(0,− γ

2ρ, 1, 0, 0, 0, 0

). (69)

Furthermore, we obtain

(l(+)2 · ∇B r(+)

2

)0

= −2(R(ε)

(d)1R(τ)(d)0 − R(ε)

(d)0

)+

(R(ε)

(v)1R(τ)(v)0 − R(ε)

(v)0

)

3ρ

0

, (70)

(l(+)2 · r(+)

2

)0

=

2(2R(ε)

(d)1 + R(ε)(v)1

)

3ρ

0

= 2(γ0

)2, (71)

ν0 = −2(R(ε)

(d)1R(τ)(d)0 − R(ε)

(d)0

)+

(R(ε)

(v)1R(τ)(v)0 − R(ε)

(v)0

)

2(2R(ε)

(d)1 + R(ε)(v)1

) , (72)

µ0 =

[l(+)2 ·

(H1 ∂ϕ

∂t∂ϕ∂x + H11 ∂2ϕ

∂x2

) (r(+)

2

)]0(

l(+)2 · r(+)

2

)0

=

(∂ϕ∂t

)0

(2R′(ε)(d)2 + R′(ε)(v)2

)

2√

3ρ0(2R(ε)

(d)1 + R(ε)(v)1

) , (73)

This example, in its simplicity, shows the influence of a tensorial internal variable onthe motion of viscoanelastic media with shape and bulk memory.

Note of L.R. The present paper is one of a series of works, in the area of asymp-totic waves in rheological media, where joint original results and previous treatmentsperformed in classical way by one of the authors (L. R), are seen in the modern lightof double-scale method, as it is presented in a monograph on asymptotic treatments ofthe other author A. G.. These works were started and planned since 2004 in occasionof a stay of A. G at the Department of Mathematics of the University of Messina, for


delivering a series of lectures on ”Asymptotic methods with applications to wavesand shocks”. They were continued during subsequent visits of A.G. at Messina in2005 and 2007 (for holding seminars and lectures on Applied Mathematics) and fin-ished in next visits of L.R. at Bucharest in 2007 and 2009. This paper was writtenin a final version in 2010. These studies gave rise from very enlightening remarksconcerning some mathematical tools, together with their practical applications indifferent fields of science. The author L. R. is very grateful to Adelina Georgescufor her deep involvement in this work regarding a systematic formalization of knownresults, obtained in the field of rheological media, and of joint new results derived onthe same subject.

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[8] J. D. Cole, Perturbation methods in applied mathematics, Blaisdell, Waltham, MA, 1968.

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[29] V. Ciancio, L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media),Physica 131 A (1985), 251-262.

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A NEW MODIFIED GALERKIN METHOD FORTHE TWO-DIMENSIONAL NAVIER-STOKESEQUATIONS

ROMAI J., 6, 2(2010), 155–179

Anca-Veronica Ion”Gh. Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of theRomanian Academy, Bucharest, [email protected]

Abstract The nonlinear Galerkin method as well as the postprocessed Galerkin method were builtupon the notion of approximate inertial manifold (a.i.m.). We conceived a modifiedGalerkin method that uses the so-called ”induced trajectories” defined in [19], insteadof the a.i.m.s. The paper presents this new method, that provides a sequence of func-tions approximating the solution of the Navier Stokes equations. The convergence ofthis sequence of functions to the exact solution is proved and some comments on therelationship between our method and the nonlinear postprocessed Galerkin method aregiven.

Keywords: Navier-Stokes equations, induced trajectories, nonlinear and post-processed Galerkin meth-ods, approximate inertial manifolds.2000 MSC: 35K55, 65M60.

1. INTRODUCTIONIn the study of dissipative dynamical systems generated by evolution partial dif-

ferential equations, the notion of inertial manifold arose. Defined in [6], this notionwas the object of a large number of works. An inertial manifold (i.m.) is a finite di-mensional manifold in the phase space, having some smoothness properties (at leastLipschitz), that is invariant to the considered dissipative dynamical system and ex-ponentially attracts all the trajectories of the system. The restriction of the (infinitedimensional) dynamical system to an i.m. is a finite dimensional dynamical systemthat has the large time asymptotic properties of the infinite dynamical system.

Since not all the dissipative dynamical systems generated by PDEs have i.m.s, andif they have one, it can not be effectively constructed otherwise than approximately,another notion was defined, that of approximate inertial manifold (a.i.m.) [5], [20],[7], [2], [13], to give only a few titles. An a.i.m. is a finite dimensional manifold thathas the property that for each trajectory there is a certain moment of time startingfrom which the trajectory is contained in a narrow neighborhood of this manifold.This property was exploited in order to construct some modified Galerkin methods

155

156 Anca-Veronica Ion

with the help of the a.i.m.s, namely the nonlinear (NL) Galerkin method [14], [3], thepostprocessed (PP) Galerkin method [8] or some combinations of these two.

In this paper we consider the Navier-Stokes equations for a two-dimensional flow,with periodic boundary conditions. For this problem we propose a new modifiedGalerkin method, that relies not on the a.i.m.s but on some more basic notion, that ofinduced trajectories, defined in [19]. In [19] a family of a.i.m.s is constructed havingas starting point these induced trajectories. Hence, the use of induced trajectories indefining a modified Galerkin method will lead to some simplifications in the numer-ical algorithm, compared to the NL Galerkin (eventually postprocessed) method.

Our work is organized as follows. The functional framework and the basic hy-pothesis are settled in Section 2. In Section 3 the projector used in the Galerkinmethod is constructed and some estimates of the ”higher modes” part of the solutionare given. In Section 4 a family of induced trajectories similar to those defined in[19], is presented. Our modified Galerkin method is presented in Section 5.

In Section 6 we give some bounds for the norms of the functions involved in ourmethod and prove that the sequence of functions provided by our algorithm convergesto the solution of the Navier Stokes equations. In Section 7 we compare our methodto the NL and PP Galerkin method, that makes use of a high accuracy a.i.m., fromthe point of view of the amount of computations involved.

2. THE EQUATIONS, THE FUNCTIONALFRAMEWORK

The plane flow of an incompressible Newtonian fluid is modelled by the Navier-Stokes equations:

∂u∂t− ν∆u + (u · ∇) u+∇p = f, (1)

divu = 0, (2)

u (0, ·) = u0 (·) , (3)

where u = u (t, x) is the fluid velocity, x ∈Ω ⊂ R2, u (., x) : [0,∞) → R2, p (., x) :[0,∞) → R is the pressure of the fluid, ν is the kinematic viscosity, and f is thevolume force. We take here Ω = (0, l) × (0, l) and consider the case of periodicboundary conditions.

We assume that f ∈[L2

per (Ω)]2

is independent of time. Since we consider periodicboundary conditions, we assume that [17], [16],

f =1l2

∫

Ω

f (x) dx = 0, (4)

that the pressure is a periodic function on Ω, and that the average of the velocity overthe periodicity cell is zero. The velocity u thus belongs to the space

A new modified Galerkin method for the two-dimensional Navier-Stokes equations 157

H =

v; v ∈

[L2

per (Ω)]2, div v = 0, v = 0

. The scalar product in H is 〈u, v〉 =∫

Ω(u1v1 + u2v2) dx, (where u = (u1, u2) , v = (v1, v2)). The induced norm is denoted

by | |.We also use the space V =

u ∈

[H1

per (Ω)]2, div u = 0,u = 0

, with the scalar

product 〈〈u, v〉〉 =∑2

i, j=1

(∂ui∂x j, ∂vi∂x j

), and the induced norm, denoted by ‖ ‖ . We denote

A = −∆. A is defined on D(A) = H∩H2 (Ω) .The classical variational formulation of the Navier-Stokes equations [17] leads to

the equationdudt

+ νAu + (u · ∇) u = f in V ′, (5)

u (0) = u0, u0 ∈ H. (6)

Consider the trilinear form b(u, v,w) = ((u · ∇) v,w) and the bilinear mappingB(u, v) defined by (B(u, v),w) = b(u, v,w). The notation B(u) = B(u,u) will be alsoused. For the bilinear mapping B(u, v) the following inequalities

|B (u, v)| ≤ c1 |u|12 |∆u| 12 ‖v‖ , (∀) u ∈D(A), v ∈V, (7)

|B (u, v)| ≤ c2 ‖u‖ ‖v‖[1 + ln

( |∆u|2λ1 ‖u‖2

)] 12

, (∀) u ∈D(A), v ∈V, (8)

hold [10], [17], [19]. We also remind the following properties of the trilinear formb(u, v,w) (valid for periodic boundary conditions [16]):

b(u, v,w) = −b(u,w, v),b(u, v, v) = 0, (9)

and the following inequalities [16]

|b(u, v,w)| ≤ c3 |u|12 ‖u‖ 1

2 ‖v‖ |w| 12 ‖w‖ 12 , (∀) u, v,w ∈V, (10)

|b(u, v,w)| ≤ c4 |u|12 ‖u‖ 1

2 ‖v‖ 12 |∆v| 12 |w| , (∀) u ∈V, v ∈ D (A) ,w ∈H. (11)

For the problem (5), (6) we have the classical existence and uniqueness results forthe Navier-Stokes equations in R2, with periodic boundary conditions.

Theorem 2.1. [17]. a) If u0 ∈ H, f ∈ H, then the problem (5), (6) has an uniqueweak solutionu ∈ C0 ([0, T ]; H) ∩ L2 (0,T ; V) .

b) If, in addition to the hypotheses in a), u0 ∈ V, then there is an unique strong so-lution u ∈ C0 ([0, T ]; V)∩L2 (0, T ; D(A)) . The solution is, in this latter case, analyticin time on the positive real axis.

The semi-dynamical system S (t)t≥0 generated by problem (5) is dissipative [18],[16]. More precisely, there is a ρ0 > 0 such that for every R > 0, there is a t0(R) > 0


with the property that for every u0 ∈ H with |u0| ≤ R, we have |S (t) u0| ≤ ρ0 fort > t0(R). In addition, there are absorbing balls in V and D (A) for S (t)t≥0, i.e.there are ρ1 > 0, ρ2 > 0 and t1(R), t2(R) with t2(R) ≥ t1(R) ≥ t0(R) such that forevery R > 0, |u0| ≤ R implies ‖S (t) u0‖ ≤ ρ1 for t > t1(R) and |AS (t) u0| ≤ ρ2 fort > t2(R).

3. THE DECOMPOSITION OF THE SPACE, THEPROJECTIONS OF THE EQUATIONS

The eigenvalues of A are λ j1, j2 = 4π2

l2

(j21 + j22

), j = ( j1, j2) ∈ N2\ (0, 0) . The

corresponding eigenfunctions are

ws±j1, j2 =

√2

l( j2,∓ j1)|j| sin

(2π

j1x1 ± j2x2

l

),

wc±j1, j2 =

√2

l( j2,∓ j1)|j| cos

(2π

j1x1 ± j2x2

l

),

where |j| =(

j21 + j22) 1

2 [19].For a fixed m ∈ N we consider the set Γm of eigenvalues λ j1, j2 having 0 ≤ j1, j2 ≤

m. We define

λ1 : = λ1,0 = λ0,1 =4π2

l2,

λm+1 : = λm+1,0 = λ0,m+1 =4π2

l2(m + 1)2 ,

δ = δ (m) :=λ1

λm+1=

1(m + 1)2 .

Obviously, λm+1 is the least eigenvalue not belonging to Γm.Let Hm be the finite dimensional space generated by the eigenfunctions corre-

sponding to the eigenvalues of Γm. We denote by P the orthogonal projection opera-tor on this subspace and by Q the orthogonal projection operator on the complemen-tary subspace, and, for the solution u of (5), we set p = Pu, q = Qu.

We project equation (5) by using P, and Q, to obtain

dpdt

+ νAp + PB(p + q) = Pf, (12)

dqdt

+ νAq + QB(p + q) = Qf. (13)


Estimates for the ”small” component of the solution

In [5] it is proved that for every R > 0, there is a moment t3 (R) ≥ t2(R) such thatfor every |u0| ≤ R,

|q (t)| ≤ K0L12 δ, ‖q (t)‖ ≤ K1L

12 δ

12 ,

|q′ (t)| ≤ K′0L12 δ, |∆q (t)| ≤ K2L

12 , t ≥ t3 (R) ,

(14)

where q′(t) is the time derivative of q(t), the coefficients K0, K′0, K1, K2 depend ofν, | f | , λ, and, for the way we chose the projection subspaces, L = L(m) = 1+ln(2m2)(see also [19]). The constant L comes from the use of inequality (8) in the proof of(14). More specific

L = supp∈PH

(1 + ln

|∆p|2λ1 ‖p‖2

)= max

λ∈Γm

(1 + ln

λ

λ1

)= 1 + ln(2m2). (15)

In [12] we improved the estimates (14) by eliminating L (which tends to infinitywith m) from the constants. We proved the following result.

Theorem 3.1. There are some positive numbers C0, C1, C ′0, C2, that depend onlyon ν, λ, |Qf| , (and not on n) and there is a moment t4(R) such that, for t ≥ t4(R) thefollowing inequalities hold

|q(t)| ≤ C0δ, (16)

‖q(t)‖ ≤ C1δ12 , (17)

∣∣∣q′(t)∣∣∣ ≤ C ′0δ, (18)

|∆q(t)| ≤ C2. (19)

4. INDUCED TRAJECTORIES FOR THENAVIER- STOKES EQUATIONS

In [19] a family of functions, uii∈N,ui : R+ → H that define the so-called in-duced trajectories, ui(t) ; t ≥ 0, is constructed. In [20] the definitions are somehowsimplified. We use here functions that share properties of both families of functionsfrom [19] and [20].

The functions have the form ui(t) = p(t) + qi(t), where, as above, p(t) = P(u(t)),and the functions qi are successively defined as:

q0(t) = (νA)−1[Qf −QB (p(t))], (20)

q1(t) = (νA)−1[Qf −QB (p(t) + q0(t))], (21)

q j(t) = (νA)−1[Qf −QB(p(t) + q j−1(t)

)− q′j−2(t)]. (22)


For the sake of some later discussions (Section 5), we denote the right-hand sidesof (22) by Ψ j(p, q j−1, q j−2), j ≥ 0, with the convention q−1 = q−2 = 0. For thefunctions q j the following inequalities are proved in [20]:

|q j − q| ≤ k jL1+ j/2δ3/2+ j/2, (23)

‖q j − q‖ ≤ k jL1+ j/2δ1+ j/2, (24)

|A(q j − q)| ≤ k jL1+ j/2δ1/2+ j/2, (25)

|(q j − q)′| ≤ k jL1+ j/2δ3/2+ j/2. (26)

They hold for t large enough (i.e. for t larger than a certain value t(ν, | f |, λ)). Theseinequalities imply similar ones for the difference u − uj.

The definitions (20) - (22) are the starting point for the construction, in [19], [20],of a family of a.i.m.s, M j j≥0. The first one of these a.i.m.s, M0, was defined in [5]independent of the notion of induced trajectories.

The a.i.m.s of this family were used in the nonlinear Galerkin method [14], [3] aswell as in the post-processed Galerkin method [15].

5. OUR MODIFIED GALERKIN METHODLet us take a T > t4. The interval [0,T ] is the interval on which we seek the

approximate solution. Obviously, inequalities (16) - (17) and (23) - (26) are valid fort ∈ [t4, T ] .

In the sequel we present our modified Galerkin method. It has several levels, ofincreasing accuracy.

Level 0. We define the first step of this level as the classical Galerkin method. Letus consider the Cauchy problem

dpdt− ν∆p + PB (p) = Pf, (27)

p(0) = Pu0.

This problem is the Galerkin approximation of our problem. We denote by p0 (t)its solution and define

q0(t) = (νA)−1[Qf −QB (p0(t))]. (28)

At this level, the exact solution u is approximated by

u0 (t) = p0 (t) + q0(t). (29)

This preliminary level differs from the PP Galerkin method of [8] only in the post-processing part, in the fact that we compute q0(t) at any moment of time and not onlyat the end of the time interval on which (27) is integrated.


Level 1. Now we consider the problem

dpdt− ν∆p + PB

(p+q0

)= Pf, (30)

p(0) = Pu0,

with q0(t) computed at the preceding level. Since q0(t) is already known, this equa-tion is not more difficult to integrate than the simple Galerkin equation attached tothe Navier-Stokes equation. It is an adjusted Galerkin equation since the nonlinearterm is adjusted by adding to p(t) the term q0(t) that approximates q(t) better than 0does. We denote by p1 (t) the solution of problem (30).

Then we define

q1(t) = (νA)−1[Qf −QB(p1(t) + q0(t)

)]

. The approximate solution will be defined at this level as

u1 (t) = p1 (t) + q1(t). (31)

This function is an approximation of the function u1 (that defines the second inducedtrajectory - see Section 4).

Level k. Let us consider a k ∈ N. We assume that we already constructed qk−2and qk−1. Now, we define pk as the solution of the problem

dpdt + νAp + PB

(p + qk−1

)= Pf,

p(0) = Pu0,(32)

and then set qk as

qk(t) = (νA)−1[Qf −QB

(pk(t) + qk−1(t)

) − dqk−2(t)dt

]. (33)

The corresponding approximate solution of (5)-(6) is defined by

uk (t) = pk (t) + qk (t) .

Remarks. 1. The right hand side of (33) is equal to Ψk(pk (t) , qk−1 (t) , qk−2 (t)),(the functions Ψ j are defined in Section 4) and, as can be proved, qk is an approxi-mation of qk .

Our functions uk are approximations of the functions uk = p + qk that gener-ate the induced trajectories. Our construction by-passes the construction of a.i.m.s.and is based directly upon that of the induced trajectories. We might name the setsuk(t); t ≥ 0

- approximate induced trajectories.

2. If k is the last level we construct, than we may compute qk only at the momentof interest (T for example) as in the PP Galerkin method of [8].

We name our method the repeatedly adjusted and postprocessed Galerkin method(abreviated as R-APP Galerkin method).


6. CONVERGENCE OF THE METHODWe now prove that the sequence of functions uk defined in the previous section

converges in the norm of V to the exact solution u of the Navier Stokes equations.We must first give some estimates for the norms of the functions pk, qk.

6.1. UPPER BOUNDS FOR pk, qk

In the sequel we assume that the initial condition of the studied problem u0

belongs to V and is taken in a radius R ball of H, that is |u0| ≤ R.

Theorem 6.1. There is a t5 ≥ t4 and there is a natural number M0, suchthat, for the functions p j, q j, defined in the previous section, the followinginequalities hold

|p j| ≤ η0, ‖p j‖ ≤ η1, |∆p j| ≤ η2, j ≥ 0,

|q j| ≤ κ0δ, ‖q j‖ ≤ κ1δ1/2, |∆q j| ≤ κ2,

∣∣∣q ′j (t)∣∣∣ ≤ κ′0δ, j ≥ 0,

if t ≥ t5 and m ≥ M0.The numbers η0, η1, η2 depend only on ν, λ1, f and κ0 =

2|Qf|νλ1

, κ1 =2|Qf|νλ1/2

1,

κ2 = 2ν|Qf|.

Proof. For p0, by using the method for the full Navier-Stokes eqns., [18], theinequality

lim supt→∞

|p0(t)| ≤ η0,0 := |Pf|/(νλ1)

may be proved. Hence for a t0(R) (the same t0(R) from the proof of the exis-tence of the absorbing ball in H for u), we have |p0| ≤ η0,0, for t > t0(R). Alsothe inequality ‖p0‖ ≤ η0,1, where η0,1 is obtained from ρ1 (given e.g. in [18]pg. 111) by replacing |f| with |Pf|. We can see that η0,0 ≤ ρ0, η0,1 ≤ ρ1.

We must remark that, since the equations for p0 have the same form as theNavier Stokes equations, we can prove that p0 is analytic in time, as is provedfor u in [17]. More than that, p0(t) is the restriction to the real axis of an ana-lytic function (a D(A) valued function) of a complex variable. The domain ofanalyticity is a neighborhood of the positive real axis (in the complex plane).

Now, as in [12], take m as an even number, put n = m/2, and considerthe space Hn defined similarly as Hm, Pn the projection operator on Hn, andQn = P − Pn. We also set pp(t) = Pnp0(t), pq(t) = Qnp0(t) (hence p0(t) =

pp(t) + pq(t)) and

δn = λ1,0/λn+1,0 = 1/(n + 1)2, Ln = 1 + ln(2n2).


By using the method of proof of Theorem 1 given in [12], we may showthat |∆pq| ≤ C2, ‖pq‖ ≤ C1δ

1/2n , |pq| ≤ C0δn. For q0 we have

|ν∆q0| ≤ |Qf| + |QB (p0) | ≤≤ |Qf| + |QB

(pp, pq

)| + |QB

(pq, pp

)| + |QB

(pq, pq

)|,

since QB(pp, pp

)= 0 [12]. By using inequalities (7), it follows that (8),

|ν∆q0| ≤ |Qf| + c2L1/2n ‖pp‖ ‖pq‖ + c1|pq|1/2|∆pq|1/2‖pp‖+

+c1|pq|1/2|∆pq|1/2‖pq‖ ≤≤ |Qf| + c2C1L1/2

n η0,1δ1/2n + c1C

1/20 C1/2

2 η0,1δ1/2n + c1C

1/20 C1/2

2 C1δn.

The function L1/2n δ1/2

n is decreasing and tends to 0 when n→ ∞ hence thereis an n0 such that for n ≥ n0 the sum of the last three terms above is less than|Qf|. For such an n,

|∆q0| ≤ 2ν|Qf| := κ2.

As consequences, by using the inequalities |Aq| > λ1/2n+1‖q‖ > λn+1|q|, that hold

for every q ∈ QD(A), we obtain

‖q0‖ ≤ 2|Qf|νλ1/2

1

δ1/2 := κ1δ1/2 and |q0| ≤ 2|Qf|

νλ1δ := κ0δ.

In order to obtain an estimate for q ′0(t), we observe that the analyticity of p0

implies, by (28), the analyticity of q0. The domain of analyticity is the sameas for p0. Then, we consider a point t ∈ [0,T ], a closed curve in the domain ofanalyticity of q0 that surrounds t, and, by using the Cauchy integral formula,we obtain an estimate of the form

∣∣∣q ′0 (t)∣∣∣ ≤ κδ.

Now we assume that for every 0 ≤ j ≤ k − 1, we have

‖∆q j‖ ≤ κ2, ‖q j‖ ≤ κ1δ1/2, |q j| ≤ κ0δ,

∣∣∣q ′j (t)∣∣∣ ≤ κδ.

The function pk is defined as the solution of equation (32). We use theclassical method for obtaining bounds for the norms of the solution of theNavier Stokes equation, as is presented in [18], that is, we take the scalarproduct of (32) with its solution, pk, and we obtain

12

ddt|pk|2 + ν‖pk‖2 = 〈Pf, pk〉 − 〈PB(pk + qk−1), pk〉 ≤ (34)


≤ |Pf| |pk| + |〈PB(pk, qk−1), pk〉| + |〈PB(qk−1, qk−1), pk〉|.Here we have used the equalities 〈PB(pk,pk), pk〉 = 〈B(pk,pk), pk〉 = 0, and〈PB(qk−1,pk), pk〉 = 〈B(qk−1,pk), pk〉 = 0, due to (9). We estimate the termsin (34):

|〈PB(pk, qk−1), pk〉| ≤ c2L1/2‖pk‖ ‖qk−1‖ |pk| ≤ ν

8‖pk‖2 +

2c22Lν‖qk−1‖2 |pk|2,

|〈PB(qk−1, qk−1), pk〉| ≤ c1 |qk−1|1/2|∆qk−1|1/2‖qk−1‖ |pk|.(34) becomes

12

ddt|pk|2 +

7ν8‖pk‖2 ≤ |Pf| |pk| +

2c22Lν

κ21δ |pk|2 + c1 |qk−1|1/2|∆qk−1|1/2‖qk−1‖ |pk|.

In the LHS of the inequality above, we use λ1|pk| ≤ ‖pk‖ and replace 7ν8 ‖pk‖2

with 7νλ18 |pk|2. Moreover, since the function of m, Lδ, is decreasing with m and

tends to 0 when m→ ∞ we may choose m1 ≥ m0 := 2n0 such that

2c22κ

21

νL(m)δ(m) ≤ νλ1

8, for m ≥ m1.

Now, the relation for |pk| becomes

12

ddt|pk|2 +

7νλ1

8|pk|2 ≤ 2

νλ1|Pf|2 +

2c21

νλ1|qk−1| |∆qk−1| ‖qk−1‖2 +

3νλ1

8|pk|2, (35)

and, thus,

12

ddt|pk|2 +

νλ1

2|pk|2 ≤ 2

νλ1|Pf|2 +

2c21

νλ1|qk−1| |∆qk−1| ‖qk−1‖2 ≤ (36)

≤ 2νλ1|Pf|2 +

2c21

νλ1κ0κ2κ

21δ

2.

For m ≥ m1 with m1 fixed above we have, by the definition of κ1, and consid-ering that ν ≤ 1,

L(m)δ(m) ≤ ν4λ21

64c22|Qf|2 ≤

ν2λ21

32c22|Qf|2 ,

hence

2c21

νλ1κ0κ2κ

21δ

2 =16c2

1

νλ31

|Qf|4δ(m)2 =16c2

1|Qf|4νλ3

1L(m)2L(m)2δ(m)2 ≤


≤ 16c21|Qf|4

νλ31L(m)2

(νλ1)4

322c42|Qf|4 ≤

c21ν

3λ1

64c42L(m)2

.

Finally we obtain

ddt|pk|2 + νλ1|pk|2 ≤ 4

νλ1|Pf|2 +

c21ν

3λ1

32c42L(m)2

, (37)

and

|pk(t)|2 ≤ |pk(0)|2e−νλ1t +4

(νλ1)2 |Pf|2 +c2

1ν2

32c42L(m)2

.

If we chose a t such that R2e−νλ1

2 t ≤ 1(νλ1)2 |Pf|2, for t > t then

|pk| ≤√

5νλ1|Pf| + c1ν

8c22

:= η0.

A short look in the proof of the existence of an absorbing ball in H for theNavier Stokes eqns., given in [18] pg 109, shows that t may be taken as equalto t0(R).

Inequalities ‖pk‖ ≤ η1 and ‖∆pk‖ ≤ η2 may be proved by methods similar tothose used in [18], III.2 combined to those above. The fact that η1, η2 dependonly on ν, λ1, f is a consequence of the uniform bounds of the various normsof q j. The presentation of the proofs here would lengthen too much the paper.

Now, in what concerns qk, we start from its definition, (33), and see that

|ν∆qk| ≤ |Qf| + |QB(pk, pk)| + |QB(pk, qk−1)|+ (38)

+|QB(qk−1, pk)| + |QB(qk−1, qk−1)| + |q′k−2|.For pk we define pkp = Pnpk, pkq = Qnpk. For pkq, by the method of [12], wecan prove inequalities of the form

|pkq| ≤ C0δn, ‖pkq‖ ≤ C1δ1/2n , |∆pkq| ≤ C2,

with C1, C2, C3 independent of k, since the estimates for the various norms ofqk−1 do not depend on k.

This and the fact that QB(pkp,pkp) = 0, enable us to write (by (8), (11))

|QB(pk, pk)| ≤ |QB(pkp, pkq)| + |QB(pkq, pkp)| + |QB(pkq, pkq)| ≤

≤ c2L1/2n η1C1δ

1/2n +c4C

1/20 C1/2

1 δ3/4n η1/2

1 η1/22 +c4C

1/20 C1C

1/22 δn → 0, when n→ ∞.


The following three terms in the RHS of (38) are of a smaller order than|QB(pk, pk)|, and, by the induction hypothesis

|q ′k−2| ≤ κ′0δ.Now, it is obvious that we can choose an m2 ≥ m1 such that the sum after |Qf|in the RHS of (38) is smaller than |Qf|, for any m > m2 such that

‖∆qk‖ ≤ 2ν|Qf|.

From here, two other inequalities of interest for qk follow:

‖qk‖ ≤ 2|Qf|νλ1/2

1

δ1/2, |qk| ≤ 2|Qf|νλ1

δ.

To obtain an estimate for q′k, we have to remark that, as for p0, it can beproved that every function pk is analytic in time, and is the restriction to thereal axis of an analytic function of complex variable defined on a neighbor-hood of the real axis, in the complex plane. Actually, the domain of analyticitymay be proved to be independent of k, since the norms of q j, 0 ≤ j ≤ k arebounded uniformly (with upper bounds not depending on j).

By using the Cauchy integral formula, we will find |q′k| ≤ κ′δ.We set κ′0 := max

(κ, κ′

), and the relation

|q′k| ≤ κ′0δfollow. It is very important to remark that, at step k + 1 (that is, for thefunctions pk+1, qk+1), the same constants as at step k would be implied in theestimates. Some differences appear only between the first two steps (0 and 1)and the following, because of the different definitions of the functions in thesedifferent cases. Hence no further increase of m besides the last one (m > m2)is necessary. We set M0 = m2.

Thus the induction hypothesis on qk is confirmed and we proved also thatthe estimates for pk hold for any k ≥ 0, completing the proof.

6.2. THE CONVERGENCE RESULTS

In the proof of the main result of this section, we need the following resultthat is a direct consequence of Lemma 1 from [8].


Lemma 6.1. Consider a function G : [0,∞) 7→ H. For a s ≥ 0, we denote byGi

j,l(s) the coordinate of G(s) with respect to the eigenfunction wij,l, such that,

G(s) =∑j,l

(4∑

i=1Gi

j,l (s) wij,l

). If

∣∣∣∣Gij,l (s)

∣∣∣∣ ≤ cij,l, f or s ∈ [0, t], 0 ≤ j, l ≤ m, 1 ≤ i ≤ 4,

then∣∣∣∣∣∣∫ t

0e−ν(t−s)APG(s)ds

∣∣∣∣∣∣ ≤1ν

∑

j,k≤m

4∑

i=1

(ci

j,l

)2

λ2j,l

12

. (39)

We state and prove our result concerning the convergence of the sequenceof approximate solutions given by the R-APP Galerkin method.

Theorem 6.2. The sequence of functions uk (t) , k ≥ 0, converges in V uni-formly with respect to t ∈ [t5, T ] to the exact solution of the problem (5)-(6). More precisely, there is a natural number M such that for m > M, andt ∈ [t5, T ], the inequalities :

∣∣∣(u − uk)

(t)∣∣∣ ≤ (1 +

1λ1δ1/4)δ5/4+k/4, (40)

‖(u − uk)(t)‖ ≤λ1/2

1 +1

λ1/21

δ1/4

δ3/4+k/4, (41)

hold for every k ≥ 0.

Proof. We prove our assertion by induction.1. We start with k = 0.Theorem 1 and Remark 1 from [9] (for f ∈

[L2

per(Ω)]2

) combined with ourestimates (16)-(18), lead to

∣∣∣(p − p0)

(t)∣∣∣ ≤ CL1/2δ3/2, (42)

for t ∈ [t3,T ], where C = C(T, ρ1). We must say that C is of the order ofeT . Since the function of m, L1/2δ1/4, tends to 0 when m → ∞ and C doesnot depend on m, we can choose a m3 ≥ m2 (m2 was defined in the proof ofTheorem 6.1) such that for m ≥ m3, the inequality CL1/2δ1/4 < 1 is true, hence

|p − p0| ≤ δ5/4, for m ≥ m3.

From here, by using the inequality ‖p‖ ≤ λ1/2m+1,0|p|, valid for every p ∈ Hm,

we obtain ‖p − p0‖ ≤ λ1/21 δ3/4, |∆(p − p0)| ≤ λ1δ

1/4, for m ≥ m3.


We estimate the various norms of(q − q0

)(t):

∣∣∣ν∆ (q − q0

)∣∣∣ =∣∣∣QB(p + q) −QB (p0) +

dqdt

∣∣∣≤

∣∣∣QB(p − p0,p

)∣∣∣ +∣∣∣QB

(p0,p − p0

)∣∣∣ ++ |QB(p,q)| + |QB(q,p)| + |QB (q,q)| +

∣∣∣ dqdt

∣∣∣ .For the first two terms in the RHS, (11), (7), and the estimates for p, imply∣∣∣QB

(p − p0,p

)∣∣∣ ≤ c4|p − p0|1/2‖p − p0‖1/2‖p‖1/2|∆p|1/2 ≤ c4δ5/8λ1/4

1 δ3/8ρ1/21 ρ1/2

2

= c4λ1/41 ρ1/2

1 ρ1/22 δ,∣∣∣QB

(p0,p − p0

)∣∣∣ ≤ c1|p0|1/2|∆p0|1/2‖p − p0‖ ≤ c1η1/20 η1/2

2 λ1/21 δ3/4.

With (7) and (17), the third term yields

|QB(p,q)| ≤ c1 |p| 12 |∆p| 12 ‖q‖ ≤ c1ρ1/20 ρ1/2

2 C1δ1/2,

and the fourth, with (11), (16), (17),

|QB(q,p)| ≤ c4|q|1/2‖q‖1/2‖p‖1/2|∆p|1/2 ≤ c4C1/20 C1/2

1 ρ1/21 ρ1/2

2 δ3/4.

The fifth term is smaller than the preceding one, and for the last term weuse (18). Putting all the above inequalities together, we obtain

∣∣∣∆ (q (t) − q0 (t)

)∣∣∣ ≤ 1ν

(c4λ1/21 ρ1/2

1 ρ1/22 δ1/2 + c1η

1/20 η1/2

2 λ1/2δ1/4+

+c1ρ1/20 ρ1/2

2 C1 + 2c4C1/20 C1/2

2 ρ1/21 ρ1/2

2 δ1/4 + C′0δ1/2)δ1/2,

for t great enough. We denote by K0 the coefficient of δ1/2 above. We thusobtained

∣∣∣∆ (q (t) − q0 (t)

)∣∣∣ ≤ K0δ1/2, and, as consequences,

∥∥∥q (t) − q0 (t)∥∥∥ ≤ K0

λ1/21

δ,∣∣∣q (t) − q0 (t)

∣∣∣ ≤ K0

λ1δ3/2. (43)

As we remarked in the proof of Theorem 6.1., q0 is the restriction to thepositive real axis of a complex variable analytic function, defined on a neigh-borhood of this axis. On the other hand, q = Qu has the same property [17],but the domain of analyticity may be different (being also a neighborhood ofthe positive real axis). We take the intersection of the two domains, and byusing the Cauchy integral formula, we obtain an inequality of the form

∣∣∣q′ (t) − q′0 (t)∣∣∣ ≤ K′0δ

3/2. (44)


Inequality (42) and the second inequality of (43) imply

∣∣∣u (t) − u0 (t)∣∣∣ ≤ (1 +

K0

λ1δ1/4)δ5/4. (45)

2. We now estimate |p − p1| and∣∣∣q − q1

∣∣∣ .We have, by subtracting (27) from(12) and by adding and subtracting in the RHS the term PB

(p1 + q0,p + q

),

ddt

(p − p1

)= ν∆ (p − p1) − PB

(p + q − (

p1 + q0),p + q

) −−PB

(p1 + q0,p + q − (

p1 + q0)).

From here, by using the semigroup of linear operators of infinitesimal gener-ator νA, we obtain

ddt

eνtA(p − p1

)(t) = eνtA

−PB(p − p1,u

) − PB(q − q0,u

)−−PB

(p1 + q0,q − q0

) − PB(p1 + q0,p − p1

),

and, by integrating and using p(0) = p1(0),

(p − p1

)(t) = −

∫ t

0e−ν(t−s)A

PB(p − p1,u

)+ PB

(p1 + q0,p − p1

)ds −

−∫ t

0e−ν(t−s)A

PB(p1 + q0,q − q0

)+ PB

(q − q0,u

)ds.

Inspired by [8] and [9], we use the inequalities [1]

∣∣∣A−αB (u, v)∣∣∣ ≤

C

∣∣∣A1−αu∣∣∣ |v| ≤ C

∣∣∣A1/2u∣∣∣ |v| ,

C |u|∣∣∣A1−αv

∣∣∣ ≤ C |u|∣∣∣A1/2v

∣∣∣ ,

valid for α ∈ (1/2, 1) and [11]∣∣∣Aαe−νtA

∣∣∣ ≤ Ct−αe−νλ2 t, and obtain

∣∣∣(p − p1)

(t)∣∣∣ ≤ C(ρ1, η1, κ1)

∫ t

0(t − s)−α e−

νλ2 (t−s)

∣∣∣(p − p1)

(s)∣∣∣ ds+

+

∣∣∣∣∣∣∫ t

0e−ν(t−s)A [

PB(p1 + q0,q − q0

)+ PB

(q − q0,p + q

)](s) ds

∣∣∣∣∣∣ .

A form of Gronwall inequality ([11], Lemma 7.1.1) implies


∣∣∣(p − p1)

(t)∣∣∣ ≤ (46)

≤ C(T, ρ1, η1, κ1)max0≤t≤T

∣∣∣∣∣∣∫ t

0e−ν(t−s)A

PB(p1 + q0,q − q0

)+ PB

(q − q0,p + q

)(s) ds

∣∣∣∣∣∣ .

We must remark that C(T, ρ1, η1, κ1) above is of the order of eT .By using the method of [8], we find estimates for the coordinates of the

several terms in the RHS of (46):∣∣∣∣ B

(q0,q − q0

)j,k

∣∣∣∣ ≤∣∣∣q0

∣∣∣∣∣∣∣A 1

2(q − q0

)∣∣∣∣ ≤ κ0δK0

λ1/21

δ = κ0K0

λ1/21

δ2, (47)

∣∣∣∣ B(q − q0,q

)j,k

∣∣∣∣ ≤∣∣∣q − q0

∣∣∣∣∣∣∣A 1

2 q∣∣∣∣ ≤ K0

λ1δ3/2C1δ

1/2 =K0

λ1C1δ

2, (48)∣∣∣∣ B

(p1,q − q0

)j,k

∣∣∣∣ ≤∣∣∣∣A 1

2(q − q0

)∣∣∣∣(∣∣∣∣(I − Pm− j

)p∣∣∣∣ + |(I − Pm−k) p|

)

≤ K0

λ1/21δρ0

(1

λm− j+1+ 1

λm−k+1

),

(49)

∣∣∣∣ B(q − q0,p

)j,k

∣∣∣∣ ≤ K0

λ1δ3/2ρ1

1

λ12m− j+1

+1

λ12m−k+1

, (50)

where Pm− j represents the projection operator on the space spanned by theeigenfunctions corresponding to the eigenvalues in Γm− j and λ j = λ j,0.

By using the inequalities (39), (47), (48) and∑

j,k≤m

λ−2j,k ≤ C,

it follows that∣∣∣∣∣∣∫ t

0e−ν(t−s)A

PB(q0,q − q0

)+ PB

(q − q0,q

)ds

∣∣∣∣∣∣ ≤1ν

(κ0K0

λ1/21

+K0

λ1C1)δ2.

In order to estimate the term∣∣∣∣∫ t

0e−ν(t−s)APB

(p1,q − q0

)ds

∣∣∣∣ we use (49) andthe inequality ∑

j,k≤m

1λ2

j,kλ2m− j+1

≤ C(m + 1)3 = Cδ3/2,

proved in [8]. It follows∣∣∣∣∣∣∫ t

0e−ν(t−s)APB

(p1,q − q0

)ds

∣∣∣∣∣∣ ≤ CK0

νλ1/21

ρ0δ1+ 3

4 .


A similar estimate can be proved for∣∣∣∣∫ t

0e−ν(t−s)A

PB(q − q0,p

)ds

∣∣∣∣ , hencefinally, by using (46), we obtain

∣∣∣p − p1

∣∣∣ ≤ C(T, ρ1, η1, κ1)1ν

(κ0K0

λ1/21

δ1/4 +K0

λ1C1δ

1/4 + CK0

λ1/21

ρ0 + CK0

λ1ρ1)δ7/4.

(51)Now, if K0 > 1, we define m4 such that, for m ≥ m4 the inequality

C(T, ρ1, η1, κ1)1ν

(κ0K0

λ1/21

δ1/4 +K0

λ1C1δ

1/4 + CK0

λ1/21

ρ0 + CK0

λ1ρ1)δ1/4 ≤ 1 (52)

holds. If K0 ≤ 1, then m4 will be chosen such that

C(T, ρ1, η1, κ1)1ν

(κ01

λ1/21

δ1/4 +1λ1

C1δ1/4 + C

1

λ1/21

ρ0 + C1λ1ρ1)δ1/4 ≤ 1. (53)

With this assumption we obtain, for m > m4 and t enough large,

∣∣∣(p − p1)

(t)∣∣∣ ≤ δ3/2. (54)

Now, we estimate the various norms of q − q1 :

∣∣∣ν∆ (q − q1

)∣∣∣ ≤ |QB (p − p1,u)| +∣∣∣QB

(q − q0,u

)∣∣∣ ++ |QB (p1,p − p1)| +

∣∣∣QB(p1,q − q0

)∣∣∣ ++

∣∣∣QB(q0,p − p1

)∣∣∣ +∣∣∣QB

(q0,q − q0

)∣∣∣ +∣∣∣dq

dt

∣∣∣ .(55)

As we did for∣∣∣q − q0

∣∣∣, we estimate one by one the terms from the rightside. For the first two, we use (11):

|QB (p − p1,u)| ≤ c4|p − p1|1/2 ‖p − p1‖1/2 ‖u‖1/2 |∆u|≤ c4δ

3/4λ1/41 δ1/2ρ1/2

1 ρ2 = c4λ1/41 ρ1/2

1 ρ2δ5/4;

∣∣∣QB(q − q0,u

)∣∣∣ ≤ c4

∣∣∣q − q0

∣∣∣ 12∥∥∥q − q0

∥∥∥ 12 ‖u‖ 1

2 |∆u| 12

= c4K0

λ3/41

ρ1/21 ρ1/2

2 δ5/4.


The following terms can be evaluated by using (7):∣∣∣QB

(p1,p − p1

)∣∣∣ ≤ c1|p1|1/2‖p1‖1/2∥∥∥p − p1

∥∥∥≤ c1η

1/20 η1/2

1 λ1/21 δ,

∣∣∣QB(p1,q − q0

)∣∣∣ ≤ c1 |p1| 12∣∣∣∆p1

∣∣∣ 12∥∥∥q − q0

∥∥∥≤ c1η

1/20 η1/2

2K0

λ1/21

δ,

∣∣∣QB(q0,p − p1

)∣∣∣ ≤ c1

∣∣∣q0

∣∣∣ 12∣∣∣∆q0

∣∣∣ 12 ‖p − p1‖

≤ c1κ1/20 κ1/2

2 λ1/21 δ3/2,

∣∣∣QB(q0,q − q0

)∣∣∣ ≤ c1

∣∣∣q0

∣∣∣ 12∣∣∣∆q0

∣∣∣ 12∥∥∥q − q0

∥∥∥≤ c1κ

1/20 κ1/2

2K0

λ1/21

δ3/2.

With (18) we obtain

∣∣∣∆ (q − q1

)∣∣∣ ≤ 1ν

c4λ1/41 ρ1/2

1 ρ2δ1/4 + c4

K0

λ3/41

ρ1/21 ρ1/2

2 δ1/4 + c1η1/20 η1/2

1 λ1/21 +

+c1η1/20 η1/2

2K0

λ1/21

+ c1κ1/20 κ1/2

2 λ1/21 δ1/2 + c1κ

1/20 κ1/2

2K0

λ1/21

δ1/2 + C′0

δ. (56)

We denote the coefficient of δ above by K1(= K1(m) since it depends on m).At this point we make a new assumption on m. That is, we consider a naturalnumber m5 such that K1(m5)δ(m5)1/4 < 1. It follows that K1(m)δ(m)1/4 < 1 forany m > m5, and for such an m,

∣∣∣∆ (q − q1

)∣∣∣ ≤ δ3/4,∥∥∥q − q1

∥∥∥ ≤ 1

λ1/21

δ5/4,∣∣∣q − q1

∣∣∣ ≤ 1λ1δ7/4. (57)

The arguments used to state the analyticity in time of q0 remain valid for q1and the following relation, that will be used later, follows

∣∣∣q ′ − q ′1∣∣∣ ≤ K′1δ

7/4.

By using (54) and (57) we now obtain

∣∣∣u − u1

∣∣∣ ≤ (1 +1λ1δ1/4)δ3/2. (58)


3. The induction step. We assume that, for every 0 ≤ j ≤ k + 1 the inequali-ties

∣∣∣p − p j

∣∣∣ ≤ δ5/4+ j/4,∣∣∣∆(q − q j)∣∣∣ ≤ δ1/2+ j/4,

hold. We prove that similar inequalities hold also for j = k + 2 :

(p − pk+2

)(t) = e−νtA

(p − pk+2

)(0)−

−∫ t

0e−ν(t−s)A

PB(p − pk+2,u

)+ PB

(pk+2 + qk+1,p − pk+2

)ds−

−∫ t

0e−ν(t−s)A

PB(pk+2 + qk+1,q − qk+1

)+ PB

(q − qk+1,u

)ds.

As we did for∣∣∣(p − p1

)(t)

∣∣∣ , we obtain

∣∣∣(p − pk+2)

(t)∣∣∣ ≤

∫ t

0C(ρ1, η1, κ1) (t − s)−α e−

νλ2 (t−s)

∣∣∣(p − pk+2)(s)∣∣∣ ds+

+

∣∣∣∣∣∣∫ t

0e−ν(t−s)A

PB(pk+2 + qk+1,q − qk+1

)+ PB

(q − qk+1,p + q

)ds

∣∣∣∣∣∣ .

The already cited Gronwall-type Lemma of [11] implies

∣∣∣(p − pk+2)

(t)∣∣∣ ≤

≤ C(T, ρ1, η1, κ1)max0≤t≤T

∣∣∣∣∣∣∫ t

0e−ν(t−s)A

PB(pk+2 + qk+1,q − qk+1

)+ PB

(q − qk+1,u

)ds

∣∣∣∣∣∣ .

Remark. The coefficients C(ρ1, η1, κ1) and C(T, ρ1, η1, κ1) from the aboveinequalities are equal to the coefficients from the similar inequalities concern-ing

∣∣∣(p − p1)

(t)∣∣∣ .

We evaluate the coordinates of each term in the brackets after e−ν(t−s)A :

∣∣∣∣ B(qk+1,q − qk+1

)j,l

∣∣∣∣ ≤∣∣∣qk+1

∣∣∣∣∣∣∣A 1

2(q − qk+1

)∣∣∣∣ ≤ κ0δ1

λ1/21

δ1+(k+1)/4

=κ0

λ1/21

δ2+(k+1)/4,


∣∣∣∣ B(q − qk+1,q

)j,l

∣∣∣∣ ≤∣∣∣q − qk+1

∣∣∣∣∣∣∣A 1

2 q∣∣∣∣ 1λ1δ3/2+(k+1)/4C1δ

12

=1λ1

C1δ2+(k+1)/4,

∣∣∣∣ B(pk+2,q − qk+1

)j,l

∣∣∣∣ ≤∣∣∣∣A 1

2(q − qk+1

)∣∣∣∣(∣∣∣∣(I − Pm− j

)pk+2

∣∣∣∣ + |(I − Pm−l) pk+2|)

≤ 1

λ1/21

δ1+(k+1)/4η0

(1/λm− j+1 + 1/λm−l+1

),

∣∣∣∣ B(q − qk+1,p

)j,l

∣∣∣∣ ≤ 1λ1ρ1δ

3/2+(k+1)/4(1/λ

12m− j+1 + 1/λ

12m−l+1

).

The arguments used for the terms involved in∣∣∣(p − p1

)(t)

∣∣∣ lead to

∣∣∣∣∣∣∫ t

0e−ν(t−s)A

PB(qk+1,q − qk+1

)+ PB

(q − qk+1,q

)ds

∣∣∣∣∣∣ ≤1ν

(κ0

λ1/21

+C1

λ1)δ2+(k+1)/4,

∣∣∣∣∣∣∫ t

0e−ν(t−s)A

PB(pk+2,q − qk+1

)+ PB

(q − qk+1,p

)ds

∣∣∣∣∣∣ ≤Cν

(η0

λ1/21

+ρ1

λ1)δk/4+2.

By putting these results together, it follows

∣∣∣(p − pk+2)

(t)∣∣∣ ≤ C(T, ρ1, η1, κ1)

1ν

(κ0

λ1/21

δ1/4 +C1

λ1δ1/4 +

Cη0

λ1/21

+Cρ1

λ1)δ5/4+(k+3)/4.

(59)By comparing the above inequality to (52), (53), we see that for m > m5, theinequality

C(T, ρ1, η1, κ1)1ν

(κ0

λ1/21

δ1/4 +C1

λ1δ1/4 +

Cη0

λ1/21

+Cρ1

λ1)δ1/4 < 1

holds. Thus ∣∣∣(p − pk+2)

(t)∣∣∣ ≤ δ5/4+(k+2)/4

that confirms our induction hypothesis in what concerns pk.Now, for

∣∣∣ν∆ (q − qk+2

)(t)

∣∣∣ we have

ν∆(q − qk+2

)= QB(p + q) −QB(pk+2 + qk+1) + q′ − q′k,


thus∣∣∣ν∆ (

q − qk+2)∣∣∣ ≤ |QB (p − pk+2,u)| +

∣∣∣QB(q − qk+1,u

)∣∣∣ ++ |QB (pk+2,p − pk+2)| +

∣∣∣QB(pk+2,q − qk+1

)∣∣∣ ++

∣∣∣QB(qk+1,p − pk+2

)∣∣∣ +∣∣∣QB

(qk+1,q − qk+1

)∣∣∣ +∣∣∣q′ − q′k

∣∣∣ .(60)

By using the induction hypotheses, we obtain∣∣∣QB(p − pk+2,u)

∣∣∣ ≤ c4|p − pk+2|1/2∥∥∥p − pk+2

∥∥∥1/2ρ1/2

1 ρ1/22 ≤

≤ c4ρ1/21 ρ1/2

2 λ1/41 δ1+(k+2)/4,

∣∣∣QB(q − qk+1,u)∣∣∣ ≤ c4

∣∣∣q − qk+1

∣∣∣ 12∥∥∥q − qk+1

∥∥∥ 12 ‖u‖ 1

2 |∆u| 12

≤ c41

λ3/41

ρ1/21 ρ1/2

2 δ1+(k+2)/4,

∣∣∣QB(pk+2,p − pk+2)∣∣∣ ≤ c1|pk+2|1/2|∆pk+2|1/2

∥∥∥p − pk+2

∥∥∥≤ c1η

1/20 η1/2

2 λ1/21 δ1+(k+1)/4,

∣∣∣QB(pk+2,q − qk+1)∣∣∣ ≤ c1 |pk+2| 12

∣∣∣∆pk+2

∣∣∣ 12∥∥∥q − qk+1

∥∥∥≤ c1η

1/20 η1/2

21

λ1/21

δ1+(k+1)/4,

∣∣∣QB(qk+1,p − pk+2)∣∣∣ ≤ c1

∣∣∣qk+1

∣∣∣ 12∣∣∣∆qk+1

∣∣∣ 12∥∥∥p − pk+2

∥∥∥≤ c1κ

1/20 κ1/2

2 λ1/21 δ3/2+(k+1)/4,

∣∣∣QB(qk+1,q − qk+1

)∣∣∣ ≤ c1

∣∣∣qk+1

∣∣∣ 12∣∣∣∆qk+1

∣∣∣ 12∥∥∥q − qk+1

∥∥∥≤ c1κ

1/20 κ1/2

21

λ1/21

δ2+(k+1)/4.

In what concerns |q′ − q ′k|, we remind the observation from the proof ofTheorem 6.1, that the complex domain of analyticity of p j must be the samefor every j ≥ 2, since the norms of q j have upper bounds that do not de-pend on j. Then, we obtain that the domain of analyticity of q j must also beindependent of j. Hence we may obtain estimates of the form

|q′ − q ′j| ≤ K′δ3/2+ j/4,

with K′ independent of j.


∣∣∣∆ (q − qk+2

)∣∣∣ ≤ 1ν

(c4ρ1/21 ρ1/2

2 λ1/41 δ1/4 + c4

1

λ3/41

ρ1/21 ρ1/2

2 δ1/4 +

+ c1η1/20 η1/2

2 λ1/21 + c1η

1/20 η1/2

21

λ1/21

+ c1κ1/20 κ1/2

2 λ1/21 δ1/2 +

+ c1κ1/20 κ1/2

21

λ1/21

δ + K′δ1/4)δ3/4+(k+2)/4.

We denote by K the coefficient of δ3/4+(k+2)/4 above. By comparing the RHS ofthe above inequality with that of (56), we find that, for m > m5, the inequalityKδ1/4 < 1 holds, thus

∣∣∣∆ (q − qk+2

)∣∣∣ ≤ δ1/2+(k+2)/4,

from where∥∥∥q − qk+2

∥∥∥ ≤ 1

λ1/21

δ1+(k+2)/4,∣∣∣q − qk+2

∣∣∣ ≤ 1λ1δ3/2+(k+2)/4.

This will imply also |q′ − q ′k+2| ≤ K′δ3/2+(k+2)/4

From (59) and the above estimates it follows that

∣∣∣u − uk+2

∣∣∣ ≤ (1 +1λ1δ1/4)δ5/4+(k+2)/4. (61)

The inequality above shows that the sequence of functions uk defined in Sec-tion 5 is strongly convergent in the norm of H to the exact solution u. We alsohave

‖u − uk‖ ≤ ‖p − pk‖ + ‖q − qk‖ ≤ λ1/2n+1|p − pk| + λ−1/2

n+1 |∆(q − qk|)≤ λ1/2

1 δ3/4+k/4 +1

λ1/21

δ1+k/4,

that proves that uk → u when k → ∞, in the norm of V. By setting M := m5,the conclusions of the theorem follow.

Remarks. 1. We could have obtained estimates of the form∣∣∣(p − pk

)(t)

∣∣∣ ≤ Ck(T )δ5/4+k/2,∣∣∣q − qk

∣∣∣ ≤ Ck(T )δ3/2+k/2

where Ck(T ), Ck(T ) depend increasingly on k and T . We had to ”sacrifice”,at every level, a δ1/4 factor in order to obtain estimates with coefficients thatdo not depend on k and on T .


2. The value of M in the above theorem might be very large. With the priceof having an even smaller exponent of δ in the previous error estimates, wecould replace δ1/4 with δ1/2 or δ3/4 in the conditions that define m3, m4, m5.This will lead to smaller values of M, thus of the dimension of HM.

3. The fact that M is large is not convenient, since one advantage of themodified Galerkin methods is that the dimension of the space Hm is smallerthan that used in the classical Galerkin method. However, the above theoremis just a result of convergence of the method, and its performances depend onthe method of proof. Whether or not the R-APP Galerkin method works wellfor a small dimension projector P is to be proved by numerical experiments.

7. COMMENTS ON THE METHOD

1. We conceived the R-APP Galerkin method in order to bring simplifica-tions to the NL Galerkin methods that use high-accuracy a.i.m.s. Hence, themethod is meaningful only if we use at least three levels (k ≥ 2).

The NL Galerkin method based on the use of high accuracy a.i.m.s [3],[15], applied to the Navier-Stokes problem and corresponding to our levelk, k ≥ 2, consists in solving the finite dimensional problem

dpdt− ν∆p+PB(p + Φk

(p)) = Pf, (61)

p (0) = Pu (0) ,

for the approximation p of p = Pu. Here Φk : PH → QH is the functiondefining an a.i.m. of high accuracy. The advantage of this method towardsours seems to be that the system of equations for p is integrated only once,while in our method several integrations are necessary. But the recursive def-inition of Φk (requiring the definitions of all Φ j with j < k) [3] makes thetotal volume of computations in the NL Galerkin method to be greater thanthat of our method. The simplification occurs in our method from the factthat q′k−2 (from the definition of qk) may be approximated by the numericalderivative (qk−2(t) − qk−2(t − h)) /h (since we have already computed qk−2(t)at every time step). This must be compared with the definitions of z′j,m in [3]or q1

j in [15].This simplification is present in the postprocessing step of every level,

hence also in the postprocessing of the last level, k. Thus, if we compareR-APP Galerkin method to the NL PP Galerkin method that uses high levelof accuracy a.i.m.s [15], we see that, because of its simpler definition, ourmethod comprises less computations than the other in the part that corre-


sponds to the NL Galerkin method (that is from level 0 up to the determi-nation of pk) as well as in the part that corresponds to the postprocessingmethod (that is the computation of qk and its summing to pk).

Also, the structure of our method, with iterative levels, makes the compu-tations easier to program, since the programs for the numerical integrationof the systems of ODEs for pk(t) should have the same structure for all k,only the coordinates of qk−1(t) remaining to be replaced in the nonlinear term.Moreover, each level represents a certain approximation of the solution, sowe can enjoy partial results.

2. The memory of the computer is better organized in our method, sinceat the beginning of the computations for the level k, we may erase from thememory the values of p j(t), j < k and of ql(t), l < k − 2, and keep only thoseof qk−1(t), qk−2(t). This feature may be improved by taking q ′k−1 instead ofq ′k−2, in the definition of qk. This will not affect the accuracy of the methodand will allow us to keep in the memory of the computer, at level k, only thepreviously computed values of qk−1 and erase all other intermediate functions.A similar procedure is not possible in the NL Galerkin method, where thedefinition of Φk requires at every step of the numerical integration the appealof all Φ j, j < k [3], [15].

3. As in all modified Galerkin methods, problems appear due to f. If thisfunction has a infinity of nonzero coefficients in its Fourier series, it will gen-erate a infinite number of non-zero coordinates in q j(t). A truncation criterionmust be applied and it will depend on f. Thus the number of coordinates ofq j(t) to be computed depends on j and on the given function f.

References

[1] P. Constantin, C. Foias, Navier-Stokes Equations, Chicago Lectures in Math., Univ. of ChicagoPress, IL, 1988;

[2] A. Debussche, M. Marion, On the construction of families of approximate inertial manifolds, J.Diff. Eqns., 100(1992), 173-201;

[3] C. Devulder, M. Marion, A class of numerical algorithms for large time integration: the nonlin-ear Galerkin methods, SIAM J. Numer. Anal., 29(1992), 462-483;

[4] C. Devulder, M. Marion, E.S.Titi, On the rate of convergence of Nonlinear Galerkin methods,Math. Comp. 60(1993), 495-515;

[5] C. Foias, O.Manley, R.Temam, Modelling of the interactions of the small and large eddies intwo dimensional turbulent flows, Math. Modelling and Num. Anal., 22(1988), 93-114, ;

[6] C. Foias, G. R. Sell, R. Temam, Varietes inertielles des equations differentielles, C.R. Acad. Sci.,Ser.I, 301(1985), 139-141;

[7] C. Foias, G. R. Sell, E. S. Titi, Exponential tracking and approximation of inertial manifolds fordissipative nonlinear equations, Journal of Differential Equations, 1(1989), 199-244;


[8] B. Garcia-Archilla, J. Novo, E.S. Titi, Postprocessing the Galerkin method: a novel approach toapproximate inertial manifolds, SIAM J. Numer. Anal., 35(1998), 941-972;

[9] B. Garcia-Archilla, Julia Novo, E.S. Titi, An approximate inertial manifolds approach to post-processing the Galerkin method for the Navier-Stokes equation, Mathematics of Computation,68(1999), 893-911;

[10] A. Georgescu, Hydrodynamic stability theory, Martinus Nijhoff Publ., Dordrecht, 1985;

[11] D. Henry, Geometric theory of semilinear parabolic equations, Springer, Berlin, 1991;

[12] A.-V. Ion, Improvement of an inequality for the solution of the two dimensional Navier-Stokesequations, ROMAI Journal, 4, 1(2008), 119-130.

[13] M. Marion, Approximate inertial manifolds for reaction-diffusion equations in high space di-mension, J. Dynamics Differential Equations, 1(1989), 245-267;

[14] M. Marion, R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26(1989), 1139-1157;

[15] J. Novo, E.S. Titi, S. Wynne, Efficient methods using high accuracy approximate inertial mani-folds, Numer. Math., 87(2001), 523-554;

[16] J. C.Robinson, Infinite-dimensional dynamical systems; An introduction to dissipative parabolicPDEs and the theory of global attractors, Cambridge University Press, 2001;

[17] R. Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Reg. Conf.Ser. in Appl. Math., SIAM, 1995;

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FORCING A CONTROLLED DIFFUSION PROCESSTO LEAVE THROUGH THE RIGHT END OFAN INTERVAL

ROMAI J., 6, 2(2010), 181–184

Mario LefebvreDepartment of Mathematics and Industrial Engineering,Ecole Polytechnique, Montreal, [email protected]

Abstract Let X(t), t ≥ 0 be a one-dimensional controlled diffusion process evolving in the in-terval [c, d]. We consider the problem of finding the control that minimizes the math-ematical expectation of a cost function with quadratic control costs on the way and aterminal cost function that is infinite if the process hits c before d. The optimal controlis obtained explicitly and particular cases are presented.

Keywords: optimal stochastic control, LQG homing, Brownian motion, first exit time, Kolmogorovbackward equation.2000 MSC: 93E20.

1. INTRODUCTIONLet X(t), t ≥ 0 be a one-dimensional controlled diffusion process defined by the

stochastic differential equation

dX(t) = m[X(t)] dt + b[X(t)] u(t) dt + v[X(t)]1/2 dB(t), (1)

where B(t), t ≥ 0 is a standard Brownian motion, u(t) is the control variable andb(·) , 0.

We define the random variable T (x) by

T (x) = inft > 0 : X(t) = c or d | X(0) = x.Our aim is to determine the value of the control u∗(t) that minimizes the expectedvalue of the cost function

J(x) =

∫ T (x)

0

12 q(x) u2(t) dt + K[X(T ), T ],

where q is a positive function and K is the termination cost function.Next, let x(t), t ≥ 0 be the uncontrolled process obtained by setting u(t) ≡ 0 in

(1), and let τ be the same as T , but for x(t), t ≥ 0. If the condition

P[τ(x) < ∞] = 1 (2)

181

182 Mario Lefebvre

holds, and if the functions b, v and q are such that

b2(x)q(x) v(x)

≡ α > 0, (3)

then making use of a result proved by Whittle (see [2], p. 289), we can state that theoptimal control u∗ [= u∗(0)] is given by

u∗ =v(x)b(x)

G′(x)G(x)

, (4)

whereG(x) := E

[exp −αK[x(τ), τ] | x(0) = x

].

In Lefebvre [1], the author solved the problem of forcing the controlled processX(t), t ≥ 0 to stay in the continuation region C := (−∞, d) until a fixed time t0 bygiving an infinite penalty if T1 < t0, where

T1(x) = inft > 0 : X(t) = d | X(0) = x < d.

In the present paper, we consider the controlled process X(t), t ≥ 0 in the interval[c, d]. We want the process to leave the continuation region through its right end. Wewill take

K[X(T ),T ] = K[X(T )],

where the function K is such that K(c) = ∞ and K(d) ∈ R. That is, we give an infinitepenalty if X(t) reaches c (before d). The constant d can be chosen as large as we want.The larger it is, the longer it will take X(t) to attain this value.

By giving an infinite penalty if the final value of X(t) is equal to c, we force theprocess to avoid this boundary. We assume that there are no constraints on the controlvariable u(t). In the next section, we will obtain an explicit formula for the optimalcontrol u∗, and we will present some particular cases.

2. OPTIMAL CONTROLLet

πd(x) := P[x(τ) = d | x(0) = x].

The function πd satisfies the Kolmogorov backward equation

v(x)2

π′′d (x) + m(x) π′d(x) = 0,

and is subject to the boundary conditions

πd(c) = 0 and πd(d) = 1.

Forcing a controlled diffusion process to leave through the right end of an interval 183

We easily find that

πd(x) =

∫ xc exp

∫ uc −

2 m(s)v(s) ds

du

∫ dc exp

∫ uc −

2 m(s)v(s) ds

du. (5)

We will prove the following proposition.

Proposition 2.1. Assume that the conditions (2) and (3) are satisfied, and that thetermination cost function is K[X(T ),T ] = K[X(T )], with K(c) = ∞ and K(d) ∈ R.Then, the optimal control is given by

u∗ =v(x)b(x)

exp∫ x

c −2 m(u)v(u) du

∫ x

c exp∫ u

c −2 m(s)v(s) ds

du

for c < x < d. (6)

Proof. We can write that P[x(τ) = c | x(0) = x] = 1 − πd(x). Hence, we deduce fromWhittle’s result that u∗ is given by (4), with

G(x) = E[exp−αK[x(τ)] | x(0) = x

]

= e−αK(c) [1 − πd(x)] + e−αK(d) πd(x).

Since K(c) = ∞, we obtain that

G(x) = e−αK(d) πd(x),

so thatG′(x) = e−αK(d) π′d(x).

Hence, the optimal solution (6) follows at once from (5).

Remarks. i) Because the interval [c, d] is bounded, the condition (2) is not restrictive.Furthermore, when b, q and v are all constant functions, then the condition (3) isautomatically fulfilled.ii) We see that the optimal control does not depend on the value of K(d).iii) In many applications, we would like to take c = 0. If the uncontrolled processx(t), t ≥ 0 can attain the boundary at the origin, then we can indeed replace c by 0in (6).Particular cases.I) First, if m(x) ≡ 0, then

πd(x) =x − cd − c

andu∗ =

v(x)b(x)

1x − c

for c < x < d.

Notice that this case includes the (controlled) standard Brownian motion, for whichm(x) ≡ 0 and v(x) ≡ 1.

184 Mario Lefebvre

II) Next, assume that q(x) ∝ b2(x). With m(x) ≡ m0 , 0 and v(x) ≡ v0 > 0, so thatthe uncontrolled process x(t), t ≥ 0 is a Wiener process with drift coefficient m0 anddiffusion coefficient v0, we find that

u∗ =2 m0

b(x)1

exp

2 m0v0

(x − c)− 1

for c < x < d.

III) Finally, if x(t), t ≥ 0 is a geometric Brownian motion defined by x(t) = eB(t),then m(x) = x/2 and v(x) = x2. The origin being a natural boundary for the geometricBrownian motion, we must take c > 0. The optimal control takes the form

u∗ =x

b(x)1

ln(x/c)for 0 < c < x < d.

Here, q(x) must be proportional to b2(x)/x2.

3. CONCLUSIONBased on the work presented in Lefebvre [1], we have solved the problem of opti-

mally controlling a general diffusion process so that it leaves the continuation region(c, d) through the right-hand side of the interval. The objective could have been toleave through the left-hand side instead. Moreover, we could assume that d = ∞. Inthat case, we could try to maximize the time spent by the controlled process in theinterval (c,∞).

Finally, the same type of problem as the one solved here could be considered intwo dimensions. For example, (X1(t), X2(t)) could be a controlled two-dimensionalBrownian motion, and T be the first time that X1(t) hits the boundary x1 = c. If thetermination cost is a function of X2(T ), then we would have to determine the distri-bution of this variable, which is a continuous rather than discrete random variable.

References[1] M. Lefebvre, Forcing a stochastic process to stay in or to leave a given region, Ann. Appl.

Probab., 1(1991), 167-172.

[2] P. Whittle, Optimization over Time, Vol. I, Wiley, Chichester, 1982.

STUDIES AND APPLICATIONS OF ABSOLUTESTABILITY IN THE AUTOMATIC REGULATIONCASE OF THE NONLINEAR DYNAMICAL SYSTEMS

ROMAI J., 6, 2(2010), 185–198

Mircea Lupu1, Olivia Florea2,Ciprian Lupu3

1Faculty of Mathematics and Computer Science, Transilvania University of Brasov, RomaniaAcademy of Romanian Scientists (Corresponding Member), Bucharest, Romania2Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Romania3Faculty of Automatics and Computer Science, Politehnica University of Bucharest, [email protected], [email protected], [email protected]

Abstract In this paper the methods of study of the automatic regulation of the absolute stability forsome nonlinear dynamical systems are presented. Two methods for the absolute stabilityare specified: a) the Lurie method with the effective determination of the Lyapunovfunction; b) the frequencies method of the Romanian researcher V. M. Popov, that usesthe transfer function in the critical cases. The applications envisaged here refer to metalcutting tools machine. Both analytical and numerical aspects are considered.

Keywords: : nonlinear systems, automatic control system, absolute stability, tools machine.2000 MSC: 34D23.

1. INTRODUCTIONThe automatic regulation for the stability of dynamical systems occupies a fun-

damental position in science and technique, and it is applied to the optimization ofthe technological process of cutting tools, of robots, of vehicles (or some machinescomponents) movement regime, of energetic radioactive regimes, chemical, electro-magnetic, thermal, hidroaerodynamic regimes, etc.

The complex technical achievements lead to complex mathematical models forclosed circuits with input - output, following for the automatic regulation the inte-gration of some mechanisms and devices with inverse reaction of response for thecontrol and the fast and efficient elimination of the perturbations which can appearsalong these processes or dynamical regimes. Generally these dynamical regimes arenonlinear and some contributions and special achievements for automatic regulation,generating the automatic regulation of absolute stability (a.r.a.s.) for these classes ofnonliniarities were necessary.

We highlight two special methods of a.r.a.s.:

1 Lyapunov’s function method discovered by A.I. Lurie [10], [12], [17] anddeveloped into a series of studies by M.A Aizerman, V.A. Iacubovici, F.R.Gantmaher, R.E. Kalman, D.R. Merkin [11] and others [1] [14];

185

186 Mircea Lupu, Olivia Florea, Ciprian Lupu

2 Frequency method developed by researcher V. M. Popov [15] generalizingNyquist’s criterion, then developed in many studies [1], [2], [12].

We note the contributions of Romanian researchers recognized by the works andmonographs on the stability and optimal control theory: C. Corduneanu, A. Halanay,V. Barbu, Vl. Rasvan, V. Ionescu, M.E. Popescu, S. Chiriacescu, A. Georgescu andalso directly on a.r.a.s.: I. Dumitrache [4] D. Popescu 13], C. Belea [2], V. Rasvan[16], S. Chiriacescu [3] and other recent works [5] [6] [7] [8] [9].

The research has shown that both methods are equivalent, and studies can be per-formed qualitatively or numerically. In this paper we present the two methods andapply them to a concrete problem.

2. A.R.A.S. USING THE LYAPUNOV’SFUNCTION METHOD

In this part we’ll present the Lurie’s ideas and the effective method for finding theLyapunov’s function. [10] [11] [2] [16]

Generally, the systems of automatic regulation (s.r.a.) consist of the controlledprocessor system, sensor elements of measurement, acquisition board, and the mech-anism feedback controller. The regulator comprises all the sensors and the acquisitionboard, but the controller is included in the feedback mechanism. Parameters charac-terizing the object control system (to control the working mode) are measured bysensors, and their records with the sensor response mechanism ζ is transmitted to theacquisition board.

This processes the commandσ, which is mechanically transmitted to the controllerwhich, on its turn, distributes the object state and interact simultaneously adjustingthe response mechanism. We highlight below the dynamic system equations.

We denote by x1, x2, . . . , xn the state parameters of the regime’s subject which mustbe controlled, i.e. the coordinates and the sensorial speeds. We recall that the vari-ation of these parameters of the open circuit (excluding the controller) system is de-

scribed by linear differential equations with constant coefficients: xk =n∑

j=1ak jx j, k =

1, . . . , n. If the system is with closed loop, then the variables x1, x2, . . . , xn will beunder the influence of the regulation body, and we denote by ξ its state. In this casefor the autonomous closed system we have the equations:

xk =

n∑

j=1

ak jx j + bkξ, k = 1, . . . , n. (1)

The relation between the output ζ and the input ξ is

ζ = kξ. (2)

The acquisition board collects the signals and transmits them to the input sensorsin order to obtain the embedded system

Studies and applications of absolute stability in the automatic regulation case ... 187

σ =

n∑

j=1

c jx j − rξ, (3)

where c j, r are transfer numbers, r > 0 is the transfer coefficient of the inverse rigidconnection (the regulator characteristics) [10], [11], [12].

The connection between the output (linear) function σ of the controller and thenonlinear input ϕ in the case of automatic regulation is expressed by the relation

ξ = ϕ(σ). (4)

The characteristic function of the controller ϕ(σ) , σ ∈ (−∞,+∞) is continuousand satisfies the conditions [11], [6], [7]:

a) ϕ(0) = 0,b) σ · ϕ(σ) > 0, ∀σ , 0,c)

∫ ±∞0 ϕ(σ)dσ = ∞.

(5)

The graph of the function ϕ lies in the quarters I, III. The functions ϕ(σ) are namedadmissible. Moreover, we assume that the sector condition

0 <ϕ(σ)σ

< k (6)

is satisfied, where k is the amplification coefficient.

Example 2.1. 1 ϕ(σ) = sgn(σ) · ln(σ2 + 1), k > 1;

2 ϕ(σ) = a(eσ − 1), k ≤ a.

We assume that the n × n square matrix A =∥∥∥ak j

∥∥∥ is nonsingular. The equations(1), (3), (4) model the perturbed system with the zeros x(0, 0, . . . , 0), ξ = 0 .

By setting B =

b1· · ·bn

, C = ( c1 . . . cn ) , C’ the transpose matrix of C, this

system becomes

X = AX + Bξ, ξ = ϕ(σ), σ = C′X − rξ, X =

x1· · ·xn

. (7)

3. THE CANONICAL FORM AND THELYAPUNOV FUNCTION OF SYSTEM (7)

We assume that A with det A = ∆0 , 0 is Hurwitz, i.e. the characteristic polyno-mial

P(λ) = (−1)n det(A − λE) (8)


has simple roots with Re(λk) < 0, k = 1, . . . , n.The system (7) is brought to the canonical form if the matrix A is brought to the

Jordan form J = diagA =

λ1 0. . .

0 λn

. There is a matrix T = (tk j) such that

T−1AT = J, (⇔ AT = T J), det T , 0. (9)

With the linear transform:

X = TY, Y =

y1· · ·yn

, (10)

from (7) we obtain: TY = ATY+Bξ, ξ = ϕ(σ), σ = C′TY − rξ , that implies:

Y = JY + B1ξ, ξ = ϕ(σ), σ = C′1Y − rξ, B1 = T−1B, C′1 = C′T . (11)

Hence, with the linear transform:

Z = JY + B1ξ, σ = C′1Y − rξ, Z =

z1· · ·zn

, (12)

the problem (1),(4) becomes

Z = JZ + B1ϕ(σ),σ = C′1Z − rϕ(σ). (13)

The disturbed system (13) with the equilibrium solution ( zk = 0, σ = 0) is equiv-alent with system (7) with the equilibrium solution (xk = 0, ξ = 0) and the transform(12) is nondegenerate if the determinant ∆ is non-null:

∆ =

∣∣∣∣∣∣J B1

C′1 −r

∣∣∣∣∣∣ , 0 ⇔ r + C′1J−1B1 , 0 (14)

Returning to J−1 = T−1A−1T, B1 = T−1B, C′1 = C′T transforms we obtain from(13) the final condition

r + C′A−1B , 0. (15)

Lurie’s problem consists in establishing the conditions for the asymptotic stabilityof the the null solution (xk = 0, ξ = 0) of system (7) (equivalent with (13) ) withsolution zk = 0, σ = 0 for the initial perturbations and for any admissible functionsϕ(σ) defined in (5), (6).

The conditions imposed on the matrix A and on the function ϕ(σ) imply the abso-lute stability (a.s) of the system. [1] [13]


We remark that if ϕ(σ) is linear, since the matrix A is Hurwitz, the systems (7),respectively (13) are asymptotic stable. The simplicity of system (13) entails imme-diate techniques for determining the Lyapunov function V = V(z1, . . . , zn, σ) attachedto the system (13).

The function V(z, σ) of class C1 is a Lyapunov function for system (13) if V(z =

0, σ = 0) = 0, it is positive defined (V(z, σ) > 0), it is radially unlimited to ∞, withthe absolute derivative V = dV

dt negative defined (dVdt < 0) for (z , 0, σ , 0) in a

vicinity of the equilibrium point. For the case of automatic regulation we search thefunction V = V(z, σ) as the sum of a quadratic form zk corresponding to the linearblock A and an integral term corresponding to the non linear part

V(z, σ) = Z′PZ +

∫ σ

0ϕ(σ)dσ = V1(z, σ) +

∫ σ

0ϕ(σ)dσ. (16)

From theory [1] [4] Z′PZ is the quadratic form defined strictly positive if the ma-trix P is symmetric (P = P′) and we have A′P + PA = −Q where Q is symmetric andpositive (with positive eigenvalues).

The integral term from (15) is strictly positive because of the conditions (5). Ob-viously, V(z = 0, σ = 0) = 0.

Next we impose the condition V < 0 and obtain conditions for parameters ck, r fora.r.a.s..

From (16), by using (13) and

Q = Q′, P = P′, B′1PZ + Z′PB1 = B′1PZ + (PB1)′Z = 2(PB1)′Z,

for

dV(z, σ)dt

= Z′(J′P + PJ)Z − rϕ2(σ) + ϕ(σ)(B′1PZ + Z′PB1) + ϕ(σ)C1,

we obtain

dVdt

= −Z′QZ − rϕ2(σ) + 2ϕ(σ)(PB1 +

12

C1

)′Z, (17)

V(z = 0, σ = 0) = 0.

The connection between the matrices P(pi j), Q(qi j) can be established. From λi +

λ j , 0, i, j = 1, . . . , n, P = P′, J = diagA and Q = Q′, we have qi j = −(λi pi j + λ j pi j

)that implies

pi j = − qi j

λi + λ j. (18)

Remark 3.1. The matrix A is stable with λi +λ j , 0 if Q is a quadratic form positivedefined.


Example 3.1. If we choose Q = E (E - the unit matrix) and P is obtained from (17),than the below observation is valid.

We must have (−V) positive defined (since V < 0 ). Apply to the RHS of (17) theSilvester criterion demanding that all diagonal minors of (17) to be positive. BecauseQ is positive as quadratic form, than the first n inequalities are satisfied and the lastinequality is

r >(PB1 +

12

C1

)/Q−1

(PB1 +

12

C1

). (19)

If the regulator parameters verify the conditions (15), (19) there are sufficient con-ditions for the asymptotic stability of the solution (x = 0, ξ = 0) of the system (1),(3), (4) [10] [16].

Remark 3.2. A choice technique of the quadratic form V1(z) for pi j according toLurie is:

V1(z) = ε

s∑

k=1

z2k−1z2k +ε

2

n−2s∑

k=1

z22s+k −

n∑

k=1

n∑

j=1

akzka jz j

λk + λ j

where a1, a2, . . . , a2s are complex conjugated, a2s+1, . . . , an are real corresponding toroots λk determining the coefficients ak .

Remark 3.3. The two transforms for the diagonal system (1), (3), (4) to obtain (13)can be replaced directly by the transform [12]

xk = −n∑

i=1

Nk(λi)D′(λi)

zi, (12′)

where from (7) P(λ) = (−1)nD(λ),Nk(λ) =n∑

i=1biDik(λ) , Dik are the corresponding

algebraic complements of (i, k) from D(λ) = A−λE. In this case the simplified systemanalogous to (13) is:

zk = λkzk + ϕ(λ), σ =

n∑

i=1

fizi − rϕ(σ), k = 1, ..., n (13′)

for which we will build easier V(z, ϕ) .

For the case when a root is null (P(0) = 0) and all the others have

Re(λk) < 0, k = 1, ..., n − 1, the system (13), with Z =

(zz1

), becomes:

˙z = JZ + B1ϕ, z1 = b0ϕ, ε = C′1Z + C0z1 − rϕ, (13′′)


where J is a (n− 1)× (n− 1) matrix, Z, B1 are (n− 1, 1) (column) matrices, while C1is a (1, n − 1) (row) matrix.

In this case the Lyapunov function is searched in the form

V(z, z1, σ) = az21 +

z′Pz +

∫ σ

0ϕ(σ)dσ

(16′).

For proofs and recent applications we recommend [2][12][11].

4. THE FREQUENCY METHOD FOR A.R.A.S.This method obtained by V. M. Popov [15] is applied to the dynamical system with

continuous nonlinearities. We present in this section the method and criteria given byAizerman, Kalman, Jakubovici [16] [11].

Consider the dynamical, autonomous, non homogeneous system

xi =n∑

l=1ailxl + biu, i = 1, ..., n,

σ =n∑

l=1clxl, u = −ϕ(σ),

(20)

where ail, bi, cl are real constants, u is the arbitrary function of input, continuous,nonlinear with ϕ(σ) and σ is the output function.

By using the Laplace transform, and by replacing the operator ddt with s we obtain

from (20):

sxi =

n∑

l=1

ailxl + biu, σ =

n∑

l=1

clxl, i = 1, ..., n. (21)

Eliminating from (21) the characteristic parameters of the regulator we obtain

σ = W(s)u, σ = W(s)(−ϕ), (22)

where W(s) =Qm(s)Qn(s) is the transfer function and Q(s) are polynomials of degrees m,

respectively n, with m < n [4] [5] [13].The transfer function connects σ and ϕ; the function ϕ satisfies the conditions (5)

and the sector condition (6) 0 < ϕ(σ)σ < k ≤ ∞ - the plot ϕ = ϕ(σ) in the plane (σ, ϕ)

will be the sector 0 ≤ ϕ(σ) ≤ kσ. The sector condition and the nonlinearity of ϕdetermine the system (σ, ϕ) with closed loop through the impulse function ϕ.

We study the absolute stability of the null solution (x = 0, u = 0) of system (20).Because the system is closed and nonlinear we can’t apply directly the Nyquist cri-terion, [4] [5] [15]. If ϕ ≡ kσ then the system is linear and this criterion can beapplied.

Since the block∑

ailxl is linear and biu is nonlinear it results that the roots of thecharacteristic polynomial P(λ) = (−1)(A − λE) = 0, P(λi) = 0, the poles of W(s)and k will influence the determination of the absolute stability criteria.


From W(s = jω) = U(ω) + jV(ω), j =√−1 we have the hodograph for the axes

(U,V) [2] [4] [5] [6] [12]:

U = U(ω), V = V(ω), 0 ≤ ω ≤ ∞. (23)

If all poles of W(s) have Re(si) < 0 then the system is uncritical; if some of thepoles of W(s) are null or on the imaginary axis and the rest have Re(si) < 0 then thesystem is in the critical case.

We enunciate the criteria for absolute stability of automatic control a.r.a.s. by thefrequency method.

Criterion 1. (the uncritical case). Assume that the following conditions are satisfiedfor system (20):

a) The function ϕ(σ) satisfy (5), (6),

b) All poles of W(s) have Re(si) < 0 ,

c) There is a q ∈ R such that for any ω ≥ 0 the condition

1k

+ Re[(1 + jωq) W( jω)

] ≥ 0 (24)

holds.Then the system (20) is automatic regulated and absolute stable for the null solu-

tion (x = 0, u = 0).

From (24) we obtain

1k

+ U(ω) − qωV(ω) ≥ 0. (24′)

From a geometric point of view, criterion (24) shows that in the plane U1 =

U, V1 = ωV there exists the line

1k

+ U1 − qV1 = 0 (24′′)

passing through(− 1

k , 0)

such that the plot of the hodograph is under this line forω ≥ 0, k > 0 .

Criterion 2. (the critical case when there is a simple null pole s0 = 0 ). Assume thatthe following conditions are satisfied:

a) The function ϕ verify (5), (6),b) W(s) has a simple null pole, and the others poles si have Re(si) < 0,


c) ρ = lims→0

sW(s) > 0 and there is a q ∈ R such that for any ω ≥ 0 condition (24)

holds. Then for the system (20) for the null solution we have a.r.a.s.

Criterion 3. (the critical case when s = 0 is a double pole). Assume that the follow-ing conditions are satisfied:

a) The function ϕ(σ) verify (5), (6) and the sector condition for k = ∞,b) W(s) has a double pole in s=0 and the other poles have Re(si) < 0,c) ρ = lim

s→0s2W(s) > 0 , µ = lim

s→0dds

[s2W(s)

]> 0 and π(ω) = ωImW( jω) < 0 for

∀ω ≥ 0.Then for the system (20) we have a.r.a.s. for the null solution.

Remark 4.1. The form of these criteria (1, 2, 3) has an analytical character but theirverification for the construction of hodograph values of the coefficients must be donenumerically. For special cases we recommend the monographs [2] [4] [12] [16].

5. THE ABSOLUTE STABILITY IN THEAUTOMATIC REGULATION OF METALCUTTING

The high precision of the metal cutting tools implies an automatic regulation ofthe processes.

In this section we present the original results for modeling and the for the studyof the nonlinear dynamics of the cutting processes (CP) with tools into metal blocks,for composite materials, blocks or hardwood, practically defined in [3].

These (CP) are: CP of drilling, CP of milling, CP of grinding, screw machine,spindle bearing. Machine tool bar is provided with an inner elastic hard metal cut-ting, cutting inside to run the required geometric rotation and advancing to step slow.Because of the variation in hardness, density, coefficient of elasticity, material compo-sition manufactured by the process disturbances will occur in work mode: transversevibration due to shaft rotation or longitudinal vibrations to advance. The automaticcontroller is equipped with sensors, micrometers, tensiometers, rigid response mech-anisms of signals output power amplifiers and accelerators. Their purpose is to adjustthe characteristics to obtain asymptotic stability of the system work, resulting in highprecision components. We apply the two methods described in the previous sections.

5.1. THE A.R.A.S. METHOD BY LYAPUNOVFUNCTION

Consider the dynamic system modeled mathematically, brought to a canonical au-tonomous form, that features automatic adjustment for absolute stability of dynamiccutting machining processes. [3] [14]


x1 = a11x1 + b1ξx2 = a23x3x3 = a31x1 + a32x2 + a33x3x4 = a44x4 + b4ξ

σ = c2x2 + c4x4 − rξ, ξ = ϕ(σ) (25)

where ai j, bi, r, ξ are constants i, j = 1, 2, 3, 4, and

a11 = −m < 0, a31 = n > 0, a32 = −εn < 0, a33 = −p < 0, a44 = −l < 0a23 = 1, c2 = 1, c4 = c < 0, b1 = b > 0, b4 = d − r > 0, r > 0. (26)

These represent mass inertia, elastic constants, strain or pressure coefficients, andσ, r, ξ are the characteristics of the controller. We assume that the input function ϕis generally nonlinear and the conditions (5) (6) hold. We observe that the linearresponse function σ of the controller evaluate the elements x2 - the speed of rotationof the cutting bar and x4 - the speed of advancing its material, [3].

We check the absolute stability of the zero solution of the system (x = 0, ξ = 0) .We have det A , 0. Moreover, Re(λi) < 0, i = 1, 2, 3, 4, as the following computationsshow:

P(λ) = D(λ) =

∣∣∣∣∣∣∣∣∣∣∣

a11 − λ 0 0 00 −λ a23 0

a31 a32 a33 − λ 00 0 0 a44 − λ

∣∣∣∣∣∣∣∣∣∣∣=

= (a11 − λ)(a44 − λ)(λ2 − λa33 − a23a32) = 0,

(27)

λ1 = a11 = −m < 0, λ4 = a44 = −l < 0, (27)

λ2,3 =12

(−p ±

√p2 − 4εn

)< 0, λi ∈ R

In this case, following the diagonalization method (Section 3) with the formulas(9) - (13) or by directly choosing the alternative from Remark 3.3 we get the diagonalsystem in zi and σ (12’), (13’):

zi = λizi + ϕ(σ); σ =

4∑

i=1

fizi − rϕ(σ), i = 1, ..., 4, (29)

f1 =b1a31

(λ1 − λ2)(λ1 − λ3), f2 =

−b1a31

(λ1 − λ2)(λ2 − λ3)(29)

f3 =b1a31

(λ1 − λ3)(λ2 − λ3), f4 = b4c4 < 0.


We observe that f1 + f2 + f3 = 0 and whatever is the choice of order quantities,λ1, λ2, λ3 are strictly negative, and always two of the functions fi, i = 1, 2, 3 have thesame sign and the third function takes opposite sign using relation (26).

In this case, if f4 < 0 we can construct such Lyapunov function:

V(z, σ) = −12

f4z24 −

12

3∑

i=1

a2i z2

i

λi−

3∑

i=1

3∑

j=1

aia j

λi + λ jziz j +

σ∫

0

ϕ(σ)dσ (31)

where the real coefficients a1, a2, a3 will be determined.From λi < 0, λi +λ j < 0, V(0) = 0 , the terms after i = 1, 2, 3 determine a positive

definite quadratic form and since the integral is positive, we have V(z, σ) > 0 in aneighborhood of the null solution.

We calculate V(z, σ) for the function V defined above:

V(z, σ) = − f4λ4z24 − (a1z1 + a2z2 + a3z3)2 − ϕ

3∑

i=1

zi

a2

i

λi+

∑

j,i

2aia j

λi + λ j− fi

. (32)

We remark that V(z = 0, σ = 0) = 0 and in order to have the strict negativity, allthe parentheses from the sum that multiplies ϕ must be null, that is

a2i

λi+

∑

j,i

2aia j

λi + λ j− fi = 0, i = 1, 2, 3 (33)

The system (33) comprises three equations Fi(a1, a2, a3) = 0, i = 1, 2, 3 with threeunknowns and the local existence of solutions is ensured by the condition on thesystem Jacobian: J =

D(F1,F2,F3)D(a1,a2,a3) , 0. If each equation from (33) is multiplied

respectively by 1λi

and all equations are summed, then condition (33) proves to beequivalent to

S :=3∑

i=1

fiλi

=

3∑

i=1

ai

λi

2

> 0. (34)

This condition indicates that the known sum (S) is strictly positive and in the para-

metric space (a1, a2, a3) the plane (π12)3∑

i=1

aiλi

= ±√S where exists a solution, does

not admit the null solution because fi , 0.The Jacobian J can be shown to be

J = ±8√

S (a1 + a2 + a3)[a1(λ2 + λ3) + a2(λ1 + λ3) + a3(λ1 + λ2)]λ1λ2λ3(λ1 + λ2)(λ1 + λ3)(λ2 + λ3)

, 0.


The condition above shows that the solution is contained in the plane (π12), but notin the intersections of this plane with the planes a1 + a2 + a3 = 0 or a1(λ2 + λ3) +

a2(λ1 + λ3) + a3(λ1 + λ2) = 0.Analyzing the system (33) by the sign of fi, we see that a1, a2, a3 can not have all

the same sign. If this would happen, then fi > 0. We proved that there exist solutionsof system (33) and, for these, V < 0 ; it follows that the Lyapunov function providethe automatic regulation of the absolute stability. For this application sufficient con-ditions of type (15), (16) with numerical data, are also obtained.

5.2. THE FREQUENCY METHOD FOR A.R.A.SIn the following study the frequency method presented in Section 4 will be applied

to problem (25), (26). Because the system (25) is equivalent with (29) the functionu = −ϕ(s) satisfies the sector conditions. By applying the Laplace transform, thetransfer function W(s) is found. So, from (29) is obtained, for σ = W(s)(−ϕ)

szi = λizi + ϕ; sσ =

4∑

i=1

fizi − rϕ. (35)

Eliminating zi from (35) we obtain the transfer function from σ = W(s)(−ϕ)

W(s) =1s

r −4∑

i=1

λi fis − λi

. (36)

Because the real roots si = λi satisfy Re(λi) < 0, the transfer function has a simplepole in s = 0 and the rest of real roots with Re(si) < 0. In this case we may usethe Criterion 2 of critical singularity from Section 4 for a.r.a.s.. Here, the conditions(15), (19) and II a), b) were verified in the preceding subsection and only conditionc) must be verified.

ρ = lims→0

sW(s) = r +4∑

i=1fi = r + f4 = r + b4c4 > 0 implies r + c(d − r) > 0 that means

r > cd1−c > 0, d < 0 . From W(s = jω) = U(ω) + jV(ω) , we have:

U(ω) =

4∑

i=1

fiλi

λ2i + ω2

, V(ω) = − rω

+1ω

4∑

i=1

λ2i fi

λ2i + ω2

.

For given k > 0, from the condition 0 < ϕ(s) < kσ with ϕ(σ) specified, someq ∈ R verifying the condition (24’) may be determined. The parameters λi, fi areknown from (28), (30), the nonlinear function ϕ is chosen with σ from (25) andfor specified numerical k and, consequently, the delimitation of q are determined.The existence of these conditions can be checked hodographically for a.r.a.s. at thisapplication.


6. CONCLUSIONIn the paper, two methods for a.r.a.s. very useful in the fundamental and applicative

research, are presented. Their use is exemplified in the application in Section 5. Forother studies the published results of the researchers, the works [11] [12] [16] [17]are recommended.

7. ACKNOWLEDGEMENTThis work was supported by CNCSIS - IDEI Program of Romanian Research, De-

velopment and Integration National Plan II, Grant no. 1044/2007 and ”Automatics,Process Control and Computers” Research Center from University ”Politehnica” ofBucharest (U.P.B.-A.C.P.C.) projects.

References[1] E. A. Barbasin, Liapunov’s function, Nauka, Moscov, 1970 (in Russian)

[2] C. Belea, The system theory - nonlinear system, Ed. Did. Ped., Bucuresti, 1985 (in Romanian)

[3] S. Chiriacescu, Stability in the Dynamics of Metal Cutting, Ed. Elsevier, 1990; Dynamics ofcutting machines, Ed. Tehnica, Bucuresti, 2004 (in Romanian)

[4] I. Dumitrache, The engineering of automatic regulation, Ed. Politehnica Press, Bucuresti, 2005.

[5] C. Lupu &co., Industrial process control systems, Ed. Printech, Bucuresti, 2004, (in Romanian).

[6] C. Lupu &co., Practical solution for nonlinear process control, Ed. Politehnica Press, Bucuresti,2010 (in Romanian)

[7] M. Lupu, O. Florea, C. Lupu, Theoretical and practical methods regarding the absorbitors ofoscillations and the multi-model automatic regulatin of systems, Proceedings of The 5th Interna-tional Conference Dynamical Systems and Applications, Ovidius University Annals Series: civilEngineering, Volume 1, Special Issue 11, pg. 63-72, 2009

[8] M. Lupu, F. Isaia, The study of some nonlinear dynamical systems medelled by a more generalReylegh - Van der Pol equation, J. Creative Math, 16(2007), Univ. Nord Baia Mare, 81-90.

[9] M. Lupu, F. Isaia, The mathematical modeling and the stability study of some speed regulatorsfor nonlinear oscilating systems, Analele Univ. Bucuresti, LV(2006), 203-212.

[10] A. Y. Lurie, Nonlinear problems from the automatic control, Ed. Gostehizdat Moskow, 1951 (inRussian).

[11] D. R. Merkin, Introduction in the movement stability theory, Ed. Nauka, Moskow, 1987 (in Rus-sian).

[12] R. A. Nalepin &co., Exact methods for the snonlinear system control in case of automatic regu-lation, M.S. Moskva, 1971.

[13] D. Popescu & co., Modelisation, Identification et Commande des Systemes, Ed. Acad. Romane,Bucuresti, 2004.

[14] E. P. Popov, Applied theory of control of nonlinear systems, Nauka, 1973, Verlag Technik, Berlin1964.

[15] V. M. Popov, The hypersensitivity of automatic systems, Ed. Acad Romane, 1966, Nauka 1970,Springer -Verlag, 1973.

[16] Vl. Rasvan, Theory of Stability, Ed. St. Enciclopedica, Bucuresti, 1987, (in Romanian)


[17] N. Rouche, P. Habets, M. Leloy, Stability theory by Lyapunov’s Direct Method, Springer Verlag,1977.

SOLVING UNDISCOUNTED INFINITE HORIZONOPTIMIZATION PROBLEMS: A NONSTANDARDAPPROACH

ROMAI J., 6, 2(2010), 199–220

Dapeng Cai1, Takashi Gyoshin Nitta2

1Institute for Advanced Research, Nagoya University, Nagoya, Japan2Department of Mathematics, Faculty of Education, Mie University, Japan

Abstract Undiscounted infinite horizon optimization problems are intrinsically difficult because(i) the objective functional may not converge; (ii) boundary conditions at the infinite ter-minal time cannot be rigorously expressed in the real number field. In this paper, by ex-tending real numbers to hyper-real numbers, we derive the optimal solution to an undis-counted infinite horizon optimization problem that has an infinite objective functional.We demonstrate that under a hyper-real terminal time, there exists a unique optimal so-lution in the hyper-real number field. We show that under fairly general conditions, thestandard part of the hyper-real optimal path is the optimum among all feasible paths inthe standard real number field, in the sense of two modified overtaking criteria. We alsoexamine the applicability of our approach by considering two parametric examples.

Keywords: infinite-horizon optimization, boundary condition, overtaking criterion.2000 MSC: C61, E21, O41.

1. INTRODUCTIONMost environmental issues, for example, global climate change, loss of biodiver-

sity, require the evaluation and comparison of policies whose effects can be expectedto spread out into the far distant future (Nordhaus, 1994; Weitzman, 1998; 1999). Ineconomics, such issues have been considered in the framework of the infinite horizonoptimization problems. However, with an infinite horizon, the objective functional ingeneral may diverge and one ends up at comparing infinity with infinity, which hasbeen impossible within the standard real number field. As a compromise, economistshave been discounting future values, although they have long been scathing aboutits ethical dimensions (see, for example, Ramsey, 1928; Pigou, 1932; Harrod, 1948;Solow, 1974; Cline, 1992; Anand and Sen, 2000; and Stern, 2007). Discountingraises moral and logical difficulties because a positive discount rate generates a fun-damental asymmetry between the treatments of the present and future generations(Heal, 1998). It is thus imperative to tackle these issues in a manner that is ethicallyand generationally equitable. This calls for the construction of the optimal solutionsto the undiscounted infinite horizon optimization problems.

The technical difficulties of not discounting lies in the fact that “there is not enoughroom in the set of real numbers to accommodate and label numerically all the different

199

200 Dapeng Cai, Takashi Gyoshin Nitta

satisfaction levels that may occur in relation to consumption programs for an infinitefuture (Koopmans, 1960; p. 288)”, when there is “a preference for postponing satis-faction, or even neutrality toward timing (ibid.).” Moreover, to rigorously express theboundary conditions at the infinite terminal time, one needs to distinguish the state atthe infinite terminal time with that at the time after the infinite terminal time, whichhas been also impossible in the real number field. To overcome these difficulties, onenatural practice would be to extend the real number field R to the hyper-real numberfield, which has been applied to analyze economic and financial issues. An excellentsurvey on the economic applications of the non-standard analysis is available in An-derson (1991), which provides ”a careful development of non-standard methodologyin sufficient detail to allow the reader to use it in diverse areas in mathematical eco-nomics” (p. 2147). Rubio has applied non-standard analysis to optimization theory(Rubio, 1994; 2000).However, by far such analyses have been relatively general andgeometrical; they are not readily applicable to most problems in economic dynamics.On the other hand, Okumura et al. (2009a, b) present the generalized transversalitycondition for infinite horizon optimization problems that may have unbounded objec-tive functionals in the real number field. This calls for the constructions of optimalsolutions to undiscounted infinite horizon optimization problems.

In this paper, we present an approach that explicitly tackles undiscounted infinitehorizon optimization problems in the hyper real number field. We first consider asimple finite horizon undiscounted optimization problem. By resorting to the argu-ments in the nonstandard analysis, we extend real numbers into hyper-real numbersand present a mathematically correct infinite horizon extension of the finite horizonproblem (especially to specify the relevant boundary condition at T = ∞). We thendemonstrate that there exists a unique optimal solution to this infinite horizon prob-lem in the hyper-real number field: the limit of the solution for the finite horizonproblem is the unique solution to the infinite horizon problem. The conjecture thatthe limit of the solutions for the finite-horizon problem is the unique solution to theinfinite-horizon problem has been around for a while (the case with a discount factorhas been examined in for example, Stokey and Lucas (1989)). We show that “thelimit of the solutions for the finite-horizon problem” is the unique “projection” of theunique hyper-real optimal solution on the real number field. We also show the con-ditions under which the unique “projection” on the real number field is the uniqueoptimum to the infinite horizon problem, in the sense of two modified overtakingcriteria, respectively.

Of the two modified overtaking criteria that we use as our intertemporal optimalitycriteria, one is common in the literature, which examines the difference betweenthe summation of the utility along the optimal path and that of any other attainablepath when T → ∞. The other one, however, is new and takes the quotient form,with its focus on the quotient of the summation of utility along the optimal pathdivided by that of any other attainable path when T → ∞. We clarify the conditionsunder which the limits of the solutions for the finite horizon problems are optimal

Solving undiscounted infinite horizon optimization problems: a nonstandard approach 201

among all attainable paths for the infinite horizon problem in the sense of the twomodified overtaking criteria. In the two simple parametric examples we consider,we show that the modified overtaking criterion in the common form is only valid forone example. On the other hand, the quotient form modified overtaking criterion isapparently weaker, and is valid under both examples. These two examples suggestthat the two overtaking criteria differ fundamentally. It can also be easily shown thatwhen discounting is incorporated and the objective functional converges, the criteriaare equivalent to each other. The difference emerges when the objective functionaldiverges.

2. THE MODEL SETTING

2.1. TWO PERPLEXITIES OF THE INFINITEHORIZON OPTIMIZATION PROBLEM

We consider an economy that is composed of many identical households, eachforming an immortal extended family. In each period, a representative householdinvests k(t) at the beginning of the period t to produce f(k(t)) amount of output.The production process is postulated as follows: f : Rm → Rm, f is continuousdifferentiable, with f(0) = 0. In each period, the output is divided between currentconsumption c(t) ≡ (ci(t)), with 1 6 i 6 m, and investment, k(t + 1) ≡ (ki(t + 1)),with 1 6 i 6 m. Furthermore, ci(t) > 0, and ki(t + 1) > 0.

The problem we consider in this paper is the following:

Maximize∞∑

t=0

U (c (t) , t) (P)

sub ject to : k (t + 1) = f (k (t)) − c (t) , given k (0) = k0,

and to the condition that goods will not be wasted.The instantaneous utility function U : Rm+1 → R is assumed to be continuously

differentiable. Notice that the objective functional of (P) is not necessarily finite. Wefirst consider the finite horizon version of problem (P). Given the planning horizonT ∈ [0,∞), the criterion for a social planner to judge the welfare of the representativehousehold takes the form

maxc(t)

T∑

t=0

U (c (t) , t), (P′)

where the instantaneous utility function U : Rm+1 → R is continuous differentiable.The household chooses the path of c(t) that maximizes (P’), which is subject to

the budget constraint : k(t + 1) = f(k(t)) − c(t), (1)

the initial capital stock : k(0) ≡ k0 > 0, and


the boundary condition : k(T + 1) = 0. (2)

An unique solution to problem (P’), subject to (2), kT (t), cT (t)Tt=0, can be foundreadily by applying the method of Lagrange, with the first-order conditions given as(below the subscripts denote the length of the planning horizon):

∂U(cT (t), t)∂ci

= λiT (t), (3)

λiT (t)∂ fi(kT (t))

∂ki= λiT (t − 1). (4)

By substituting (1), (3), into (4) and by rearranging terms, we have

∂ fi(kT (t))∂ki

∂U((f(kT (t)) − kT (t + 1)), t)∂ci

=∂U((f(kT (t − 1)) − kT (t)), t)

∂ci. (5)

We proceed to extend the planning horizon of this problem to infinity.

Two perplexities arise immediately. First of all, the infinite series∞∑

t=0U(c(t), t) will

in general diverge and the maximization of which may be meaningless in the realnumber field. Second, it would also be imperative to present an appropriate analogueof the boundary condition (2) for the infinite-horizon case, the naıve extension ofwhich would be k(∞+1) = 0. However, as∞ has not been defined in R, k(∞+1) = 0is indefinable, and we can only state that “goods will not be wasted” in the contextof real number field, as in Stokey and Lucas (1989). It seems that the boundarycondition can be stated instead as lim

T→∞k(T + 1) = 0, however, which also implies

limT→∞

k(T ) = 0. Since the capital stock k at the infinite time is not specified as zero, we

have arrived at a contradiction and it is inappropriate to state the boundary conditionwith the concept of limit in R.

2.2. A REFORMULATION OF THE PROBLEMInstead, we extend the field of real numbers so that different “levels of infinity”

can be accommodated and we can distinguish “∞” and “∞+1”. Following Anderson[2] (1999, pp. 2150-2151), we extend real numbers into hyper-real numbers.

We define hyper-real number as ∗R ≡ RN/u, where N is a set of natural numbers,

u is a free ultrafilter that is a maximal filter containing the Frechet filter.The elements of ∗R are represented by sequences and are denoted as [< an >],

where an ∈ R. Instead of T , we consider an infinite star finite number T ≡ [< Tn >].We extend c(t) to C(t) ≡ [< Cn(t) >], and k(t) to K(t) ≡ [< Kn(t) >].

Moreover, U : Rm+1 → R and f : Rm → Rm are extended to ∗U : ∗Rm+1 → ∗R and∗f : ∗Rm → ∗Rm, respectively.


Accordingly,T∑

t=0U(c(t), t) is extended to∗

T∑[<tn>]=0

∗U(C(t), t), with its elements rep-

resented by[<

Tn∑t=0

U(Cn(t), t) >]. Hence, the maximization problem can be reformu-

lated as

max ∗T∑

[<tn>]=0

∗U(C(t), t), ∗(P)

sub ject to C([< tn >]) = ∗f(K([< tn >])) −K([< tn >] + 1), ∗(1)

given K(0), and

the f act that goods will not be wasted : K(T + 1) = 0. ∗(2)

Next, we show how to solve the optimization problem in the hyper-real field.For [< λn >] : 1, 2, . . . , T → ∗Rm, we form the Lagrangean,

∗L = ∗T∑

[<tn>]=0

(∗U(C([< tn >]), t) − [< λn >] (C([< tn >])−

−∗f(K([< tn >])) + K([< tn >] + 1))).

The first-order conditions are familiar:

∗ ∂U(C([< tn >]), t)∂ci

= [< λin >]([< tn >]), ∗(3)

[< λin >]([< tn >])∗∂fi(K([< tn >]))

∂ki= [< λin >]([< tn >] − 1). ∗(4)

Combining ∗(2), ∗(3), and ∗(4), we have

∗ ∂fi(K([< tn >]))∂ki

∗ ∂U(∗f(K([< tn >])) −K([< tn >] + 1), t)∂ci

=

= ∗∂U(∗f(K([< tn >] − 1)) −K([< tn >]), t)

∂ci, ∗(5)

where t = 1, 2, . . . , T .Equation ∗(5) is a second-order difference equation in K(t); hence it has a two-

parameter family of solutions. The unique optimum of interest is the one solution inthis family that in addition satisfies the two boundary conditions

given K(0), K(T + 1) = 0

.


For each index n, equation ∗(5) is equivalent to

∗ ∂ fi(Kn(tn))∂ki

∗ ∂U(∗ f (Kn(tn)) −Kn(tn + 1), tn)∂ci

= ∗∂U(∗ f (Kn(tn − 1)) −Kn(tn), tn)

∂ci,

∗(5′)where tn = 1, 2, . . . ,Tn, and an unique solution (KT (t),CT (t)) can be obtained byapplying the following boundary conditions:

given Kn(0), Kn(Tn + 1) = 0.

All the variables in equations ∗(3), ∗(4), and ∗(5) depend on T . In particular, [<λn >]([< tn >]) stands for the shadow price for capital at time [< tn >], given theplanning horizon T . In other words, the value of λ at each time changes with thelength of the planning horizon. In what follows, we assume that there exists a uniqueoptimal solution K(t),C(t)Tt=0 to problem ∗(P), subject to ∗(1) and ∗(2).

With a finite horizon T , because ci(t) > 0 and ki(t + 1) > 0, 0 6 ki(t + 1) 6 fi(k(t)),and the set of sequences k(t)Tt=0 satisfying (1) and (2) is a closed, bounded, andconvex subset of Rm. Moreover, the objective function (P’) is continuous and strictlyconcave. It is well-known that there is exactly one solution to (P’) (Stokey and Lucas,1989). Hence, the set of sequences K(t)Tt=0 satisfying ∗(2) is a *closed, *bounded,and *convex subset of ∗Rm, and the objective function *(P) is continuous and strictlyconcave. Therefore, there exists a unique solution K(t),C(t)Tt=0 to ∗(P) in the hyper-real number field.

2.3. THE LEGITIMACY OF THE LIMIT OF THESOLUTIONS FOR THE FINITE-HORIZONPROBLEM

Denote by st the standard mapping from ∗R → R ∪ ±∞. As is widely known inthe nonstandard analysis, if there exist lim

T→∞kT (t) and lim

T→∞cT (t), then

limT→∞

kT (t) = st(KT (t)), limT→∞

cT (t) = st(CT (t)), where t ∈ R.

Hence, from the nonstandard optimal conditions, we have specified a unique path(st(KT (t)),st(CT (t))) in R2m, although we still need to verify whether such a path isindeed the optimal solution in R2m.

2.4. TWO OVERTAKING CRITERIAIn what follows, we show that under fairly general conditions, the optimum ob-

tained by applying Theorem 2.1 (below) is indeed the unique optimum among allfeasible paths in R2m, under two modified overtaking criteria, modified to incorpo-rate the boundary condition that goods are not to be wasted. One is common in theliterature, which examines the difference between the summation of the utility alongthe optimal path and that of any other attainable path as T → ∞. The other one,


however, is new and takes the quotient form, with its focus on the quotient of thesummation of utility along the optimal path divided by that of any other attainablepath when T → ∞. For simplicity, we only consider reversible investment.

For two attainable paths (k1, c1) and (k2, c2) that satisfy the following conditions:the budget constraint: k(t + 1) = f (k(t)) − c(t),the initial capital stock: k(0) ≡ k0, andthe boundary condition that no goods should be wasted; we give the following

definitions:

Definition 2.1. (Modified Overtaking Criterion I) (k1, c1) and (k2, c2) are two infinitehorizon attainable paths that satisfy conditions (i)-(iii). For all t, (k2, c2) overtakes(k1, c1) if

limT→∞

T−1∑

t=0

U (c2 (t) , t) + U (f (k2 (T )) , T ) −

T−1∑

t=0

U (c1 (t) , t) + U (f (k1 (T )) ,T )

> 0.

Definition 2.2. (Modified Overtaking Criterion II) (k1, c1) and (k2, c2) are two in-finite horizon attainable paths that satisfy conditions (i)-(iii). For U(c1(t)) > 0,U(c2(t)) > 0 (resp. U(c1(t)) < 0, U(c2(t)) < 0), all t, (k2, c2) overtakes (k1, c1)if

limT→∞

T−1∑t=0

U (c2 (t) , t) + U (f (k2 (T )) , T )

T−1∑t=0

U (c1 (t) , t) + U (f (k1 (T )) , T )> 1

resp. lim

T→∞

T−1∑t=0

U (c1 (t) , t) + U (f (k1 (T )) , T )

T−1∑t=0

U (c2 (t) , t) + U (f (k2 (T )) , T )< 1

.

Based on the notion of weak maximality [14], the optimality criteria are definedas follows:

Definition 2.3. For all t, an infinite horizon attainable path (k, c) is optimal if noother infinite horizon attainable path (k1, c1) overtakes it:

limT→∞

T−1∑

t=0

U (c2 (t) , t) + U (f (k2 (T )) , T ) −

T−1∑

t=0

U (c1 (t) , t) + U (f (k1 (T )) ,T )

6 0.

Definition 2.4. For U(c(t)) > 0, U(c1(t)) > 0 (resp. U(c(t)) < 0, U(c1(t)) < 0),all t, an infinite horizon attainable path (k, c) is optimal if no other infinite horizon


attainable path (k1, c1) overtakes it:

limT→∞

T−1∑t=0

U (c1 (t) , t) + U (f (k1 (T )) , T )

T−1∑t=0

U (c (t) , t) + U (f (k (T )) , T )6 1

resp. lim

T→∞

T−1∑t=0

U (c1 (t) , t) + U (f (k1 (T )) , T )

T−1∑t=0

U (c (t) , t) + U (f (k (T )) , T )> 1

.

Both (k1, c1) and (k, c) are extended to hyper-real number field and are reformu-lated to satisfy the boundary condition that no goods should be wasted. The resultanttwo paths are then compared by using the notion of weak maximality. According tothe non-standard argument, the standard parts of the two derived paths are exactlytheir original paths when 0 < t < ∞, respectively. Notice that our optimal programsare not restricted to optimal stationary programs.

2.5. THE PROOF OF THE CONJECTURETo establish the legitimacy of the conjecture, we also need the following lemmas:

Lemma 2.1. Let aT and bT , T ∈ [0,∞), be two sequences. If limT→∞

bT = 0, then

limT→∞

(aT + bT ) = limT→∞

aT + limT→∞

bT = limT→∞

aT

(lim

T→∞(aT + bT ) = lim

T→∞aT + lim

T→∞bT = lim

T→∞aT

).

Lemma 2.2. If limt→∞

a(t) > 0, limt→∞ b(t) > 0, then lim

t→∞(ab)(t) = lim

t→∞a(t) · lim

t→∞ b(t).

Proof. Let limt→∞

b(t) = β, then for an arbitrary ε > 0, there exists a T1 > 0, such that

if t > T1, β − ε < b(t) < β + ε. We denote infa(s) |s > t as a(t). Since limt→∞

a(t) > 0,

there exists a T2, such that if t > T2, then a(t) > 0, that is, if s > T2, then a(s) > 0.Hence, if s > max(T1,T2), then a(s)(β − ε) < (ab)(s) < a(s)(β + ε). Assuming thatt > max(T1, T2), we have

infa(s)(β − ε) |s > t 6 inf(ab)(s) |s > t 6 infa(s)(β + ε) |s > t ,which implies

a(t)(β − ε) 6 (ab)(t) 6 a(t)(β + ε).


Hence, limt→∞

a(t)(β − ε) 6 limt→∞

(ab)(t) 6 limt→∞

a(t)(β + ε).

Since ε is arbitrarily chosen, limt→∞

ab(t) = limt→∞

a(t)β.

Theorem 2.1. We consider:

Condition (i). For an arbitrary infinite number ˜T,˜T−1∑t=0

(U(C ˜T (t), t) − U(C(t), t)) +

U(C ˜T ( ˜T ), ˜T ) − U(f(K( ˜T )), ˜T ) is infinitesimal.If Condition (i) is satisfied, then for an arbitrary (k1, c1), (k, c) overtakes (k1, c1)

in the sense of the modified overtaking criterion as defined in Definition 2.3.

Proof. For an arbitrary infinite number ˜T , C ˜T is an optimal solution. Hence, for anarbitrary admissible path (k1, c1), we have

˜T∑

t=0

U(C ˜T (t), t) −

˜T−1∑

t=0

U(C1(t), t) + U(f(K1( ˜T )), ˜T )

> 0.

From the Nonstandard Extension Theorem, there exists a finite T such that if T < T ′,

thenT ′∑t=0

U(cT ′(t), t) −(

T ′−1∑t=0

U(c1(t), t) + U(f(k1(T ′)), T ′))> 0.

We then have

limT→∞

(T−1∑t=0

(U(c(t), t) − U(c1(t), t)) + U(f(k(T )),T ) − U(f(k1(T )), T ))

=

= limT→∞

(T−1∑

t=0

(U(cT (t), t) − U(c1(t), t)) + U(f(kT (T )),T ) − U(f(k1(T )),T )

︸︷︷︸(A)

+

+

T−1∑

t=0

(U(c(t), t) − U(cT (t), t)) + U(f(k(T )),T ) − U(f(kT (T )),T )

︸︷︷︸(B)

)

By Lemma 2.1, the above equation can be further restated as

limT→∞

(T−1∑

t=0

(U(cT (t), t) − U(c1(t), t)) + U(f(kT (T )), T ) − U(f(k1(T )),T )

︸︷︷︸(A)

+

+

T−1∑

t=0

(U(c(t), t) − U(cT (t), t)) + U(f(k(T )),T ) − U(f(kT (T )), T )

︸︷︷︸(B)

)


= limT→∞

(T−1∑

t=0

(U(cT (t), t) − U(c1(t), t)) + U(f(kT (T )),T ) − U(f(k1(T )), T )

︸︷︷︸(A)

)+

+ limT→∞

(T−1∑

t=0

(U(c(t), t) − U(cT (t), t)) + U(f(k(T )), T ) − U(f(kT (T )), T )

︸︷︷︸(B)

).

From Condition (i), we have

limT→∞

T−1∑

t=0


= 0.

Hence,

limT→∞

(T−1∑

t=0

(U(cT (t), t) − U(c1(t), t)) + U(f(kT (T )),T ) − U(f(k1(T )),T )

︸︷︷︸(A)

+

+

T−1∑

t=0


︸︷︷︸(B)

)

= limT→∞

(T−1∑

t=0

(U(cT (t), t) − U(c1(t), t)) + U(f(kT (T )), T ) − U(f(k1(T )),T )

︸︷︷︸(A)

) =

= limT→∞

(T∑

t=0(U(cT (t), t) −

T−1∑t=0

(U(c1(t), t)) + U(f(k1(T )), T )))> 0.

q.e.d.

Note that under this definition, we are able to compare all paths. In Theorems2.2 and 2.4, we consider the case for U(c1(t)) < 0, U(c2(t)) < 0, all t, results forU(c1(t)) > 0, U(c2(t)) > 0 can be obtained similarly.

Theorem 2.2. : We consider:Condition (ii). For an arbitrary infinite number ˜T,

˜T−1∑t=0

(U(C ˜T (t), t) − U(C(t), t)

)+ U(C ˜T ( ˜T ), ˜T ) − U(f(K( ˜T )), ˜T )

˜T∑t=0

U(C ˜T (t), t)

is in f initesimal.


If Condition (ii) is satisfied, then for an arbitrary (k1, c1), (k, c) overtakes (k1, c1) inthe sense of the modified overtaking criterion as defined in Definition 2.4.

Proof. For an arbitrary infinite number ˜T , C ˜T is an optimal solution. Hence, for anarbitrary admissible path (k1, c1), we have

˜T−1∑t=0

U(C1(t), t) + U(f(K1( ˜T )), ˜T )

˜T∑t=0

U(C ˜T (t), t)

> 1

.From the Nonstandard Extension Theorem, there exists a finite T such that T < T ′

implies

T ′−1∑t=0

U(c1(t),t)+U(f(k1(T ′)),T ′)

T ′∑t=0

U(cT ′ (t),t)> 1. Hence, lim

T→∞

T−1∑t=0

U(c1(t),t)+U(f(k1(T )),T )

T∑t=0

U(cT (t),t)

> 1.

We then have˜T∑

t=0U(C ˜T (t),t)

˜T−1∑t=0

U(C(t),t)+U(f(K( ˜T )), ˜T )

=

˜T∑t=0

U(C ˜T (t),t)

˜T∑t=0

U(C ˜T (t),t)−(˜T−1∑t=0

U(C ˜T (t),t)−˜T−1∑t=0

U(C(t),t))−U(C ˜T ( ˜T ), ˜T )+U(f(K( ˜T )), ˜T )

.

Condition (i) implies that the standard part of

˜T∑t=0

U(C ˜T (t),t)

˜T−1∑t=0

U(C(t),t)+U(f(K( ˜T )), ˜T )

is 1. But, since

T−1∑t=0

U(c1(t),t)+U(f(k1(T )),T )

T−1∑t=0

U(c(t),t)+U(f(k(T )),T )=

T−1∑t=0

U(c1(t),t)+U(f(k1(T )),T )

T∑t=0

U(cT (t),t)

·

T∑t=0

U(cT (t),t)

T−1∑t=0

U(c(t),t)+U(f(k(T )),T )

,

from Lemma 2.1, we have

limT→∞

T−1∑t=0

U(c1(t),t)+U(f(k1(T )),T )

T−1∑t=0

U(c(t),t)+U(f(k(T )),T )=

= limT→∞

T−1∑t=0

U(c1(t),t)+U(f(k1(T )),T )

T∑t=0

U(cT (t),t)

· limT→∞

T∑t=0

U(cT (t),t)

T−1∑t=0

U(c(t),t)+U(f(k(T )),T )

> 1.

q.e.d.

Hence we have shown that the path obtained is indeed optimal among all feasiblepaths in R, if the two conditions listed in Theorem 2.1 are simultaneously satisfied. Ifthe maximand is finite, we can always resort to the standard Lagrange method to findthe optimum. On the other hand, Condition (ii) requires that the loss accompanyingthe limit practice is infinitesimal as compared to the sum of the utility sequence to bemaximized.


Finally, we explore the implications of these assumptions. For simplicity, we setm = 1. Our results can be easily extended to cases in which m takes other values.Note that when t = T , kT (T + 1) = 0, (5) is reduced to

f ′ (kT (T )) U′ ( f (kT (T ))) = U′ ( f (kT (T − 1)) − kT (T )) , (6)

and we see that kT (T − 1) can be uniquely determined given kT (T ). In other words,kT (0) can be expressed as a function of kT (T ). Letting g(x) ≡ 1

U′(x) , ϕ(x) ≡ 1f ′(x) , (??)

can be restated as

ϕ (kT (T )) g ( f (kT (T ))) = g ( f (kT (T − 1)) − kT (T )) .

We then have

f (kT (T − 1)) = kT (T ) + g−1 (ϕ (kT (T )) g ( f (kT (T )))) ,

implying

kT (T − 1) = f −1(kT (T ) + g−1 (ϕ (kT (T )) g ( f (kT (T ))))

). (7)

On the other hand, when t = T − 1, we have

f (kT (T − 2)) = g−1 (ϕ (kT (T − 1)) g ( f (kT (T − 1)) − kT (T ))) + kT (T − 1) (8),

implying

kT (T − 2) = f −1(g−1 (ϕ (kT (T − 1)) g ( f (kT (T − 1)) − kT (T ))) + kT (T − 1)

).

Letting ψT,T (k) ≡ g−1 (ϕ (k) g ( f (k)))+k, we then have f (kT (T − 1)) = ψT,T (kT (T )),i.e., kT (T ) = ψ−1

T,T ( f (kT (T − 1))), and, by substituting it into (8), we have

f (kT (T − 2)) = g−1(ϕ (kT (T − 1)) g

(f (kT (T − 1)) − ψ−1

T,T ( f (kT (T − 1)))))

+kT (T − 1) .

The first subscript T in ψ−1denotes the length of the planning horizon, whereas thesecond T is used to denote time.

From (8), we see that

f (kT (T − 1)) − kT (T ) = g−1 (ϕ (kT (T )) g ( f (kT (T )))) =

= g−1(ϕ(ψ−1

T,T ( f (kT (T − 1))))

g(

f(ψ−1

T,T ( f (kT (T − 1))))))

.

Hencef (kT (T − 2)) =

= g−1 (ϕ (kT (T − 1)))ϕ(ψ−1

T,T ( f (kT (T − 1))))

g(

f(ψ−1

T,T ( f (kT (T − 1)))))

+kT (T − 1) .

Letting ψT,T−1(k) ≡ g−1 (ϕ(k))ϕ(ψ−1

T,T ( f (k)))

g(

f(ψ−1

T,T ( f (k))))

+ k, we see thatf (kT (T − 2)) = ψT,T−1 (kT (T − 1)) , i.e., kT (T − 1) = ψ−1

T,T−1 ( f (kT (T − 2))).


In general, we have

f (kT (t)) = g−1 (ϕ (kT (t + 1))) g ( f (kT (t + 1)) − kT (t + 2)) + kT (T + 1) =

= g−1 (ϕ (kT (t + 1))) g(ψ−1

T,t+2 ( f (kT (t + 1))))

g(

f(ψ−1

T,t+2 ( f (kT (t + 1)))))

+

+kT (T + 1),

implying

ψT,t+1 (k) = g−1 (ϕ (k))ϕ(ψ−1

T,t+2 ( f (k)))

g(

f(ψ−1

T,t+2 ( f (k))))

+ k.

Hence, we define ψT,t (0 6 t 6 T ) as

ψT,t (k) = g−1 (ϕ (k))ϕ(ψ−1

T,t+1 ( f (k)))

g(

f(ψ−1

T,t+1 ( f (k))))

+ k,

and we havekT (t) = ψ−1

T,t ( f (kT (t − 1))) ,

cT (t) = f (kT (t)) − ψ−1T,t+1 ( f (kT (t))) ≡ φT,t (kT (t)) ,

In other words,

kT (t) = ψ−1T,t ( f (kT (t − 1))) = ψ−1

T,t f ψ−1T,t−1 f · · · ψ−1

T,0 f (k (0)) ,

cT (t) = φT,t

(ψ−1

T,t f ψ−1T,t−1 f · · · ψ−1

T,0 f (k (0))).

As ψT,t is derived inductively from ψT,t+1, if there exists limT→∞

ψT,t, then limT→∞

ψT,t does

not depends on t. We define ψ ≡ limT→∞

ψT,t. Similarly, we define φ ≡ limt→∞ φT,t, which

also does not depend on t. We then have

k (t) = ψ−1 ( f (k (t − 1))) ,

c (t) = f (k (t)) − ψ−1 ( f (kT (t − 1))) ≡ φ (kT (t)) .

k (t) = ψ−1 f ψ−1 f · · · ψ−1 f (k (0)) ,

c (t) = φ(ψ−1 f ψ−1 f · · · ψ−1 f (k (0))

).

Therefore, we can restate the conditions in Theorem 2.1 and 2.2, using the above.The above analysis reveals that if there exists lim

T→∞ψT,t, then k(t) and c(t) exist.

Most importantly, as ψ and φ do not depend on t, and k (t) = ψ−1 ( f (k (t − 1))),c (t) = φ (k (t)), hence we see that both the saving function ψ−1 and the consumptionfunction φ does not depend on t. This result indicates that for the optimal path, therelationship between saving and consumption is unchanging over time.


3. EXAMPLESExample 1. We consider the following social planner’s problem, in which the

social planner treats the future equally by assuming a zero rate of time preference.The planner’s objective is

maxc(t)

∞∑

t=0

(c (t))α, α > 0,

subject to

k (t + 1) = f (k (t)) − c (t) , where f (k (t)) = ak (t), a > 0, k (0) > 0 given,0 < k (t) < 1, and the fact that goods shall not be wasted.

We first solve the finite horizon version of problem:

maxc(t)

T∑t=0

(c (t))α, where T ∈ [0,∞), α > 0,

subject to

c (t) + k (t + 1) = f (k (t)) , where f (k (t)) = ak (t), k (0) > 0 given,

0 < k (t) < 1, k (T + 1) = 0.

We extend T → T , the finite horizon problem is reformulated as

maxC(t)

∗T∑

[<tn>]=0

∗ (C (t))α,

subject toC([< tn >]) + K([< tn >] + 1) = a(K([< tn >])),

where 0 < α < 1, K(0) given,

0 < K([< tn >]) < 1, K(T + 1) = 0.

The paths of C(t) and K(t) are uniquely determined given k(0) > 0 and k(T + 1) = 0.The solution (KT (t),CT (t)) is given by

KT (t) = atK (0) −aT+1K (0)

(a

t(α−1)−tα−1 − a

−t−αα−1

)

(a

(T+1)(α−1)−αα−1 − a

−T−1−αα−1

) ,

CT (t) =K (0)

(1 − a

−αα−1

)

a−αα−1 − a

−αT−2αα−1

a−t

α−1 .


Obviously, a−T−2αα−1 is infinitesimal when (i) α > 1 and a > 1, or when

(ii) α < 1 and a < 1. We have

K (t) = K (0) a−1(a−

1α−1

)t,

C (t) = K (0)(a

αα−1 − 1

) (a−

1α−1

)t.

Because aαα−1 > 1 under both case (i) and case (ii), when see that a

αα−1 > 1 and

a−1α−1 < 1, indicating that C(t) is decreasing in t. On the other hand, K(t) is also

decreasing in t.Moreover,

C (t)α −CT (t)α =

= K (0)α(1 − a−

αα−1

)α ((1

a−αα−1

)α−

(1

a−αα−1 − a−

T−1−αα−1 −T−1

)α) (a−

1α−1

)t.

Hence,

T−1∑t=0

(CT (t)α −C (t)α

)+

(CT

(T)α − f

(K

(T))α)

= K (0)α(1 − a−

αα−1

)α ((1

a−αα−1 − a

αT−21−α

)α−

(1

a−αα−1

)α)

︸︷︷︸(I)

(1−a−

Tα−1

1−a−1α−1

)

+

K (0)

(1 − a−

αα−1

)

a−αα−1 − a−

αT−2α−1

a−Tα−1

α

︸︷︷︸(II)

−(aK (0)

(a−1

)a−

Tα−1

)α︸︷︷︸

(III)

.

Because 0 < a < 1, aαT−21−α , a−

αTα−1 , and a−

Tα−1 are all infinitesimals, we see that (I), (II),

and (III) are infinitesimals. Hence,T−1∑t=0

(CT (t)α −C (t)α

)+

(CT

(T)α − f

(K

(T))α)

is infinitesimal and Condition (i) in Theorem 2.1 is satisfied.Example 2. We consider the following social planner’s problem, in which the

social planner treats the future equally by assuming a zero rate of time preference.The planner’s objective is

maxc(t)

∞∑

t=0

ln(c(t)),

subject to

c(t) + k(t + 1) = f (k(t)), where f (k(t)) = k(t)α, 0 < α < 1, k(0) given,0 < k(t) < 1, and the fact that goods shall not be wasted.


We first solve the finite horizon version of problem:

maxc(t)

T∑t=0

ln(c(t)), where T ∈ [0,∞),

subject toc(t) + k(t + 1) = f (k(t)), where f (k(t)) = k(t)α, 0 < α < 1, k(0) given,

0 < k(t) < 1, k(T + 1) = 0.

We extend T → T , the finite horizon problem is reformulated as

maxC(t)

∗T∑

[<tn>]=0

∗ ln(C(t)),

subject toC([< tn >]) + K([< tn >] + 1) = (K([< tn >]))α, where 0 < α < 1, K(0) given,

0 < K([< tn >]) < 1, K(T + 1) = 0.

As 0 < k(t) < 1, 0 < c(t) < 1. The paths of c(t) and k(t) are uniquely determinedgiven k(0) > 0 and k(T + 1) = 0, with k(t + 1) = α 1−αT−t

1−αT−t+1 kα(t), t = 0, 1, . . . ,T .Also, the solution (kT (t), cT (t)) is

kT (t) =

(1 − αT−t+1

)α

1−αt1−α k(0)α

t

(1 − αT−t+2)1−α (

1 − αT−t+3)α(1−α) · · · (1 − αT+1)αt−1 ,

cT (t) =(1 − α)α

α(1−αt )1−α k(0)α

t

(1 − αT−t+1)1−α (

1 − αT−t+2)α(1−α) · · · (1 − αT )αt−1(1−α) (1 − αT+1)αt .

Next we extend T → T , where T is an infinite star finite number. Accordingly, weextend t ∈ 0, 1, · · · ,T → t ∈ 0, 1, · · · , T . We have

limT→∞

kT (t) = st(KT (t + 1)) = α(st

(KT (t)

))α, t ∈ [0,∞).

Hence

st(KT (t)) = (α)αt−1+···+1Kαt

0 = (α)1/(1−α)(

k(0)(α)1/(1−α)

)αt

,

st(CT (t)) = (1 − α)(α)α/(1−α)(

k(0)(α)1/(1−α)

)αt+1

,

st(λT (t)) = (1 − α)−1(α)−α/(1−α)(

k(0)(α)1/(1−α)

)−αt+1

,

where λT (t) is the Lagrange multiplier for time horizon T . It stands for the shadowprice of capital at time t.


Denote st(KT (t)) ≡ K(t), st(CT (t)) ≡ C(t), st(λT (t)) ≡ λ(t). The followingproposition shows that (k(t), c(t)) is optimal in R under Definition 2.3.

Proposition 3.1. Let k(t) = K(t), c(t) = C(t). For an arbitrary (k1, c1),

limT→∞

T−1∑t=0

ln(c1(t)) + ln(k1(T ))α

T−1∑t=0

ln(c(t)) + ln(k(T ))α> 1,

i.e. Condition (ii) in Theorem 2.2 is satisfied and (k(t), c(t)) is optimal in R underDefinition 2.3.

Proof. For an arbitrary infinite number ˜T ,

limT→∞

T−1∑t=0

ln(c1(t)) + ln((k1(T ))α)

T−1∑t=0

ln(c(t)) + ln((k(T ))α)= st

˜T−1∑t=0

ln(∗c1(t)) + ln((∗k1( ˜T ))α)

˜T−1∑t=0

ln(∗c(t)) + ln((∗k( ˜T ))α)

.

In what follows, we neglect the notation * if doing so does not cause confusion.We see that˜T−1∑t=0

ln(c1(t)) + ln((k1( ˜T ))α)

˜T−1∑t=0

ln(c(t)) + ln((k( ˜T ))α)

=

˜T−1∑t=0

ln(c1(t)) + ln((k1( ˜T ))α)

˜T∑t=0

ln(c ˜T (t))

·

˜T∑t=0

ln(c ˜T (t))

˜T−1∑t=0

ln(c(t)) + ln((k( ˜T ))α)

.

(9)Furthermore, for the sake of simplicity, we use lowercase letters c ˜T to stand for C ˜T .Here c ˜T is the optimal solution for t ∈ [0, ˜T ], hence we have

˜T−1∑t=0

ln(c1(t)) + ln((k1( ˜T ))α)

˜T∑t=0

ln(c ˜T (t))

> 1.

From the Nonstandard Extension Theorem, there exists a finite T such that if T < T ′,

then

T ′−1∑t=0

ln(c1(t))+ln((k1(T ′))α)

T ′∑t=0

ln(cT ′ (t))> 1. Hence, lim

T→∞

T−1∑t=0

ln(c1(t))+ln(k1(T )α)

T∑t=0

ln(cT (t))

> 1.

Next, we consider the difference between (c(t), k(t)) and (c ˜T (t), k ˜T (t)). First,

˜T−1∑

t=0

ln(c ˜T (t)) −˜T−1∑

t=0

ln c(t) =

˜T−1∑

t=0

(ln(c ˜T (t)) − ln c(t)),


hereln(c ˜T (t)) − ln c(t) =

= −(1 − α)α−( ˜T−t+1)α ˜T−t+1 ln(1 − α ˜T−t+1) + · · · + α˜T ln(1 − α ˜T ) − αt ln(1 − α ˜T+1).

Let x = αs, ˜T − t + 1 6 s 6 ˜T , the right-hand side of the above equality is

−(1 − α)α−( ˜T−t+1)˜T∑

s= ˜T−t+1

(αs ln(1 − αs)) − αt ln(1 − α ˜T+1),

the absolute value of which is bounded by the absolute value of

−(1 − α)α−( ˜T−t+1)∫ ˜T

˜T−t+1αs ln(1 − αs)ds − αt ln(1 − α ˜T+1) ≡ (∗).

Let αs ≡ x, we see that lnα · αsds = dx, hence

(∗) = −(1 − α)α−( ˜T−t+1)∫ ˜T

˜T−t+1ln(1 − x)

dxlnα− αt ln(1 − α ˜T+1).

Let −dx = dy, (∗) can be further rewritten using

−α−( ˜T−t+1)∫ 1−α ˜T

1−α ˜T−t+1ln ydy − αt ln(1 − α ˜T+1)

=

(α−

( ˜T−t+1)− 1

)ln

(1 − α

( ˜T−t+1))

+ 1︸︷︷︸

(I)

−(α−

( ˜T−t+1)− αt−1

)ln

(1 − α ˜T

)

︸︷︷︸(II)

−αt−1︸︷︷︸(III)

+lnα

1 − ααt ln

(1 − α ˜T+1

)

︸︷︷︸(IV)

.

We observe that˜T−1∑t=0

(ln(c ˜T (t)) − ln c(t)) is bounded by the absolute value of

−1−αlnα

˜T−1∑t=0(I) + (II) + (III) + (IV) . Next, we show that

˜T−1∑t=0

(ln(c ˜T (t)) − ln c(t)) is

bounded.First, for term (I), letting s = ˜T − t + 1, we can rewrite the sum of (I) as

2∑

s= ˜T+1

(α−s − 1

)ln

(1 − αs) + 1

.


Letting αs = x, we see 0 < α˜T+1 6 x 6 α2 < 1. The sum of (I) can be further

rewritten as∫ α2

α˜T+1

(x−1 − 1

)ln (1 − x) + 1

dxlnα · x =

1lnα

∫ α2

α˜T+1

(x−1 − 1

)ln (1 − x) + 1

x−1dx.

Letting y(x) =((

x−1 − 1)

ln (1 − x) + 1)

x−1, we see that

dydx = (−2x−3 + x−2) ln(1 − x) + (x−2 − x−1) −1

1−x= −2+x

x3 ln(1 − x) − 2x2

= −2+xx3

(−x − 1

2 x2 − 13 x3 − · · ·

)− 2

x2

= 1x3

(16 x3 + 2

4·3 x4 + 35·4 x5 + · · · + n−2

n(n−1) xn + · · ·)> 0.

Hence, we see that y(x) is monotonically increasing in x, and we have

limx→α2

y(x) =((α2 − 1

)ln

(1 − α2

)+ 1

)α2 > 0.

Next, we consider limx→0

y(x). It can be easily shown that

y(x) =((

x−1 − 1)

ln (1 − x) + 1)

x−1

=(

11 − 1

2

)+

(12 − 1

3

)x +

(13 − 1

4

)x2 + · · · +

(1n − 1

n+1

)xn−1 + · · · .

Hence, we see that limx→0

y(x) = 12 , which implies y(x) is bounded in (0, α2).

Therefore, we have 1lnα

∫ α2

α˜T+1 y(x)dx < +∞.

For term (II), we see that

the sum of (II) = −α−( ˜T−1)

(1−α ˜T

)

1−α −α−1

(1−α ˜T

)

1−α

ln(1 − α ˜T

)

=− α1−αα

− ˜T ln(1 − α ˜T

)+ α+α−1−α ˜T−1

1−α ln(1 − α ˜T

).

Let α˜T = x, since ˜T is an infinite number, x is also an infinitesimal. Obviously, the

second term is also an infinitesimal. For the first term, as

st(α−

˜T ln(1 − α ˜T

))= lim

T→∞αT ln

(1 − αT

)= lim

x′→0

ln (1 − x′)x′

= limx′→0

−x′1−x′

x′= −1,

we see that the sum of (II) is bounded by α1−α .

For term (III), we see that˜T−1∑t=0

(−αt−1

)= −α−1(1−α ˜T )

1−α , hence, the sum of (III) is

bounded by − α−1

1−α .On the other hand, for term (IV), we see that

˜T−1∑

t=0

lnα1 − αα

t ln(1 − α ˜T+1

)=

lnα1 − α ln

(1 − α ˜T+1

) 1 − α ˜T

1 − α ,


which is an infinitesimal.

Hence, we have shown that˜T−1∑t=0

(ln(c ˜T (t)) − ln c(t)) is bounded.

Next, we consider ln(c ˜T ( ˜T )) and ln(k( ˜T )α). We see that ,

ln(c ˜T ( ˜T )) = ln(1 − α) +α(1−α ˜T )

1−α lnα − (1 − α) ln(1 − α)− 1−α

α α2 ln(1 − α2) + · · · + α˜T+1 ln(1 − α ˜T+1)

−α ˜T+1 ln(1 − α ˜T+1) + α˜T ln(k(0)).

As˜T+1∑t=2

αt ln(1 − αt) is bounded by˜T+1∫2αx ln(1 − αx)dx, let αx = y,

˜T+1∫

2

αx ln(1 − αx)dx =

α˜T+1∫

α2

ln(1 − y)dy

lnα

= −1lnα ((1 − α ˜T+1) ln(1 − α ˜T+1) − (1 − α ˜T+1) − (1 − α2) ln(1 − α2) + (1 − α2)),which is bounded by −1

lnα (−1 − (1 − α2) ln(1 − α2) + (1 − α2)).

On the other hand, ln(k( ˜T )α) = ln(α

1−α ˜T1−α k(0)α

t)α

, which is bounded by ln(αα

1−α ).

We see that the last term of the right-hand side of equation (9) is equal to

˜T∑t=0

ln(c ˜T (t))

˜T−1∑t=0

ln(c(t)) + ln((k( ˜T ))α)

×

×

˜T−1∑t=0

ln(c ˜T (t)) + ln c ˜T ( ˜T )

˜T∑t=0

ln(c ˜T (t)) − (˜T−1∑t=0

ln(c ˜T (t)) −˜T−1∑t=0

ln(c(t))) − ln c ˜T ( ˜T ) + ln(k( ˜T )α)

.

As ln(c ˜T ( ˜T ))is bounded, not 0 and not infinitesimal, and as ln(c ˜T (t)) (for all t ∈ [0, ˜T ])

is always negative, we see that˜T∑

t=0ln(c ˜T (t)) is negative infinite.

In summary, since ln c ˜T ( ˜T ) and

−(˜T−1∑

t=0

ln(c ˜T (t)) −˜T−1∑

t=0

ln(c(t))) − ln c ˜T ( ˜T ) + ln(k( ˜T )α)


are both bounded, and˜T∑

t=0ln(c ˜T (t)) is an infinite value, we see that the standard part

of the last term in the right-hand side of equation (9) is 1.

Hence, for an arbitrary infinite number ˜T , as˜T−1∑t=0

ln(c(t)) is infinite and ln(k( ˜T )α) is

bounded, st

˜T∑t=0

ln(c ˜T (t))

˜T−1∑t=0

ln(c(t))+ln(k( ˜T )α)

= 1, that is, limT→∞

T∑t=0

ln(cT (t))

T−1∑t=0

ln(c(t))+ln(k(T )α)

= 1.

Let

T−1∑t=0


T∑t=0

ln(cT (t))≡ a(T ),

T∑t=0

ln(cT (t))

T−1∑t=0

ln(c(t))+ln(k(T )α)≡ b(T ).

From Lemma 2.2, we have

limT→∞

T−1∑t=0


T−1∑t=0

ln(c(t))+ln(k(T )α)= lim

T→∞

T−1∑t=0


T∑t=0

ln(cT (t))

· limT→∞

T∑t=0

ln(cT (t))

T−1∑t=0

ln(c(t))+ln(k(T )α)

> 1.

References

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[2] Anderson, R. M., Chapter 39: Non-standard analysis with applications to economics, in Hilden-brand, W. Sonnenschein, H. (Eds.) Handbook of Mathematical Economics, Vol. 4, (1991)2145-2208. North-Holland.

[3] Arrow, K. J., Discounting, Morality and Gaming, In Portney, P. R. Weyant, J. P. (Eds.) Discount-ing and Intergenerational Equity, Resources for the Future, Washington, D.C., 1999.

[4] Brock, W.A., On the Existence of Weakly Maximal Programmes in Multi-Sector Economy, Re-view of Economic Studies, textbf37(1970), 275-280.

[5] doi:10.1016/j.na.2009.02.110.

[6] Cline, W. R., The Economics of Global Warming, Institute for International Economics, Wash-ington, D.C., 1992.

[7] Harrod, R. F. (1948), Towards a Dynamic Economics. London: Macmillan.

[8] Heal, G. (1998), Valuing the Future: Economic Theory and Sustainability, New York: ColumbiaUniversity.

[9] Koopmans, T., Stationary Ordinal Utility and Impatience, Econometrica, 28(1960), 287-309.

[10] Nordhaus, W. D., Managing the Global Commons: The Economics of Climate Change, MITPressCambridge, 1994.

[11] Okumura R., Cai, D., Nitta, G. T. , Transversality Conditions for Infinite Horizon OptimizationProblems: Three Additional Assumptions, ROMAI Journal, 5(2009), 105-112.,

[12] Okumura R., Cai, D., Nitta, G. T., Transversality Conditions for Infinite Horizon Optimality:Higher Order Differential Problems, Nonlinear Analysis: Theory, Methods and Applications,2009.

[13] Pigou, A. C., The Economics of Welfare. 4th ed., Macmillan, London, 1932.


[14] Ramsey, F. P., A Mathematical Theory of Saving, Economic Journal, 38(1928), 543-559.

[15] Rubio, J. E., Optimization and Nonstandard Analysis. Marcel Dekker, 1994.

[16] Rubio, J. E., Optimal Control Problems with Unbounded Constraint Sets, Optimization,31(2000), 191-210.

[17] Stern, N. (2007), The Economics of Climate Change: The Stern Review. Cambridge: CambridgeUniversity Press.

[18] Solow, R. M., Richard T. Ely Lecture: The Economics of Resources or the Resources of Eco-nomics, American Economic Review, 64, 1-14(1974).

[19] Solow, R. M., Forward, in Portney, P. R. & Weyant, J. P. (Eds.) Discounting and IntergenerationalEquity, : Resources for the Future, Washington, D.C., 1999.

[20] Stokey, N., Lucas R. (1989), Recursive Methods in Economic Dynamics, Harvard UniversityPress, Cambridge, 1989.

[21] Weitzman, M. L., Why the far-distant future should be discounted at its lowest possible rate”,Journal of Environmental Economics and Management, 36(1998), 201-208.

[22] Weitzman, M. L. (1999), Just Keep Discounting, but . . ., in Portney, P. R. & Weyant, J. P. (Eds.),Discounting and Intergenerational Equity, Resources for the Future,Washington, D.C..

THE FIRST EXAMPLE OF MAXIMALITERATIVE ALGEBRA OF THE FUNCTIONSOF TOPOLOGICAL BOOLEAN ALGEBRA OFORDER 16 WITH 3 OPEN ELEMENTS

ROMAI J., 6, 2(2010), 221–229

Mefodie T. RatiuInstitute of Mathematics and Computer Science, Academy of Sciences of Moldova, Chisinau,Republic of [email protected]

Abstract A.I. Mal’tsev [6] proposed the problem to obtain the description of iterative algebrasof functions in propositional logics. From functional point of view a special interestrepresent the separation and description of maximal iterative algebras. L. Esakia andV. Meshi [1] and independly L.Maksimova [5] discovered the existence of pre-tabularmodal logic EM4. It is approximated by the logics of a series of topological Booleanalgebras ∆i(i = 1, 2, . . . ) of order 2i with 3 open elements. In this work, for first time, amaximal subalgebra of iterative algebra of functions of topological Boolean algebra oforder 16 with one open atom is built.

Keywords: iterative algebra, maximal algebra, topological boolean algebra.2000 MSC: 03B45, 03B50.

1. INTRODUCTIONThe study of iterative algebras, initiated by E. Post [7] and A.I. Mal’tsev [6], is

closely related to functional problems of the theory of multivalent logics and oflogical calculi. Any iterative algebra is supported by a closed class of k-valent(k = 2, 3, . . . ) logic functions or by a functions of a logics defined by logical cal-culi. The superposition, in its diverse variations, plays an essential role as signatureof these algebras. But let us remember that any superpositions may be expressedeasily by a finite number of special superposition indicated, for example, in the work[6, p.25]. We can conclude that iterative algebras have finite signature. Thereforethe research in terms of such algebras is equivalent to the investigation of their bases,which, in most cases, are closed classes of k-valent logic functions.

The general problem of description of closed classes for k-valent logic, with k ≥ 3,is equivalent to the problem of the description of corresponding iterative algebras, butit is complicated in principle by the existence of subalgebras with infinite numerablebases as well as by the existence of subalgebras without base [9, 11]. Consequently,the general problem of the description of iterative algebras remains open. In moredetails was examined the problem of (functional) completeness in some iterative al-

221

222 Mefodie T. Ratiu

gebra. It requires to be clarified necessary and sufficient conditions that an arbitrarysystem of its elements to generate full algebra. A proper subalgebra of a given alge-bra is called maximal if supplementing it with any element of the difference of thesealgebras generate the whole given algebra [6]. The maximal subalgebras play a veryspecial role in the completeness problem of systems in the iterative algebras. Remem-ber that in any k-valent logic to any maximal subalgebra it corresponds respectivelysome pre-complete class of functions or formulas.

In this work for first time it is built one maximal subalgebra of iterative algebraof topological Boolean algebra of order 16 with 3 open elements including 2 trivialitems and one atom.

2. PRELIMINARIESBy a topological Boolean algebra [8] we mean any universal algebra < M; &,∨,⊃

,¬, >, such that the system, < M; &,∨,⊃,¬ > is a Boolean algebra, and thesymbol represents the operation of taken inside. In this work a significant rolewill be played by the series of following topological boolean algebras: ∆1 with 2elements, ∆2 with 4 elements, . . . , ∆n with 2n elements, . . . , every of them having2 trivial open elements and one open atom. This algebras can be represented by aseries of diagrams, from which we present the following four (see the Figures 1 and2):

u

u

u

u

u

u u

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eee¿

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u

u

uuc

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,,,l

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½½

½½ cc

cc1

0 0 0

11

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The elements 0 and 1 represent the smallest and respectively the greatest elementsof algebra, while other elements of ∆i (i = 2, 3, 4) are denoted by small letters ofGreek alphabet. The open elements of algebra are marked by (square). The con-junction of two elements of algebra is determined on the diagram by the lowest edgeup and the disjunction - by the biggest bound down to those elements. The inside ofany element α, i.e. α, is the greatest open element that is less than α. In [1, 5] itwas remarked that the series of algebras ∆1, ∆2, . . . generates a variety of topologicalBoolean algebras. It is known [2] that the modal logics represent an important familyof logics which, unlike k-valent logic, not every of them can be described by finitemodels. Between topological Boolean algebras and modal logics there is a closely

The first example of maximal iterative algebra... 223

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relation. Namely, for any modal logic L there exists a topological Boolean algebrawhose logic coincides with L.

Modal formulas are traditionally built from variables p, q, r, . . . (sometimes withindexes) with logical operators & (conjunction), ∨ (disjunction), ⊃ (implication), ¬(negation), (necessity) and parentheses. Operator ¬¬ (possibility) is denoted bydiamond . Let us notice that the logical operators and the operations of topologicalBoolean algebra signature are denoted by the same symbol, which facilitates theinterpretation of formulas on algebras. Formulas (p &¬p) and (p ⊃ p) are calledconstants and we note them by ciphers 0 and 1, respectively. The formulas usuallywe denote by capital letters of the Latin alphabet. The formula (A ⊃ B) &(B ⊃ A)is called the equivalence of formulas A and B and it is denoted by symbol A ∼ B.An arbitrary function which is realized by a formula F after its interpretation on atopological Boolean algebra ∆ will be called a function F of the algebra ∆.

When interpreting modal formulas on a topological Boolean algebra ∆, the setof all modal formulas identically equal to 1 on ∆ constitutes a modal logic that isdenoted by L∆. The best-known modal logics are the logic S 4 and the logic S 5 [2],the latter being simpler and it responding better to ancient thoughts about modalities.

A modal logic is called tabular if there is a finite topological Boolean algebra ∆

such that L = L∆. A modal logic is called pre-tabular, if it is not tabular, but any ofits proper extensions is already a tabular logic. The logic S 5 is a well known example[5] of pre-tabular modal logic. In the works [1, 5] one new pre-tabular modal logic -the logic of EM4 was presented. This logic is obtained by adding to S 4 the followingthree formulas as new axioms:

p ⊃ p, T (Z), T (Y),


whereZ = ((p ⊃ q) ∨ (q ⊃ p)), Y = ((¬¬p &((p ⊃ q) ⊃ q)) ⊃ q),

and the operator T means prescribing square before of each subformula. Thismodal logic corresponds fully to the variety of topological Boolean algebras gener-ated by the series of algebras ∆1, ∆2, . . . [1, 5].

Following A.V. Kuznetsov [3, 4] , a formula F is called expressible in a logic Lby a system of formulas Σ, if F can be obtained from the variables and formulas ofΣ with a finite number of applications of weak rule of substitution which allows thetransition from two formulae by substitution of one of them into the another insteadof all entries of a variable, and the rule replacement by an equivalent in the logic Lwhich allows the passage of one formula to any another formula equivalent to it inL. A formula is called direct expressible by Σ if it can be obtained from the variablesand formulas of Σ with a finite number of applications of weak rule of substitution.

We will say that a modal formula F(p1, . . . , pn) preserves a predicate R(x1, . . . , xm)on topological Boolean algebra ∆ if, for any elements αi j ∈ ∆ (i = 1, . . . ,m; j =

1, . . . , n), from the fact that are true the sentences

R(α11, . . . , αm1), . . . ,R(α1n, . . . , αmn)

it result that the statement

R(F[α11, . . . , α1n], . . . , F[αm1, . . . , αmn])

are true. If the predicate R is defined on a finite algebra of a degree not too large, thenR often can be given by its determining matrix

βi j (i = 1, . . . ,m; j = 1, . . . , l)

whose elements belong to the algebra ∆, so that following sentence is true

R(β1k, . . . , βmk) for k = 1, . . . , l.

For example, each of formulae ¬p, p and (p & q) preserves on the algebra ∆4 thefollowing matrix that will be used below

0 α ϕ µ ε τ γ ρ σ δ θ ω ν ψ β 10 α µ ϕ τ ε γ ρ σ δ ω θ ψ ν β 10 ϕ α µ ε γ τ ρ σ θ δ ω ν β ψ 10 ϕ µ α γ ε τ ρ σ θ ω δ β ν ψ 10 µ α ϕ τ γ ε ρ σ ω δ θ ψ β ν 10 µ ϕ α γ τ ε ρ σ ω θ δ β ψ ν 1

. (1)

Taking into account the fact that disjunction and implication are expressible in classi-cal logic by means of conjunction and negation, it follows that any formula preservesthe matrix (1) on ∆4 algebra.


3. THE MAIN RESULTFor any function f (p1, . . . , pn) of algebra ∆4 let us consider it n-dimensional (i.e.

with n entries) table. This picture (i.e. the part where are located function values andnot the argument values) is divided into sectors so as be satisfied that: two arbitrarycells with two sets of coordinates < α1, . . . , αn > and < β1, . . . , βn > belong to thesame sector then and only then when it take place the equality

αi = βi (i = 1, . . . , n).

For example, on the tables below of some functions (i.e. derivative operations) ofalgebra ∆4 the unique lines just divides the table in such sectors.

Table 1

p 0 α ϕ µ ε τ γ ρ σ δ θ ω ν ψ β 1

p 0 0 0 0 0 0 0 0 σ σ σ σ σ σ σ 1p 0 ρ ρ ρ ρ ρ ρ ρ 1 1 1 1 1 1 1 1p 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

¬p & p 0 γ τ ε µ ϕ α 0 ρ γ τ ε µ ϕ α 0p ⊃ p 1 β ψ ν ω θ δ σ 1 β ψ ν ω θ δ 1p ⊃ p 1 δ θ ω ν ψ β 1 σ δ θ ω ν ψ β 1

A cell of sector is called main, if all its coordinates are equal to 0 or 1. Obviously,any sector has a single main cell. If the coordinates of the main cell of a sectorconstitute the collection < α1, . . . , αn >, then we will denote this sector by symbolQ < α1, . . . , αn >.

Theorem 3.1. The iterative algebra of the functions of algebra ∆4, which preservesthe following line matrix with 12 elements

(0 α µ ε γ ρ σ δ ω ν β 1

)(2)

is maximal in the iterative algebra of all functions of ∆4 algebra.

Indeed, we denote by Γ the iterative algebra of all functions of algebra ∆4, whichpreserve the matrix (2). The fact that algebra Γ is proper subalgebra of iterative alge-bra of all functions of ∆4 algebra results from fact that formula p∨q does not preservematrix (2), because there is equality ψ = δ ∨ ω which shows that the disjunction of 2elements δ and ω of the matrix (2) is equal to the element ψ that does not belong tothe matrix (2).

Let us notice now that any element of the matrix (2) belongs to one of the following2 isomorphic subalgebras of algebra ∆4:

< 0, α, γ, ρ, σ, δ, β, 1; Ω >, < 0, µ, ε, ρ, σ, ω, ν, 1; Ω >,


Table 2 p ∨ q

p\q 0 α ϕ µ ε τ γ ρ σ δ θ ω ν ψ β 1

0 0 α ϕ µ ε τ γ ρ σ δ θ ω ν ψ β 1α α α ε τ ε τ ρ ρ δ δ ν ψ ν ψ 1 1ϕ ϕ ε ϕ γ ε ρ γ ρ θ ν θ β ν 1 β 1µ µ τ γ µ ρ τ γ ρ ω ψ β ω 1 ψ β 1ε ε ε ε ρ ε ρ ρ ρ ν ν ν 1 ν 1 1 1τ τ τ ρ τ ρ τ ρ ρ ψ ψ 1 ψ 1 ψ 1 1γ γ ρ γ γ ρ ρ γ ρ β 1 β β 1 1 β 1ρ ρ ρ ρ ρ ρ ρ ρ ρ 1 1 1 1 1 1 1 1σ σ δ σ ω ν ψ β 1 σ δ θ ω ν ψ β 1δ δ δ ν ψ ν ψ 1 1 δ δ ν ψ ν ψ 1 1θ θ ν θ β ν 1 β 1 θ ν θ β ν 1 β 1ω ω ψ β ω 1 ψ β 1 ω ψ β ω 1 ψ β 1ν ν ν ν 1 ν 1 1 1 ν ν ν 1 ν 1 1 1ψ ψ ψ 1 ψ 1 ψ 1 1 ψ ψ 1 ψ 1 ψ 1 1β β 1 β β 1 1 β 1 β 1 β β 1 1 β 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

where Ω = &,∨,⊃,¬,. Therefore, by applying any unary formula to the elementsof matrix (2), only elements of this matrix can be obtained. Thus, any unary formulabelongs to subalgebra Γ. It can be easily checked that the formulas p & q and p∨qbelong to subalgebra Γ.

Let us denote by S 0(p, q) the following binary formula

(p ∨ q) ∨ (p ⊃ q) ∨ (q ⊃ p) ∨ (¬p ∨ ¬q).

Notice that this formula belongs to algebra Γ because the formula S 0 can take only thevalues 0, 1 orσwhich belong to the matrix (2). Let present the table of 2-dimensionalfunction S 0.

Next we need following formula:

T0(p, q) = (p ∨ q ∨ S 0[¬(p ⊃ p),¬(q ⊃ q)])&&(p ∨ ¬q ∨ S 0[¬(p ⊃ p), q])&&(¬p ∨ q ∨ S 0[p,¬(q ⊃ q)])&&(¬p ∨ ¬q ∨ S 0(p, q)).

For use this formula we present below her 2-dimensional table on algebra ∆4.On the basis of the tables of p ∨ q and of T0 it is not difficult to verify that the

following lemma holds.


Table 3 S 0


0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1α 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1ϕ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1µ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1ε 1 1 1 1 1 σ σ 1 1 σ σ 1 1 1 1 1τ 1 1 1 1 σ 1 σ 1 1 σ 1 σ 1 1 1 1γ 1 1 1 1 σ σ 1 1 1 1 σ σ 1 1 1 1ρ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1σ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1δ 1 1 1 1 σ σ 1 1 1 1 σ σ 1 1 1 1θ 1 1 1 1 σ 1 σ 1 1 σ 1 σ 1 1 1 1ω 1 1 1 1 1 σ σ 1 1 σ σ 1 1 1 1 1ν 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1ψ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1β 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Lemma 3.1. FormulaD = (T0(p, q) ⊃ (p ∨ q))

belongs to subalgebra Γ.

The table of formula D on algebra ∆4 can be described by the following scheme

D(p, q) =

ρ in 6 cells in which the value of T0 is σ of thesector Q < 0, 0 >,

1 in the other 18 cells where the value of T0 is σ,p ∨ q in remaining cells.

Next we need the following formula:

T1(p, q) = (p ∨ ¬q ∨ S 0(p, q))&&(p ∨ ¬q ∨ S 0[p, (q ⊃ q)])&&(¬p ∨ q ∨ S 0[(p ⊃ p), q])&&(¬p ∨ ¬q ∨ S 0[(p ⊃ p), (q ⊃ q)]).

Lemma 3.2. Formula K = T1 ⊃ (p & q) belongs to subalgebra Γ.

228 Mefodie T. RatiuTable 4 T0


0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1α 1 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1ϕ 1 σ 1 σ 1 1 1 1 1 σ 1 σ 1 1 1 1µ 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1 1ε 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1τ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1γ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1ρ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1σ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1δ 1 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1θ 1 σ 1 σ 1 1 1 1 1 σ 1 σ 1 1 1 1ω 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1 1ν 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1ψ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1β 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

By using the Lemma 3.1 and Lemma 3.2 it is not difficult to finish the proof ofTheorem 3.1.

References[1] Esakia L., Meshi V., Five critical modal systems, Theoria, 43(1977), 52–62.

[2] Feys R., Modal Logics, Paris, 1965.

[3] Kuznetsov A.V., On functional expressibility in the superintuitionistic Logics. Math. issled.,Kishinev, 6, 4(1971), 75–122 (in Russian).

[4] Kuznetsov A.V., On tools for the discovery of non-deducibility or non-expressibility, Logicaldeduction, Nauka, Moscow, 1979, 5–33 (in Russian).

[5] Maksimova L., Pre-tabular extensions of Lewis’s Logic, Algebra and Logic, 14, 1(1975), 28–55(in Russian).

[6] Mal’tsev A.I., Post’s iterative algebras. Novosibirsk, 1976 (in Russian).

[7] Post E.L., Two-valued iterative systems of mathematical logic, Princeton, 1941.

[8] Rasiowa E., Sikorski R., The Mathematics of metamathematics, Warszawa, 1963.

[9] Ratsa M. Iterative chain classes of pseudo-boolean functions, Kishinev, 1990 (in Russian).

[10] Scroggs S. I., Extensions of the Lewis system S5, J.Simb.Logic, 6, 2 (1951), 112–120.

[11] Yanov Y.I., Muchnik A.A., About the existence of k-valued closed classes having not finite bases,Doklady AN SSSR, 127, 1(1959), 44–46 (in Russian).

ON MIDDLE BOL LOOPS

ROMAI J., 6, 2(2010), 229–236

Parascovia SarbuState University of Moldova, Chisinau, Republic of [email protected]

Abstract A loop Q(·) is middle Bol if every its loop isotope has the anti-automorphic inverseproperty: (xy)−1 = y−1 x−1. Every middle Bol loop can be obtained using left (right) Bolloops ([2]). Connections between structure and properties of middle Bol loops and ofthe corresponding left Bol loops are studied in the present work.

Keywords: left Bol loop, middle Bol loop, isotopy, autotopism.2000 MSC: 20N05.

1. INTRODUCTIONMiddle Bol loops were defined by V. Belousov ([1]). A loop Q(·) is called middle

Bol if the primitive loop Q(·, /, \) satisfies the identity

x(yz \ x) = (x/z)(y \ x). (1)

The identity (1) corresponds to the third closure condition B3 in algebraic netsso it is universal in loops, i.e. any loop isotope of a middle Bol loop is middle Bol([1]). In particular, every Moufang loop is middle Bol. Note that the identity (1)is a generalization of the Moufang identity (x · yz)x = xy · zx. It was proved byGwaramija in [2] that: if Q(·) is a left (right) Bol loop then the groupoid Q(), wherex y = y(y−1x · y) (respectively, x y = (y · xy−1)y), ∀x, y ∈ Q, is a middle Bol loopand, conversely, if Q() is a middle Bol loop then there exists a left (right) Bol loopQ(·) such that x y = y(y−1x · y) (respectively, x y = (y · xy−1)y), ∀x, y ∈ Q. Thisresult implies that if Q(·) is a left Bol loop and Q() is the corresponding middle Bolloop then x y = x/y−1 and x · y = x//y−1, where ”/” (”//”) is the left division in Q(·)(resp. in Q()). The class of left (right) Bol loops is intensively developed at present(see, for example [4])

Connections between structure and properties of middle Bol loops and of the cor-responding left Bol loops are studied in the present work. It is noted that two middleBol loops are isomorphic if and only if the corresponding left (right) Bol loops areisomorphic, and a general form of the autotopisms of middle Bol loops is deduced.Relations between different sets of elements, such as nucleus, left (right, middle) nu-clei, the set of Moufang elements, the center, etc. of a middle Bol loop and of thecorresponding left Bol loop are established.

229

230 Parascovia Sarbu

2. MIDDLE BOL LOOPS AND THECORRESPONDING LEFT BOL LOOPS

Proposition 2.1. Two middle Bol loops Q() and Q(⊗) are isomorphic if and only ifthe corresponding left Bol loops are isomorphic.

Proof. Let Q(·) and Q(∗) be two left Bol loops and let Q() and Q(⊗) the correspond-ing middle Bol loops, respectively, i.e. x y = y(y−1x ·y) and x⊗y = y∗ ((y−1 ∗ x)∗ y),∀x, y ∈ Q. If Q(·) Q(∗) and if ϕ is an isomorphism from Q(·) to Q(∗), then forevery x, y ∈ Q:

x y = y(y−1x · y) = ϕ−1(ϕ(y) ∗ ((ϕ(y−1) ∗ ϕ(x)) ∗ ϕ(y))) =

ϕ−1(ϕ(x) ⊗ ϕ(y)),

so Q() Q(⊗). Conversely, if Q() and Q(⊗) are isomorphic and if ϕ is an isomor-phism from Q() to Q(⊗), then ϕ(x y) = ϕ(x) ⊗ ϕ(y) ⇒ ϕ(x/y−1) = ϕ(x)//ϕ(y)−1,where ”/” (”//”)is the left division in Q(·) (resp., in Q(∗)). Denoting x/y−1 by z weget x = z · y−1, so the last equality implies:

ϕ(z) = ϕ(z · y−1)//ϕ(y)−1 ⇒ϕ(z · y−1) = ϕ(z) ∗ ϕ(y)−1 ⇒ϕ(z · y−1) = ϕ(z) ∗ ϕ(y−1),

∀x, y ∈ Q, i.e. ϕ is an isomorphism from Q(·) to Q(∗).Remark 2.1. It is known ([2]) that:

a) The unit in a left Bol loop Q(·) and in the corresponding middle Bol loop Q()is the same. Moreover, the left (right) inverse of an element x in Q(·) coincide withthe left (right) inverse of x in Q().

b) Since both left and middle Bol loops are power-associative, the equality x y =

y(y−1x · y) implies that each element x has the same order in Q(·) and in Q(). It isknown ([3]) that the order of each element in a finite left Bol loop divides the orderof the loop. So the same property is valid in finite middle Bol loops.

Let Q(·) be a loop and let consider:

i. the set of all Moufang elements of Q(·):M(·) = a ∈ Q| ax · ya = (a · xy)a,∀x, y ∈ Q,

ii. the set of all Bol elements of Q(·):B(·) = a ∈ Q| a(x · ay) = (a · xa)y,∀x, y ∈ Q,

iii. the left nucleus of Q(·):N(·)

l = a ∈ Q| a · xy = ax · y,∀x, y ∈ Q,iv. the right nucleus of Q(·):

N(·)r = a ∈ Q| xy · a = x · ya,∀x, y ∈ Q,

On middle Bol loops 231

v. the middle nucleus of Q(·):N(·)

m = a ∈ Q| xa · y = x · ay,∀x, y ∈ Q.vi. the Moufang center of Q(·):

C(·) = a ∈ Q| a2 · xy = ax · ay,∀x, y ∈ Q

Lemma 2.1. Let Q(·) be a left Bol loop and let Q() be the corresponding middle Bolloop. Then T1 = (α, β, γ) ∈ S 3

Q is an autotopism of Q(·) if and only if T2 = (γ, IβI, α)(where I(x) = x−1,∀x ∈ Q) is an autotopism of Q().Proof. As mentioned above, x · y = x//y−1 = x//I(y), ∀x, y ∈ Q, where ”//” is theleft division in Q(). If T1 = (α, β, γ) ∈ S 3

Q is an autotopism of Q(·) then γ(xy) =

α(x) · β(y), ∀x, y ∈ Q, i.e. γ(x//I(y)) = α(x)//Iβ(y), ∀x, y ∈ Q, which is equivalentto α(x) = γ(x//y) IβI(y), ∀x, y ∈ Q. Taking x//y = z in the last equality, we getα(z y) = γ(z) IβI(y), ∀x, y ∈ Q, i.e. (γ, IβI, α) is an autotopism of the loop Q().Corollary 2.1. The group of autotopisms of a middle Bol loop is isomorphic to thegroup of autotopisms of the corresponding left Bol loop.

Proof. Indeed, the mapping σ : (α, β, γ) → (γ, IβI, α) is an isomorphism betweentwo groups.

Proposition 2.2. Let Q(·) be a left Bol loop and let Q() be the corresponding middleBol loop. The following sentences are true:1. N()

l = N()r j M(), 2. N(·)

m = N(·)l = N()

l , 3. N(·)r = N()

m , 4. N(·) = N(),5. M(·) ∩ N(·)

l ⊆ N()m , 6. C(·) j B(), 7. C(·) = C().

Proof. 1. The equality N()l = N()

r is known (see, for example, [2]). If a ∈ N()l =

N()r then T1 = (La, ε, La) and T2 = (ε,Ra,Ra) are autotopisms of Q(), so T1T2 =

(La,Ra,RaLa) is an autotopism of Q(), i.e. a ∈ M().

2. If a ∈ N ·m then L()a = a x = x(x−1a · x) = x(x−1 · ax) = ax = L(·)

a (x), ∀x, y ∈ Q,so L()

a = L(·)a . Hence a ∈ N(·)

l = N(·)m if and only if T = (L(·)

a , ε, L(·)a ) is an autotopism

of Q(·) so, according to Lemma 2.1, if and only if T = (L()a , ε, L()

a ) is an autotopismof Q(), i.e. if and only if a ∈ N()

l .

3. If a ∈ N(·)r then R()

a (x) = x a = a(a−1x · a) = a(a−1 · xa) = x · a = R(·)a (x),

∀x ∈ Q, so R(·)a = R()

a . According to Lemma 2.1, a ∈ N(·)r if and only if (ε,R(·)

a ,R(·)a )

is an autotopism of Q(·), i.e., if and only if (R(·)a , IR(·)

a I, ε) = (R()a , IR()

a I, ε) is anautotopism of Q().

But IR()a I(x) = IR()

a (x−1) = I(x−1 a) = a−1 x = L()a−1(x), ∀x ∈ Q, so IR()

a I =

L()a−1 . Hence, if a ∈ N(·)

r then (R()a , L()

a−1 , ε) is an autotopism of Q(), i.e. x y =

(xa) (a−1y), ∀x, y ∈ Q. Taking x = e in the last equality, we have y = a (a−1y),∀x, y ∈ Q so, for y = a z the same equality implies x (a z) = (x a) z, ∀x, z ∈ Q,i.e. a ∈ N()

m and then N(·)r j N()

m .


On the other hand, a ∈ N()m ⇔ (xa)y = x (ay), ∀x, y ∈ Q. Taking y = a−1z,

we get (xa)(a−1z) = x(a(a−1z)) = xz, so (R()a , L()

a−1 , ε) = (R(·)a , IR(·)

a I, ε) is an

autotopism of Q(). Now, according to Lemma 2.1, (ε,R()a ,R()

a ) is an autotopism ofQ(·), which implies R()

a (x · y) = x · R()a (y) and, taking y = e we get: R()

a (x) = R(·)a (x),

∀x ∈ Q, so R()a = R(·)

a , hence (ε,R(·)a ,R

(·)a ) is an autotopism of Q(·), which means that

a ∈ N(·)r and N()

m j N(·)r . So, we proved that N()

m = N(·)r .

4. Remind that N = Nl ∩ Nr ∩ Nm, so the proof follows from 1.-3.

5. If a ∈ M(·) then R()a (x) = x a = a(a−1x · a) = x · a = R(·)

a (x), ∀x ∈ Q, soR()

a = R(·)a .

On the other hand, a ∈ M(·) if and only if (L(·)a ,R

(·)a , L

(·)a R(·)

a ) is an autotopism ofQ(·), hence (L(·)

a R(·)a , IR(·)

a I, L(·)a ) is an autotopism of Q().

So as IR(·)a I = IR()

a I ⇒ IR()a I(x) = I(x−1 a) = a−1 x, ∀x ∈ Q, i.e. IR()

a I = L(·)a−1 ,

we get that a ∈ M(·) if and only if,

T = (L(·)a R()

a , L()a−1 , L

(·)a ) = (L(·)

a , ε, L(·)a )(R()

a , L()a−1 , ε)

is an autotopism of Q(). If a ∈ M(·) ∩ N(·)l then T and (L(·)

a , ε, L(·)a ) are autotopisms

of Q() so (R()a , L()

a−1 , ε) is an autotopism of Q(), which implies (x a) (a−1 y) =

x y, ∀x, y ∈ Q. Taking x = e in the last equality, it follows that L()a−1 = L()−1

a , so

(R()a , L()−1

a , ε), i.e. a ∈ Nm.

6. It follows from the definition of the Moufang center that

C(·) = a ∈ Q|a2 · xy = ax · ay,∀x, y ∈ Q =

a ∈ Q|a · (a · xy) = ax · ay,∀x, y ∈ Q.Let us remark now that if a ∈ C(·) than a · x = x · a, for every x ∈ Q. Indeed, taking

y = e in the equality a · (a · xy) = ax ·ay we get a · (a · x) = ax ·a, ∀x ∈ Q and denotingax by z: az = za, ∀z ∈ Q.

Also, if a ∈ C(·) then, taking x → a−1x and y → a−1y in the equality a · (a · xy) =

ax · ay, we have a[a · (a−1x · a−1y)] = xy, which implies a−1x · a−1y = a−1(a−1 · xy),i.e. a−1 ∈ C(·). Here we used the equalities a−1 · ax = x and xa · a−1 which are truefor all a ∈ C(·) and all x ∈ Q.

Now, using Lemma 2.1 and the equality L(·)a = R(·)

a , we get that a ∈ C(·) if and onlyif (R(·)

a ,R(·)a ,R

(·)2a ) is an autotopism of Q(·), i.e. if and only if (R()2

a , IR()a I,R()

a ) is anautotpism of Q().

So as IR()a I(x) = I(x−1 a) = a−1 x = L()

a−1(x) we get that a ∈ C(·) if and only ifthe identity

(x y) a = [(x a) a] (a−1 y) (2)


holds for every x, y ∈ Q. Now, taking x = a−1 in (2), we have (a−1y)a = a(a−1y)so, denoting a−1 y by z, the last equality means z a = a z, for every z ∈ Q, i.e.R()

a = L()a . Also, taking y = e in (2), we get x a = [(x a) a] a−1 so, denoting

x a by z, it implies z = (z a) a−1, ∀z ∈ Q.Remark that if a ∈ C(·) then R()

a−1 = L()a−1 so a−1 x = x a−1, ∀x ∈ Q. Hence, from

the equality z = (z a) a−1 it follows z = a−1 (a z), ∀z ∈ Q.Now, taking y = c y in (2), we get a (x (a y)) = (a (x a)) y, ∀x, y ∈ Q,

i.e. a ∈ B(), so C(·) j B().

7. Using 6. and Lemma 2.1, we get that a ∈ C(·) if and only if the identity (2)holds. But the equality (2) is equivalent to a (x y) = (a (x a)) (a−1 y), so it isequivalent to the equality a (a (x y)) = a ((a (x a)) (a−1 y)). Now, usingthe properties of the elements of C(), deduced in 6., we have:

a ((a (x a)) (a−1 y)) = a (((x a) a) (a−1 y)) =

(((x a) a) (a−1 y)) a = (x a) ((a (a−1 y)) a) =

(x a) (y a) = (a x) (a y),

∀x, y ∈ Q, i.e. a ∈ C(). So, C(·) = C().

Proposition 2.3. The Moufang center of a left Bol loop is a subloop.

Proof. Indeed, let Q(·) be a left Bol loop and let C(·) be its Moufang center. So as forevery a, b ∈ C(·) we have x · a = b⇒ x = b · a−1 and a · x = b⇒ x = a−1 · b, C(·) is asubloop if and only if it contains the product of every two its elements. For a, b ∈ C(·)we have:

ab · (ab · xy) = ab · [ab · (ax′ · ay

′)] = ab · [ab · a(a · x′y′)] =

ab · a2[b(a · x′y′)] = a2[b · a(b · (a · x′y′))] = a2 · [(b · ab)(a · x′y′)] =

a2 · [ab2 · (a · x′y′))] = a2 · [a2 · (b2 · x′y′))] = a4 · (b2 · x′y′),where x

′= ax and y

′= ay. On the other hand,

(ab · x) · (ab · y) = (ab · ax′) · (ab · ay

′) = (a2 · bx

′)(a2 · by

′) =

a4(bx′ · by

′) = a4(b2 · x′y′),

where x′

= ax and y′

= ay. Hence ab · (ab · xy) = (ab · x) · (ab · y), for every x, y ∈ Q,i.e. ab ∈ C(·), so C(·) is a subloop of Q(·).Proposition 2.4. If L(·) is a subloop of a left Bol loop Q(·) then L() is a subloop ofthe corresponding middle Bol loop Q().Proof. Indeed, if L(·) is a subloop of a left Bol loop Q(·) then for every x, y ∈ L,x y = y(y−1x · y) ∈ L. More, if a, b ∈ L then each of the equations a x = b and


y a = b has its solution in L: a x = b ⇔ x = (b \ a)−1 ∈ L and, analogously,y a = b⇔ y/a−1 = b⇔ y = b · a−1 ∈ L.

Corollary 2.2. From first two remarks and the equality C(·) = C() it follows that theMoufang center C() is a subloop of the middle Bol loop Q().

Let Q(·, /, \) be an arbitrary loop . An element a ∈ Q is called a middle Bolelement if the equality

a(yz \ a) = (a/z) · (z \ a),

holds for every y, z ∈ Q ([2]). So, we can consider the mapping

Ia : Q→ Q, Ia(x) = x \ a,

which is a bijection. From this definition it follows that I−1a = a/x, ∀x ∈ Q, and that

a ∈ Q is a middle Bol element if and only if LaIa(xy) = I−1a (y) · Ia(x), x, y ∈ Q, i.e. if

and only if (I−1a , Ia, LaIa) is an anti-autotopism of Q(·).

Lemma 2.2. If Q(·) is a middle Bol loop then, for every a ∈ Q, the triple T =

(I−1a I, IaI, LaIaI) is an autotopism of Q(·).Proof. Indeed, so as (I−1

a , Ia, LaIa) and (I, I, I) are anti-autotopisms of Q(·) it fol-lows that T is an autotopism of Q(·).

Remind that if Q(·) is a quasigroup then a bijection τ : Q → Q is called a left(right) pseudo-automorphism if there exists an element c ∈ Q such that c · τ(xy) =

(c · τ(x))τ(y) (respectively, τ(xy) · c = τ(x)(τ(y) · c)), ∀x, y ∈ Q. In this case theelement c ∈ Q is called a left (resp. right) companion of τ. So, it follows from thedefinition that τ is a left (right) pseudo-automorphism with the companion c if andonly if (Lcτ, τ, Lcτ) (resp., (τ,Rcτ,Rcτ)) is an autotopism of Q(·).

Proposition 2.5. If Q(·) is a left Bol loop and c ∈ Q then a bijection τ ∈ S Q is aleft pseudo-automorphism with the companion c if and only if IτI is a left pseudo-automorphism with the companion c of the corresponding middle Bol loop.

Proof. The mapping τ ∈ S Q is a left pseudo-automorphism of Q(·) with the compan-ion c if and only if (L(·)

c τ, τ, L(·)c τ) is an autotopism of Q(·) so, according to Lemma

2.1, if and only if (L(·)c τ, IτI, L(·)

c τ) is an autotopism of the corresponding middle Bolloop Q(), i.e. if and only if the equality

L(·)c τ(x y) = L(·)

c τ(x) IτI(y),

holds for every x, y ∈ Q. Taking x = e in the last equality we get L(·)c τ(y) = L()

c IτI(y),where L()

c is the left translation in the loop Q(), i.e.

(L(·)c τ, IτI, L(·)

c τ) = (L()c IτI, IτI, L()

c IτI),


which means that IτI is a left pseudo-automorphism of Q() with the companion c.

Corollary 2.3. The group of left pseudo-automorphisms of a left Bol loop Q(·) isisomorphic to the group of left pseudo-automorphisms of the corresponding middleBol loop.

Proof. Indeed, the mapping τ→ IτI is an isomorphism between two groups.

It is known ([1]) that every autotopism T = (α, β, γ) of a left Bol loop Q(·) can berepresented as follows:

T = (L(·)k τ, τ, L

(·)k τ)(R(·)−1

b L(·)−1b , Lb, L

(·)−1b ), (3)

where τ is a left pseudo-automorphism with the companion k of the loop Q(·) andb = β(e), e being the unit of Q(·).

Proposition 2.6. Every autotopism T = (α, β, γ) of a middle Bol loop Q() can berepresented in the form:

(L()k τ, τ, L()

k τ)(IIa, II−1a , IIaL()−1

a ), (4)

where τ is a left pseudo-automorphism of Q() with the companion k, a = β(e).

Proof. If T = (α, β, γ) is an autotopism of a middle Bol loop Q(, //, \\) then, ac-cording to Lemma 2.1, T1 = (γ, IβI, α) is an autotopism of the corresponding left Bolloop Q(·, /, \).

According to the general form (3), there exists a left pseudo-automorphism τ′

ofQ(·) with the companion k and an element b = β(e) ∈ Q, where e is the unit of Q(·),such that

(γ, IβI, α) = (L(·)k τ

′, τ′, L(·)

k τ′)(R(·)−1

b L(·)−1b , Lb, L

(·)−1b ). (5)

From the equality x ·y = x//I(y) it follows x · I(y) = x//y = I−1x (y), so L·xI(y) = I−1

x (y),∀y ∈ Q, i.e.

L·x = I−1x I, (6)

for every x ∈ Q. So as Q() is a middle Bol loop, according to Lemma 2.2, the tuplesT = (I−1

a I, IaI, L()a IaI) and T−1 = (IIa, II−1

a , II−1a L()−1

a ) are autotopisms of Q() forevery x ∈ Q.

On the other hand, if (R(·)−1b L(·)−1

b , L(·)b , L

(·)−1b ) is an autotopism of Q(·), then T =

(L(·)−1b , IL(·)

b I,R(·)−1b ) is an autotopism of Q(·). Now, taking b = a and using the equal-

ity (6), we get that

T1 = (IIa, II−1a ,R(·)−1

a IIa)


is an autotopism of Q(). So as the autotpisms T−1 and T1 have the same two com-ponents, we get T−1 = T and

R(·)−1a IIa = II−1

a L()−1a (7)

So, from (5), it follows

(α, β, γ) = (L(·)k τ

′, Iτ

′I, L(·)

k τ′)(L(·)−1

b , IL(·)b I,R(·)−1

b L(·)−1b ). (8)

Remark now that the mapping Iτ′I is a left pseudo-automorphism of Q() with the

companion k. Indeed, so as (L(·)k τ

′, Iτ

′I, L(·)

k τ′) is an autotopism of Q(), we have

L(·)k τ

′(x y) = L(·)

k τ′(x) Iτ

′I(y), ∀x, y ∈ Q, which implies for x = e: L(·)

k τ′(y) =

L()k Iτ

′I(y), ∀y ∈ Q, i.e. L(·)

k τ′

= L()k Iτ

′I. Hence

(L(·)k τ

′, Iτ

′I, L(·)

k τ′) = (L()

k Iτ′I, Iτ

′I, L()

k Iτ′I) (9)

is an autotopism of Q(), i.e. Iτ′I is a pseudo-automorphism of Q() with the com-

panion k. Denoting Iτ′I by τ, b by a and using (6), (7) and (9), from (8) we get:

(α, β, γ) = (L()k τ, τ, L()

k τ)(IIa, II−1a , IIaL()−1

a ),

where a = β(e).

Corollary 2.4. If T = (α, β, γ) is an autotopism of a middle Bol loop Q() then thereexists a left pseudo-automorphism τ of Q() with a companion k such that

T = (I−1k τI, τ, IkτI)(IIa, II−1

a , IIaL()−1a )

where a = β(e).

Proof. Let Q() be a middle Bol loop and let Q(·) be the corresponding left Bol loopthen, for a ∈ Q, we have Ia(x) = x \ \a and I−1

a (x) = a//x, so Ia = IL(·)−1a i.e.

L(·)a = I−1

a I.

This work was partially supported by the CSCDT ASM institutional project No. 06.411.027F.

References[1] V. Belousov, Foundations of the theory of quasigroups and loops, Nauka, Moscow, 1967.(Rus-

sian)

[2] A. Gwaramija, On a class of loops, Uch. Zapiski MGPI., 375, (1971), 25-34. (Russian)

[3] P. Syrbu, Loops with universal elasticity, J. Quasigroups and Rel. Syst., 1, 1(1994), 57-65.

[4] G. Nagy, A class of simple proper Bol loops, Manuscripta Math., 1, 127(2008), 81-88.

ALTMAN ORDERING PRINCIPLESAND DEPENDENT CHOICES

ROMAI J., 6, 2(2010), 237–244

Mihai Turinici”A. Myller” Mathematical Seminar; ”A. I. Cuza” University;Iasi, [email protected]

Abstract Some ordering principles related to Altman’s [Nonlinear Analysis, 6 (1982), 157-165]are equivalent with the Dependent Choices Principle. This is also true for many inter-mediary statements, including the Szaz unboundedness criterion [Math. Moravica, 5(2001), 1-6].

Keywords: Quasi-order, pseudometric, (ascending) Cauchy/asymptotic sequence, regularity, maximalelement, sequential inductivity, monotone function, unboundedness.2000 MSC: 49J53, 47J30.

1. INTRODUCTIONLet M be some nonempty set; and (≤), some quasi-order (i.e.: reflexive and tran-

sitive relation) over it. Denote M(x,≤) := t ∈ M; x ≤ t, x ∈ M; the subsets M(x,≥),x ∈ M, are introduced in a similar way, modulo (≥) (=the dual quasi-order). Further,take some application F : M × M → R with the properties

(a01) F is (≤)-pseudometric: F(x, y) ≥ 0, when x ≤ y

(a02) ∀y ∈ M, F(·, y) is decreasing on M(y,≥).

Let ψF stand for the function (from M to R+ ∪ ∞)ψF(x) = supF(x, y); y ∈ M(x,≤), x ∈ M.

From (a02), ψF is (≤)-decreasing; for, if x1 ≤ x2, then (denoting for simplicity Mi =

M(xi,≤), i ∈ 1, 2),ψF(x1) = supF(x1, y); y ∈ M1 ≥supF(x1, y); y ∈ M2 ≥ supF(x2, y); y ∈ M2 = ψF(x2).

On the other hand, the alternative ∞ ∈ ψF(M) cannot be avoided, in general. But,under a regularity condition like

(a03) F(x, ·) is bounded above over M(x,≤), ∀x ∈ M,

this happens [in the sense: ψF(M) ⊆ R+].

237

238 Mihai Turinici

Call z ∈ M, (≤, F)-semi-maximal when: z ≤ w ∈ M =⇒ F(z,w) = 0; note that, interms of the above introduced function, this amounts to: ψF(z) = 0. The following1982 Altman’s ordering principle [1] (in short: AOP) is our starting point:

Theorem 1.1. Let (a01)–(a03) hold, as well as

(a04) (M,≤) is sequentially inductive:each ascending sequence is bounded above

(a05) (M,≤, F) is semi regular:((xn)=ascending) =⇒ lim infn F(xn, xn+1) = 0.

Then, for each u ∈ M there exists a (≤, F)-semi-maximal v ∈ M with u ≤ v.

[In fact, the original statement was formulated in terms of the dual quasi-order (≥)and the function Φ(x, y) = −F(y, x), x, y ∈ M. Moreover (in terms of this translation),(a02) and (a05) were taken – respectively – in their stronger form

(a06) ∀y ∈ M, F(·, y) is decreasing [x1 ≤ x2 =⇒ F(x1, y) ≥ F(x2, y)]

(a07) (M,≤, F) is regular: [(xn)=ascending] =⇒ limn F(xn, xn+1) = 0.

But, the author’s reasoning also works in this relaxed setting].This result includes the 1976 Brezis-Browder ordering principle [2] (in short: BB).

Moreover, as precise by the quoted authors, their statement includes Ekeland’s vari-ational principle [5] (in short: EVP); hence, so does AOP. Now, BB and EVP foundsome basic applications to control and optimization, generalized differential calcu-lus, critical point theory and global analysis; see the quoted papers for a survey ofthese. So, it must be not surprising that AOP was subjected to different extensions.For example, a pseudometric enlargement of this ordering principle was obtained byTurinici [11]. Further contributions in the area were provided by Kang and Park [7];see also Zhu and Li [15]. The obtained facts are interesting from a technical view-point. However, we must emphasize that, whenever a maximality principle of thistype is to be applied, a substitution of it by the Brezis-Browder’s is always possible.This raises the question of to what extent are these extensions of BB effective. Asalready shown in Turinici [12], the answer is negative for most of these. It is our aimin the following to provide a simpler proof of this fact, with respect to AOP; detailswill be given in Section 3. Note that, as a consequence, AOP is reducible to the De-pendent Choices Principle (in short: DC) due to Tarski [10]; for a direct verificationwe refer to Section 2.

Now, in a close connection with Theorem 1.1, the following Szaz unboundednesscriterion [9] (in short: S-U) is available:

Theorem 1.2. Suppose that (in addition to (a01)+(a02))

(a08) each ascending sequence (xn) with supn F(x0, xn) < ∞is bounded above and lim infn F(xn, xn+1) = 0

Altman ordering principles and dependent choices 239

(a09) ∀x ∈ M: ψF(x) > 0 [i.e.: ∃y ∈ M(x,≤) with F(x, y) > 0]

Then, ψF(x) = ∞, for all x ∈ M.

This statement is shown to imply a related one in Brezis and Browder [2]; which,as precise there, includes BB. Moreover [according to the author’s observation] hisresult includes AOP as well. It is our aim in the following to show (in Section 4) thatthe reciprocal inclusion is also true; i.e., that S-U is nothing but a logical equivalentof AOP. Further aspects will be delineated elsewhere.

2. DC IMPLIES AOPIn the following, a direct proof of AOP is provided, via DC.(A) Let M be a nonempty set; and R ⊆ M × M stand for a relation on it. For each

x ∈ M, denote M(x,R) = y ∈ M; xRy (the section of R through x).

Proposition 2.1. (Dependent Choices Principle) Suppose that

(b01) M(c,R) is nonempty, for each c ∈ M.

Then, for each a ∈ M there exists (xn) ⊆ M with x0 = a and xnRxn+1, for all n.

This principle, due to Tarski [10] is deductible from the Axiom of Choice (in short:AC), but not conversely; cf. Wolk [14]. Moreover, it suffices for constructing a largepart of the ”usual” mathematics; see Moore [8, Appendix 2, Table 4].

(B) We may now pass to the promised proof of AOP

Proof. Let the premises of the quoted statement be in use. If, by absurd, its conclu-sion is not true, there must be some u ∈ M with (cf. Section 1)

(b02) ψF(x) > 0, for all x in U := M(u,≤).

Let γ be arbitrary fixed in ]0, 1[. We introduce a relation R over U as:

(b03) xRy iff x ≤ y, F(x, y) > γψF(x).

From (b02), U(x,R) is nonempty, for each x ∈ U. So, by (DC), there must be asequence (xn) in U with (x0 = u and) xnRxn+1, ∀n; i.e.,

xn ≤ xn+1 and F(xn, xn+1) > γψF(xn), for all n.

Let v be an upper bound (in U) of this (ascending) sequence (assured by (a04)). AsψF is decreasing, we have [by the above relations] F(xn, xn+1) > γψF(v), ∀n. Passingto lim inf as n→ ∞ yields ψF(v) = 0; in contradiction to (b02). Hence, this workingassumption cannot be accepted; and the conclusion follows.

Note that only the values of F on gr(≤) := (x, y) ∈ M×M; x ≤ y were consideredhere. So, we may substitute (a01) with its extended version

240 Mihai Turinici

(b04) F(x, y) ≥ 0, for all x, y ∈ M (i.e.: F(M × M) ⊆ R+).

For, otherwise, the truncated function F∗(x, y) = max0, F(x, y), x, y ∈ M, (fulfilling(b04)) has all properties of F; and, from the conclusion of Theorem 1.1 written forF∗ we derive the same fact (modulo F); because F∗ = F on gr(≤). Suppose thatthis extension was effectively performed. In this case, we claim that the boundednesscondition (a03) may be dropped. For, the associated function G(x, y) = F(x, y)/(1 +

F(x, y)), x, y ∈ M, fulfills such a property; as well as all remaining ones (involvingF). This, along with the observation that the notions of (≤, F)-semi-maximal and(≤,G)-semi-maximal are identical, proves our claim. Further technical aspects maybe found in Turinici [11]; see also Kang and Park [7].

3. AOP =⇒ (S-U) =⇒ BBUnder these preliminary facts, we may now return to our initial setting.(A) Concerning the former inclusion, its original proof uses Altman’s ideas [1] for

establishing Theorem 1.1. A slightly different reasoning is that given below.

Proof. Assume that conclusion of Theorem 1.2 would be false:

(c01) ψF(x0) < ∞, for some x0 ∈ M.

(Here, ψF : M → R ∪ ∞ is that of Section 1). Denote for simplicity M0 =

M(x0,≤). Given the ascending sequence (yn) in M0, we have F(y0, yn) ≤ ψF(y0) ≤ψF(x0) < ∞, ∀n; wherefrom (by (a08)), (yn) is bounded above in M (hence in M0)and lim infn F(yn, yn+1) = 0; so that, (a04)+(a05) hold over M0. Moreover, (a03) isclearly true (over M0) in view of (c01). Summing up, Theorem 1.1 applies to (M0,≤)and F. It gives us, for the starting point x0 ∈ M0, some z ∈ M0 with the property:z ≤ w ∈ M0 =⇒ F(z,w) = 0; or, equivalently: ψF(z) = 0. But then, (a09) cannothold; contradiction. Hence (c01) is false; and the conclusion follows.

(B) Let again (M,≤) be a quasi-ordered structure; and ϕ : M → R, some function.By the construction

(c02) F(x, y) = ϕ(x) − ϕ(y), x, y ∈ M,

one gives, via (S-U), a basic result in Brezis and Browder [2, Theorem 1] (called:Brezis-Browder’s unboundedness criterion; in short: BB-U).

Theorem 3.1. Assume that ϕ is (≤)-decreasing and

(c03) each ascending sequence (xn) with infn ϕ(xn) > −∞ is bounded above

(c04) ∀x ∈ M, ∃y ∈ M(x,≤) with ϕ(x) > ϕ(y).

Then, inf[ϕ(M(x,≤))] = −∞, ∀x ∈ M.


A basic application of it is to be given under the lines below. Let the triplet (M,≤;ϕ) be introduced as before. Call z ∈ M, (≤, ϕ)-maximal when: z ≤ w ∈ M implyϕ(z) = ϕ(w). The following 1976 Brezis-Browder ordering principle [2] (in short:BB) about the existence of such points, is available.

Theorem 3.2. Let (M,≤) be sequentially inductive and ϕ be (≤)-decreasing, boundedfrom below. Then, for each u ∈ M there exists a (≤, ϕ)-maximal v ∈ M with u ≤ v.

Proof. Assume this is not true; then, there must be some u ∈ M such that (c04)holds with Mu := M(u,≤) in place of M. But then, the preceding statement yieldsinf[ϕ(Mu)] = −∞; in contradiction with the choice of ϕ.

Note that the boundedness from below condition is not essential for the conclusionabove; see Carja, Necula and Vrabie [4, Ch 2, Sect 2.1] for details. Moreover (cf.Zhu and Li [15]), (R,≥) may be substituted by a separable ordered structure (P,≤)without altering the conclusion above; see also Turinici [13].

For the moment, BB follows from AOP, via (S-U). A direct deduction of this isavailable, by the construction (c02). In fact, (a01)+(a02) hold, from the monotonicityof ϕ. On the other hand, (a03) holds too, by the boundedness from below of ϕ; and,finally, (a05) is obtainable from both these. Summing up, Theorem 1.1 is applicableto our data. This, and the observation that (≤, ϕ)-maximal is identical with (≤, F)-semi-maximal (in the context of (c02)) ends the argument.

Finally, note that BB (hence, a fortiori, AOP) includes EVP. Further aspects maybe found in Hyers, Isac and Rassias [6, Ch 5].

4. (BB-LC) IMPLIES (DC)By the developments above, we have the (chain of) implications (DC) =⇒ (AOP)

=⇒ (S-U) =⇒ (BB). So, it is natural asking whether these may be reversed. Thenatural setting for solving such a problem is (ZF)(=the Zermelo-Fraenkel system)without (AC); referred to in the following as the reduced Zermelo-Fraenkel system.

Let X be a nonempty set; and (≤) be an order (i.e.: antisymmetric quasi-order)over it. We say that (≤) has the inf-lattice property, provided: x∧ y := inf(x, y) exists,for all x, y ∈ X. Further, call z ∈ X, (≤)-maximal if X(z,≤) = z; the class of all thesepoints will be denoted as max(X,≤). Accordingly, (≤) is called a Zorn order whenmax(X,≤) is nonempty and cofinal in X [for each u ∈ X, there exists a (≤)-maximalv ∈ X with u ≤ v].

Now, the statement below is a particular case of BB:

Theorem 4.1. Let the partially ordered structure (X,≤) be such that

(d01) (X,≤) is sequentially complete

(d02) (≤) has the inf-lattice property.

In addition, suppose that there exists a function ϕ : X → R+ with

242 Mihai Turinici

(d03) ϕ is strictly decreasing and ϕ(X) is countable.

Then, (≤) is a Zorn order.

We shall refer to it as: the lattice-countable version of BB (in short: BB-Lc).Clearly, (BB) =⇒ (BB-Lc); since (by (d03)) x ≤ y and ϕ(x) = ϕ(y) imply x = y. Theremarkable fact to be added is that this last principle yields (DC); and so, it completesthe circle between all these results.

Proposition 4.1. We have (in the reduced Zermelo-Fraenkel system) (BB-Lc) =⇒(DC). So (by the above) the maximal/unboundedness results (AOP), (S-U) and (BB)are all equivalent with (DC); hence, mutually equivalent.

Proof. Let M be some nonempty set; and R stand for some relation over M withthe property (b01). Fix in the following a ∈ M; as well as some b ∈ M(a,R). Foreach n ≥ 2 in N (= the set of natural numbers), let N(n, >) := 0, ..., n − 1 stand forthe initial segment determined by n; and Xn denote the class of all finite sequencesx : N(n, >) → M with: x(0) = a, x(1) = b and x(m)Rx(m + 1) for 0 ≤ m ≤ n − 2.In this case, N(n, >) is just Dom(x) (the domain of x); and n = card(N(n, >)) will bereferred to as the order of x [denoted as ω(x)]. Put X = ∪Xn; n ≥ 2. Let stand forthe partial order (on X)

(d04) x y iff Dom(x) ⊆ Dom(y) and x = y|Dom(x);

and ≺ denote its associated strict order. All we have to prove is that (X,) has strictlyascending infinite sequences.

(A) Let x, y ∈ X be arbitrary fixed. Denote

K(x, y) := n ∈ Dom(x) ∩ Dom(y); x(n) , y(n).If x and y are comparable (i.e.: either x y or y x; written as: x <> y) thenK(x, y) = ∅. Conversely, if K(x, y) = ∅, then x y if Dom(x) ⊆ Dom(y) and y x ifDom(y) ⊆ Dom(x); hence x <> y. Summing up, we have the characterization

(x, y ∈ X): x <> y if and only if K(x, y) = ∅.The negation of this property means: x and y are not comparable (denoted as: x||y).By the property above, it means: K(x, y) , ∅. Note that, in such a case, k(x, y) :=min(K(x, y)) is well defined; and N(k(x, y), >) it is the largest initial interval of Dom(x)∩Dom(y) where x and y are identical.

Lemma 4.1. The order () has the inf-lattice property. Moreover, the map x 7→ ω(x)is strictly increasing: x ≺ y implies ω(x) < ω(y).

Proof. Let x, y ∈ X be arbitrary fixed. The case x <> y is clear; so, without loss, onemay assume that x||y. Note that, by the remark above, K(x, y) , ∅ and k := k(x, y)exists. Let the finite sequence z ∈ Xk be introduced as z = x|N(k,>) = y|N(k,>). For the


moment z x and z y. Suppose that w ∈ Xh fulfills the same properties. Then,the restrictions of x and y to N(h, >) are identical; wherefrom (see above) h ≤ k andw z; which tells us that z = x ∧ y. The last part is obvious.

(B) Denote ϕ(x) = 3−ω(x), x ∈ X. Clearly, ϕ(X) = 3−n; n ≥ 2; hence, ϕ hascountably many strictly positive values. Moreover, if x, y ∈ X are such that x ≺ y,then (by Lemma 4.1) ϕ(x) > ϕ(y); which tells us that ϕ is strictly decreasing.

(C) From (b01), max(X,) = ∅; i.e.: for each x ∈ X there exists y ∈ X withx ≺ y. This, along with (BB-Lc), tells us that (X,≤) is not sequentially complete:there exists at least one ascending sequence (xn) in X which is not bounded above.By this last property, B(k) := n ∈ N(k, <); xk ≺ xn is nonempty, for all k ∈ N.Define g(n) = min[B(n)], n ∈ N; hence n < g(n) and xn ≺ xg(n), for all n. Theiterative process [p(0) = g(0), p(n + 1) = g(p(n)), n ≥ 0] is therefore constructiblewithout any use of (DC); and gives a strictly ascending sequence of ranks (p(n)). So,(yn := xp(n); n ∈ N) is a subsequence of (xn) with yn ≺ ym (hence ω(yn) < ω(ym)) forn < m; moreover, as ω(y0) > 1, we must have ω(yn) > n + 1, for all n. But then, thesequence (cn = yn(n); n ∈ N) is well defined (in M) with c0 = a and cnRcn+1, for alln. The proof is thereby complete.

In particular, when the specific assumptions (d02) and (d03) (the second half) areignored in Theorem 4.1, Proposition 4.1 is comparable with a statement in Brunner[3]. Further aspects may be found in Turinici [13].

References

[1] M. Altman, A generalization of the Brezis-Browder principle on ordered sets, Nonlinear Analy-sis, 6 (1982), 157-165.

[2] H. Brezis, F. E. Browder, A general principle on ordered sets in nonlinear functional analysis,Advances Math., 21 (1976), 355-364.

[3] N. Brunner, Topologische Maximalprinzipien, Zeitschr. Math. Logik Grundl. Math., 33 (1987),135-139.

[4] O. Carja, M. Necula, I. I. Vrabie, Viability, Invariance and Applications, North Holland Mathe-matics Studies vol. 207, Elsevier B. V., Amsterdam, 2007.

[5] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (New Series), 1 (1979),443-474.

[6] D. H. Hyers, G. Isac, T. M. Rassias, Topics in Nonlinear Analysis and Applications, World Sci.Publ., Singapore, 1997.

[7] B. G. Kang, S. Park, On generalized ordering principles in nonlinear analysis, Nonlinear Anal-ysis, 14 (1990), 159-165.

[8] G. H. Moore, Zermelo’s Axiom of Choice: its Origin, Development and Influence, Springer, NewYork, 1982.

[9] A. Szaz, An Altman type generalization of the Brezis-Browder ordering principle, Math. Morav-ica, 5 (2001), 1-6.

244 Mihai Turinici

[10] A. Tarski, Axiomatic and algebraic aspects of two theorems on sums of cardinals, Fund. Math.,35 (1948), 79-104.

[11] M. Turinici, A generalization of Altman’s ordering principle, Proc. Amer. Math. Soc., 90 (1984),128-132.

[12] M. Turinici, Metric variants of the Brezis-Browder ordering principle, Demonstr. Math., 22(1989), 213-228.

[13] M. Turinici, Brezis-Browder Principles and Applications, in (P. M. Pardalos et al., eds.) ”Nonlin-ear Analysis and Variational Problems” (Springer Optimiz. Appl., vol. 35), pp. 153-197, SpringerScience+Business Media, LLC, 2010.

[14] E. S. Wolk, On the principle of dependent choices and some forms of Zorn’s lemma, Canad.Math. Bull., 26 (1983), 365-367.

[15] J. Zhu, S. J. Li, Generalization of ordering principles and applications, J. Optim. Th. Appl., 132(2007), 493-507.

NOTE

ROMAI J., 6, 2(2010), 245–245

We were informed by Prof. Francesco J. Aragon Artacho from Univ. of Alicante,Spain, about the following unpleasant situation: the paper ”Proximal point methodsfor variational inequalities involving regular mappings” signed by Corina L. Chiriacin ROMAI Journal v. 6, 1(2010), 41-45, was also published in Analele UniversitatiiOradea, facultatea Matematica, TXVII (2010), 65-69 under the title ”Convergence ofthe proximal point algorithm variational inequalities with singular mappings”. More-over, the paper has a consistent intersection with the paper ”Convergence of the prox-imal point method for metrically regular mappings”, by F. J. Aragon Artacho, A. L.Donchev and M. H. Geoffroy, published in ESAIM: Proceedings 17 (2007), 1-8,without even citing this work. Prof. F. J. Aragon Artacho concluded this to be a caseof plagiarism.

The Directory Council of ROMAI compared the three papers involved in the aboveassertions and found the conclusion formulated by Prof. Aragon Artacho as justified.

As a consequence, we, the members of the Directory Council decide to excludethe paper signed by Corina L. Chiriac from the issue v. 6, 1(2010), posted on thewebsite of ROMAI Journal http://rj.romai.ro/. We also consider this paper morallyexcluded from the printed issue (even if we have not the means to do this physically).

We will inform the mathematical databases that are reviewing our journal aboutthis decision.

Corina L. Chiriac will not be allowed to publish again in ROMAI Journal.

Directory Council of ROMAI

245

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