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Springer Optimization and Its Applications

VOLUME 55

Managing EditorPanos M. Pardalos (University of Florida)

Editor–Combinatorial OptimizationDing-Zhu Du (University of Texas at Dallas)

Advisory BoardJ. Birge (University of Chicago)C.A. Floudas (Princeton University)F. Giannessi (University of Pisa)H.D. Sherali (Virginia Polytechnic and State University)T. Terlaky (McMaster University)Y. Ye (Stanford University)

Aims and ScopeOptimization has been expanding in all directions at an astonishing rateduring the last few decades. New algorithmic and theoretical techniqueshave been developed, the diffusion into other disciplines has proceeded ata rapid pace, and our knowledge of all aspects of the field has grown evenmore profound. At the same time, one of the most striking trends in opti-mization is the constantly increasing emphasis on the interdisciplinary na-ture of the field. Optimization has been a basic tool in all areas of appliedmathematics, engineering, medicine, economics, and other sciences.

The series Springer Optimization and Its Applications publishes under-graduate and graduate textbooks, monographs and state-of-the-art exposi-tory work that focus on algorithms for solving optimization problems andalso study applications involving such problems. Some of the topics coveredinclude nonlinear optimization (convex and nonconvex), network flow prob-lems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximationtechniques and heuristic approaches.

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D.A. Klyushin • S.I. Lyashko • D.A. NomirovskiiYu.I. Petunin • V.V. Semenov

Generalized Solutionsof Operator Equationsand Extreme Elements

123

D.A. KlyushinDepartment of CyberneticsKyiv National Taras Shevchenko University01601 [email protected]

S.I. LyashkoDepartment of CyberneticsKyiv National Taras Shevchenko University01601 [email protected]

D.A. NomirovskiiDepartment of CyberneticsKyiv National Taras Shevchenko University01601 [email protected]

Yu.I. PetuninDepartment of CyberneticsKyiv National Taras Shevchenko University01601 [email protected]

SemenovDepartment of CyberneticsKyiv National Taras Shevchenko University01601 [email protected]

ISSN 1931-6828ISBN 978-1-4614-0618-1 e-ISBN 978-1-4614-0619-8DOI 10.1007/978-1-4614-0619-8Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011935362

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In memory of my parents, Nina Andreevnaand Anatoliy Arkhipovich.

Dmitry Klyushin

Dedicated in memory of my father, IvanIvanovich and to my dear family: mother,Vera Stepanovna; wife, Natalie; and children,Lena, Viktor, and Vera.

Sergey Lyashko

To my daughter, Alina.

Dmitry Nomirovskii

In memory of my parents, Zoya Ivanovna andIvan Petrovich.

Yuriy Petunin

To my son, Andrey.

Vladimir Semenov

Preface

“F-friends,” said Fyodor Simeonovich ...“But this is the Ben B-Betzalel’s p-problem.

C-Cagliostro has proved that it does not have a s-solution indeed.”“We do know that it does not have a solution,” said Junta...

“We wish to know how to solve it.”“You are somehow arguing oddly, C-Christo. . .

H-how to s-search for a s-solution, when it does not exist?It’s a nonsense.

“I am sorry, Fyodor, but it’s you who are arguing strangely.The nonsense is to search for a solution when it exists anyway.

The question is how to deal with a problem that does not have a solution.This is a profoundly principled question ...”

A. Strugatsky and B. Strugatsky, “Monday Begins on Saturday”

At the International Mathematical Congress in Paris (1900), D. Hilbert put forthhis famous 23 problems. In Hilbert’s opinion, these problems had to predefine themainstream of mathematics in the twentieth century. By now, most of Hilbert’s prob-lems have been solved successfully. However, despite the fact that many mathe-matical disciplines have arisen and new important problems were put forth in thetwentieth century, Hilbert’s problems remain fundamental [3].

Among Hilbert’s problems, the 20th problem – “the general problem of bound-ary values” – takes its deserved place. This problem is formulated in the followingway: “has not every regular variation problem a solution, provided certain assump-tions regarding the given boundary conditions are satisfied (say that the functionsconcerned in these boundary conditions are continuous and have in sections one ormore derivatives) and provided also if need be that the notion of a solution shall besuitably extended?” (see [25]).

The 20th problem is outstanding because D. Hilbert put it on extending the clas-sical solution when there was neither the concept of completion of metric space northe concept of normed space that serves as a basis of such a notion as “general-ized solution of operator equations”. The idea of the generalized solution is quite

vii

viii Preface

simple: consider an operator equation A(x) = y, where A is a continuous opera-tor (linear or nonlinear) from metric or Banach space E into F . Operator equationscover wide classes of differential equations (including boundary value problems),integral equations, integro-differential equations and more. In many situations, theoperator equation A(x) = y does not have a classical solution, since the right-handside y does not belong to the range R(A)⊂ F of the operator A, but we can introducea weaker topology in E , so that the completion ˜E of E in this topology is a widerspace: E ⊂ ˜E and the operator A can be extended by continuity to ˜E , so that theright-hand side y belongs to the range R(Ã) of the extended operator Ã. Thus, theoperator equation Ã(x) = y (x ∈ ˜E , y ∈ F , Ã : ˜E → F) has a classical solution x ∈ ˜Ecalled a generalized solution of the original equation A(x) = y. This is exactly suchan extension of the concept of solution about which D. Hilbert wrote.

The concept of generalized solution is closely related to the concept of a near-solution xε of the operator equation A(x) = y; this is such an element in E thatA(xε) = yε differs less than ε from y: ρ(y,yε )< ε. In some cases, xε may be consid-ered as an approximate solution of the equation A(x) = y. If we put ε = εn → 0 asn→ ∞ and consider a sequence of the near-solutions xεn , then in ˜E (but not in E!)the sequence xεn converges to a generalized solution x. In the case of linear operatorA, the computation of the near-solution is reduced to the problem of computation ofthe approximate (or precise) solution of a system of linear algebraic equations. Thisis why we give so much attention to these issues and propose various methods forsolving this problem.

Along with the investigation of generalized solutions, we study the so-called gen-eralized extreme elements which are closely related to this concept. Let D be a re-gion in a Banach or metric space E and a continuous functional f (x) is defined on D.As a rule, the region D is non-compact in an infinite dimensional space, thereforethe extreme element x∗ from D, at which f (x) attains its minimum or maximumvalue may not exist. Determination of a “generalized” extreme element resemblesthe construction of generalized solution. We introduce a weaker topology TD on theset D, such that the completion ˜D of D with respect to the topology TD is a compacttopological space, and the functional f may be extended on ˜D by continuity, suchthat there is a classical extreme element x∗ in ˜D. This element is considered as ageneralized extreme element, since x∗ /∈ D. Note that the concept of a generalizedextreme element may be defined in other ways. These ways are considered in thebook as well.

By an operator equation we will always mean an equation where some knownoperator L from E into F acts on an unknown element u (a vector, sequence orfunction), where F may differ from E . The spaces E and F may be finite or infinitedimensional spaces, normed spaces (in particular, Banach), metric spaces, topolog-ical vector spaces, topological or differentiable manifold, and so on. In a generalway, an operator equation has the following form

L u = f ,

Preface ix

where u is an unknown element in E , f is the known element in F , and L is theknown operator which acts from E into F . The most important problems related tooperator equations are the existence and uniqueness of a solution. The uniquenessof a solution is ensured by the condition of invertibility of the operator L , that maybe satisfied by the corresponding factorization of the space E (at least theoretically).It is clear that a solution of the equation L u = f exists iff the right-hand side fbelong in the range R(L ) of the operator L . Thus, if f ∈ R(L ) then the issueof the existence of a solution of the equation L u = f has, in principle, a positiveanswer. However, in many cases the right-hand side f does not belong to the setR(L ), so this equation does not have a solution in a classical sense. Nevertheless,from the practical point of view such equations may have “intuitive solutions”, thatmust be defined correctly. The problem of construction of a generalized solution ofthe operator equation is closely related with the problem of introducing the “natural”notion of a generalized solution of the equation L u = f for all f ∈ F ; in particular,when f ∈ F \R(L ), and with the investigation of the properties of such generalizedsolutions. The point is that the description of a function set of R(L ) is extremelydifficult. Therefore it is impossible to establish the criteria for the solvability of theequation L u = f . We could say that it is possible to formulate the criterion of thesolvability of the equation L u = f only in exceptional cases. For example, evenin the simplest case of the investigation of the classical solvability of an ordinarydifferential equation u′(t) = f (t) when 1 > t > 0 and u(0) = 0, it is necessary to testthe convergence of an integral (possibly improper integral)

∫ 1

0f (t)dt.

However, as is well known, there are no general effective criteria for testing theconvergence of improper integrals.

Consider one of the approaches to the formalization of such solutions. Supposethat in any ε-neighborhood f (in topological space F – in any neighborhood f ) thereexists such an element fε , that L uε = fε for some uε ∈ E . Then for small ε > 0one could think that fε ≈ f , since the distance ρ( fε , f ) < ε , therefore the elementuε can be accepted as a “generalized” solution of the operator equation L u = f(if topological space F is non-metrizable, then these reasonings must be slightlymodified, but this is not a principal issue).

Consider the issue of the existence of classical and generalized solutions on con-crete examples. Suppose that we want to obtain the best unbiased linear estima-tion x∗ of an unknown mathematical expectation of a continuous random processx(t) (t ∈ [0,T ]) with a constant mathematical expectation and a correlation functionK(t,s). If we look for this estimation in the form

x∗ =∫ T

0x(t)u(t)dt,

x Preface

then the problem is reduced to looking for the solution u(t) of the integral equation

∫ T

0K(t,s)u(t)dt = 1 (P.1)

in the function class L2(0,T ). In general case, the matter concerns the equation

∫

DK(t,s)u(t)dt = f (s), t ∈ D. (P.2)

However, solutions of such equations have the square integrability property veryseldom (see example [23]). For example, it is shown in [36] that (P.1) never has aclassical solution if a correlation function K(τ) of the stationary random process x(t)has a spectral density. Nevertheless, it has the generalized solution. In some cases,the fact that the integral equation (P.1) does not have a solution in the class of square-integrable functions can be proved directly. For example, if a correlation functionhas the form K(t,s) = e−β |t−s|, which corresponds to a stationary Markov processwhen all probability distributions are normal, then it is impossible to construct afunction u(t) that sets the best unbiased estimation x∗ of an unknown mathematicalexpectation. To prove this statement let us consider the integral equation

∫ T

0e−β |t−s| dF(t) =

22+βT

.

It is easy to examine that this equation is satisfied by the following function ofbounded variation

F(t) =Θ(t)+Θ(t−T )+β t

2+βT,

where∫ T

0 dF(t) = 1 and Θ (t) is the Heaviside function:

Θ (t) =

{

0, if t < 01, if t ≥ 0.

Hence, the expression

x∗ =x(0)+ x(T )+β

∫ T0 x(t)dt

2+βT

defines an unbiased estimation x∗ having the least variance in the class of unbiasedlinear estimations (actually, this estimation is also the best in a much more widerestimation class [23]). Since the estimation x∗ is unique and the formula for x∗ con-tains Dirac delta-functions δ (t) and δ (t − T ), that do not belong to L2(0,T ), it isimpossible to construct the function u(t) from L2(0,T ), that defines the estimationx∗ and is a solution of (P.1). Therefore, (P.1) does not have the classical solutionin L2(0,T ). The issues related with the problem described above are listed in [96]:“The problems are: in which functional spaces should one look for the solution?

Preface xi

Is the solution unique? Is the solution of the equation also is the solution of theestimation problem? Does the solution depend continuously on the initial data, forexample, on f and K? How can the solution be found analytically and numerically?What are the properties of the solutions? For example, what is the order of singu-larity? How can the properties of the integral operator be described, for example, inL2(D)?”

Another possible example that requires the introduction of a generalized solutionis the problem of optimal control of a system with a generalized external impact

L u = f (h), (P.3)

J(h) = Φ(u(h),h)→minh, h ∈U, (P.4)

where h is a control from an admissible set U , L : E → F is some operator, J is aperformance functional. To express this problem correctly it is necessary to ensurethe solvability of (P.3) for all h ∈ U , i.e., it is necessary to ensure the inclusionf (U)⊂ R(L ). However, generally it is very difficult to describe the range of f andL ; therefore, it is very hard to check the condition f (U)⊂ R(L ). Moreover, oftensuch an inclusion does not occur at all (in spite of the fact that a physical interpre-tation of the equation is natural and reasonable from the practical point of view).Thus, we must develop a theory of generalized solvability of (P.3) for an arbitraryright-hand side f from the set f (U), or (much better) for all f ∈ F . In a generalsense, (P.3) has a solution u(h) for an arbitrary control h ∈ U . It is clear, that wemust know peculiarities of these generalized solutions to prove some meaningfulstatements about the problem of the minimization of (P.4).

Now, problems of complex system control with singular impacts have a funda-mental importance. For example, simulation of devices with laser and pulse impacts,correction of space vehicles movement, modelling of water transport in porous me-dia with point sources and sinks are closely related with the equations with a singularright-hand side. The singularity of a control impact means that a control map f takeson a value in a space of generalized function. Traditionally, the natural range of theoperator L does not contain generalized functions. So, lumped singularity in spaceand time bring us outside of the classical problem definitions. So, we face with theneed to develop a theory of generalized solvability of (P.3).

The problem of construction of generalized solutions becomes the most im-portant in the case of linear operator L (e.g., differential or integral) which actsbetween linear topological spaces E,F , in particular, between Banach or Hilbertspaces. Note that the “naturalness” of generalized solution means the conservationof the main properties of operator L (linearity, continuity, injectivity and so on)under extension on the class of generalized solutions. Thus, the offered problemfundamentally differs from various definitions of approximate solutions, pseudo-solutions, quasi-solution, and so on. [47, 107, 112].

The problems of construction of generalized solutions of equations with lineardifferential and integral operators are quite typical. They have been investigatedsuccessfully for a long time. For example, this problem for the classical operatorof differentiation d

dt : C1([0,1])→ L2(0,1) may be solved by introducing of the

xii Preface

Sobolev generalized derivative and corresponding Sobolev spaces. In this sense,the theory of generalized functions may be considered as the first step in solving theposed problem.

The method of a priori inequations (e.g. [5, 39, 54]) is a very effective tool forthe investigation of existence and uniqueness of solutions of various classical lin-ear problems with generalized impacts. It was often used in the context of riggedHilbert spaces. In [5], the theory of generalized solvability for equations with el-liptic differential operators acting in the Sobolev discrete scale of Hilbert spaceswas constructed. This theory is based on the concept of weak solutions (in the con-text of the theory of generalized functions). Berezanskii proved the theorems of aunique generalized solvability of elliptic operator equations for major problems ofmathematical physics and investigated the smoothness of the generalized solutions.The theorems proved are the criteria of solvability (i.e., an operator determines atopological isomorphism). For example, the theorems of a unique solvability (in L2

and other spaces) of the equations of mathematical physics of various types wereproved in [44, 45]. Some criteria of solvability of parabolic equations are describedin [2]. The issues of generalized solvability for pseudo-parabolic equations of or-der more than two were investigated in [58, 100], for pseudo-hyperbolic equations– in [73, 79, 85, 100], for Sobolev type system – in [59, 76, 100], for wave systemsof fifth order – [60, 61, 64, 80, 82], and in many other papers (see also [62, 63]).Note that in these papers were used a priori inequalities in negative norms when ageneralized solution belongs to Sobolev type spaces.

The generalized solvability of linear integral equations is closely related withFredholm and Volterra; integral equations of the first kind [23] and [68, 87, 88, 92].It must be stressed that in many above-mentioned papers the proofs of existenceand uniqueness of a generalized solution are based on the classical idea of relationsbetween direct and “adjoint” equations and the coercive inequality. Therefore, thesetheorems can be considered as the developing of classical results of S.G. Krein (e.g.,see [39]).

There is one more important aspect of the theory of generalized solutions. It isrelated to the problem of optimal control (P.3), (P.4), rather than with only (P.3).As it is well known, there are problems of calculus of variations and optimal con-trol which have no solutions in “traditional” sets of curves (in spaces of smoothfunctions). This problem was solved in classical papers on optimal control theoryin the generalized statement. For example, the general plan of looking for gener-alized extreme curves is described in [116]. The plan involves the following ac-tivities: to densely embed the control space (and therefore an admissible set ofcontrols) in a new topological space such that a functional in question is still se-quentially continuous and an admissible set is sequentially compact. This idea nat-urally connects the optimal control problems with the Schwarz distributions spaces.We have to mention L. Young [118] among the authors who began to apply theideas of the theory of generalized functions to the calculus of variations problemsand the optimal control problems. From the Young’s point of view, the spaces ofcurves with “traditional” topologies are poorly adaptable for the calculus of vari-ations. More convenient are the topologies which induce so-called “generalized

Preface xiii

curves” (by Young) that are equivalent to the concepts of weak controls and glidingregimes. In the optimal control theory for ordinary and partial differential equations,Filippov and Gamkrelidze considered the weak solutions and analogous construc-tions [17, 22] (“gliding regimes”), Warga studied the “generalized curves” [115],McShane investigated the “generalized controls” [70, 71], Chouila-Houri consid-ered “boundary controls” [10], and this list might be continued (see, e.g. books ofA. Chikrii [9], M. Zgurovsky, V. Mel’nik [120], V. Kuntsevich [42], J.-L. Lions[51, 52], and B. Mordukhovich [72]).

These results naturally pose the general problem of looking for generalized ex-treme elements in various classes of functionals. These problems are interestingeven in the simplest case when we look for an extremum of a continuous functionaldefined on a bounded set in a Banach space.

Thus, there are many papers, where existence and uniqueness of generalized so-lution of an operator equation or extremal problem solutions were investigated. Mul-tiplicity and similarity of these papers suggest that there is a general approach to theconstruction of the concept of generalized solvability. The major elements of thisapproach are described in our book.

The book consists of the preface, eight chapters, divided into sections, and abibliography. The numbering of definitions, lemmas, theorems, and so on, is con-tinuous. Chapter 1 contains major definitions, concepts, and auxiliary facts used inthe book. Chapter 2 is an introduction to the theory of generalized solutions of op-erator equations. It describes the simple schemes of generalized solutions for linearoperator equations. In Chap. 3, we investigate the method of a priori estimates forgeneralized solutions. Chapter 4 describes some applications of the theory of gen-eralized solvability of linear equations. Chapter 5 is devoted to numerical aspectsof the theory. Chapter 6 describes the general topological method of construction ofgeneralized solutions of linear operator equations. In Chap. 7, the issues of general-ized solvability of nonlinear operator equations are considered. Chapter 8 is devotedto the generalized solvability of extreme problems.

Kiev, Ukraine Dmitry KlyushinSergey Lyashko

Dmitry NomirovskiiYuriy Petunin

Vladimir Semenov

Yuriy Ivanovich Petunin

Our co-author, friend and master Yuriy Ivanovich Petunin suddenly died on June 1,2011.

Yuriy Ivanovich Petunin was born on September 30, 1937, in Michurinsk(Russia). He graduated Tambov State Pedagogical Institute and passed Ph.D. de-fense under the supervision of S.G. Krein in 1962. He became a Doctor of Scienceon 1968. Since 1970 Yuriy Ivanovich has been the professor of the department ofcomputational mathematics of the faculty of cybernetics of the Kiev National TarasShevchenko University.

Yu.I. Petunin started his scientific activity in the area of functional analysis underthe supervision of S.G. Krein. His main achievements in this field are the creationof the theory of scales of Banach spaces, development of the theory of interpolationof linear operators (with S.G. Krein and E.M. Semenov) and the theory of char-acteristics of linear manifolds in conjugate Banach spaces (with A.M. Plichko). Inaddition he was the first who rigorously justified the empirical three-sigma rule forunimodal distributions, proving the famous problem posed by K.F. Gauss more than150 years ago (the classical Vysochanskii-Petunin inequality), developed the theoryof confidence intervals for a bulk of general population and parameters using orderstatistics, developed the theory of linear estimations of unknown mathematical ex-pectation, the theory of quadratic estimations of unknown variance, statistical testswhich use the procedure of indecision and individual statistical tests. Yu.I. Petuninsolved the Banach’s problem of norming subspaces in conjugate Banach spaces andthe Calderon-Lions problem of interpolation in factor spaces. He also developed thelattice approach to solving the sixth Hilbert’s problem (with D.A. Klyushin). Yu.I.Petunin had significant achievements in the theory of pattern recognition, in partic-ular, in its application to differential diagnostics of oncological diseases (with B.V.Rublev, D.A. Klyushin, K.P. Ganina, N.V. Boroday, and R.I. Andrushkiw). It shouldbe stressed that one of the main ideas in this book – the concept of generalized so-lution of operator equations in Banach space – was developed by Yuriy Ivanovich.

xv

xvi Yuriy Ivanovich Petunin

Yuriy Ivanovich was notable for his exceptional honesty, nobility, devotion toscience and religiosity. His sincerity and helpfulness have always attracted manypeople. Death of such a man is a great grief for the whole mathematical community.

Cherished memories of our colleague, friend and master will always stay in ourhearts!

Dmitry KlyushinSergey Lyashko

Dmitry NomirovskiiVladimir Semenov

Acknowledgements

We are very grateful to our colleagues, especially to the correspondent mem-ber of the National Academy of Sciences of Ukraine, A.A. Chikrii and academi-cians I.N. Kovalenko, V.M. Kuntsevich, I.V. Sergienko, Yu.M. Yermolyev, M.Z.Zgurovsky. We want to mention also our teachers and colleagues who are not withus: Prof. V.P. Didenko, Prof. S.G. Krein, Academician I.I. Lyashko, Dr. D.L. Pikus,Academician B.N. Pshenichny, Academician N.Z. Shor, and correspondent memberV.V. Skopetsky. We thank Yuriy Malitsky and Varvara Obolonchikova for the helpin the preparation of this manuscript.

We would like to express our sincere thanks to Prof. Panos Pardalos for his kindlysupport of our book.

Special thanks go to Elizabeth Loew (Senior Editor, Mathematics, Springer),Nathan Brothers (Assistant Editor) and Jacob Gallay (Editorial Assistant) for alltheir help with the cover, production and manufacturing of our book.

xvii

Contents

1 The Major Definitions, Concepts and Auxiliary Facts . . . . . . . . . . . . . . 1

2 The Simplest Schemes of Generalized Solutionof Linear Operator Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Strong Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Strong Near-Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Weak Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Weak Near-Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Existence and Uniqueness of a Weak Generalized Solution

of a Linear Operator Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Relation Between Weak and Strong Solutions of a Linear Operator

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 A Priori Estimates for Linear Continuous Operators . . . . . . . . . . . . . . . 173.1 A Priori Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 A Generalized Solution of an Operator Equation in Banach Spaces . 183.3 A Generalized Solution in Locally Convex Linear Topological

Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Relation Between Generalized Solutions in Banach and Locally

Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Applications of the Theory of Generalized Solvabilityof Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1 Application to the Equations with Hilbert–Schmidt Operator . . . . . . 294.2 Generalized Solutions of an Infinite System of Linear Algebraic

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Application to Volterra Integral Equation of the First Kind . . . . . . . . 444.4 Application to the Statistics of Random Processes . . . . . . . . . . . . . . . 474.5 Application to Parabolic Differential Equation in a Connected

Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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xx Contents

4.5.2 Properties of Operators Associated with a Boundary ValueProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5.3 Generalized Solvability of the Boundary Value Problem . . . . 544.6 Application to Parabolic Differential Equation in a Disconnected

Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6.1 Main Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6.2 Properties of Operators Associated with a Boundary Value

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.6.3 Generalized Solution of a Parabolic System

with Discontinuous Coefficients and Solutions . . . . . . . . . . . . 674.6.4 Approximate Method for Solving the Boundary Value

Problem for a Parabolic Equation with InhomogeneousTransmission Conditions of Non-ideal Contact Type . . . . . . . 69

4.7 On the Unique Solvability of Wave Systems . . . . . . . . . . . . . . . . . . . . 804.7.1 Basic Notation and Statement of the Operator Equation . . . . 814.7.2 A Priori Inequalities: Main Case . . . . . . . . . . . . . . . . . . . . . . . . 844.7.3 Analysis of the System on the Basis of a Single Chain

of a Priori Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.7.4 Construction of a “Scale” of Solvability Theorems . . . . . . . . 93

4.8 Projection Theorem for Banach and Locally Convex Spaces . . . . . . . 97

5 Computation of Near-Solutions of Operator Equations . . . . . . . . . . . . . 1035.1 Construction of Near-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 Method of Neumann Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 The Condition Number of Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Hotteling Method for Correction Inverse Matrix . . . . . . . . . . . . . . . . . 1115.5 Exact Solving a System of Linear Algebraic Equations . . . . . . . . . . . 1125.6 Solving a System of Linear Algebraic Equations with Guarantee

Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.7 Characterization of a Classic Solution Using Neumann Series . . . . . 117

6 General Scheme of the Construction of Generalized Solutionsof Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.1 Generalized Solution of Linear Operator Equations in Locally

Convex Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2 Examples of Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2.1 Classical Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.2.2 Generalized Strong Solvability . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2.3 Generalized Weak Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2.4 A Priori Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3 Properties of the Generalized Solvability in the Spaces E1,E2 . . . . . 135

7 Concept of Generalized Solution of Nonlinear Operator Equation . . . 1377.1 Generalized Solution of Nonlinear Operator Equation . . . . . . . . . . . . 1377.2 Near-Solution of Nonlinear Operator Equation . . . . . . . . . . . . . . . . . . 138

Contents xxi

7.3 Existence and Uniqueness of a Generalized Solution . . . . . . . . . . . . . 1397.4 Correctness of Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.5 Pseudo-Generalized and Essentially Generalized Solutions . . . . . . . . 1407.6 Relation Between Pseudo-Generalized and Generalized Solutions . . 1447.7 Example of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.8 Computation of Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.9 Uniform Structures and Generalized Solutions

of Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527.9.1 Definition of a Generalized Solution of Operator Equation . . 1527.9.2 Generalized Solutions and Embeddings

of Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.9.3 Examples of Generalized Solutions . . . . . . . . . . . . . . . . . . . . . 1587.9.4 Generalized Solution of Operator Equation

in Proximity Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8 Generalized Extreme Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.1 Examples of Generalized Extreme Elements . . . . . . . . . . . . . . . . . . . . 1638.2 Generalized Extreme Elements for Linear and Positively

Homogeneous Convex Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.3 On Compact Embedding into a Banach Space . . . . . . . . . . . . . . . . . . . 1738.4 Generalized Extreme Elements for General Convex Functionals . . . . 1778.5 Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Chapter 1The Major Definitions, Conceptsand Auxiliary Facts

Let E be a linear (vector) space on the field R, i.e. on the set E an operation ofaddition of every two elements x,y from E (denoted as x+ y) and an operation ofmultiplication of every element x from E on an arbitrary λ from R (denoted as λ x)are defined as follows:

(1) (x+ y)+ z = x+(y+ z).(2) x+ y = y+ x.(3) there exists θ ∈ E , such that 0x = θ for x ∈ E .(4) (λ + μ)x = λx+μx.(5) λ (x+ y) = λx+λy.(6) (λ μ)x = λ (μx).(7) 1x = x.

In the linear space E the difference x− y means x+(−1)y.A map f : E → R is called a linear functional on E (when E is a linear space)

if the following conditions are satisfied:

1 (Additivity). For all x,y ∈ E

f (x+ y) = f (x)+ f (y).

2 (Homogeneity). For all λ ∈R,x ∈ E

f (λ x) = λ f (x).

The algebraically conjugate to a linear space E is a space E ′ of linear functionals fdefined on E . The set E ′ is a linear space if f +g and λ f are the functionals definedas follows:

(1) ( f +g)(x) = f (x)+ g(x) for all f ,g ∈ E ′, x ∈ E .(2) (λ f )(x) = λ f (x) for all f ∈ E ′, λ ∈R, x ∈ E .

In a similar way, the second algebraically conjugate space E ′′ = (E ′)′ is definedand so on. Every x ∈ E uniquely defines some element of E ′′ – a linear functional

D.A. Klyushin et al., Generalized Solutions of Operator Equations and Extreme Elements,Springer Optimization and Its Applications 55, DOI 10.1007/978-1-4614-0619-8 1,© Springer Science+Business Media, LLC 2012

1

2 1 The Major Definitions, Concepts and Auxiliary Facts

Lx ∈ E ′′ – according to the following rule: Lx( f ) = f (x) for all f ∈ E ′. Thus, it ismeaningful to say about embedding E ⊂ E ′′. A functional f : E→R is called con-vex if λ f (x)+ (1−λ ) f (y) ≥ f (λx+(1−λ )y), for every x,y ∈ E and an arbitraryreal number λ ∈ [0,1].

Let us consider two linear spaces E,F and their Cartesian product in which abilinear form 〈x,y〉, x ∈ E,y ∈ F is defined. By a bilinear form we mean a two-variate function which is linear in each argument separately. The linear spaces E,Fare said to be dual with respect to this bilinear form (in other words, (E,F) is a dualpair) if:

(1)For all x ∈ E (x = 0) there exists y ∈ F such that 〈x,y〉 = 0.(2)For all y ∈ F (y = 0) there exists x ∈ F such that 〈x,y〉 = 0.

If E is a linear space and E ′ is an algebraic conjugate to E then the expressionf (x) = 〈 f ,x〉, where f ∈ E ′, x ∈ E defines a bilinear form over E ′ and E . In otherwords, the spaces E ′ and E are dual each other.

If linear spaces E and F are dual to each other with respect to the bilinear form〈x,y〉, where x ∈ E,y ∈ F , then every element y ∈ F can be identified with somelinear functional f ∈ E ′ using the equality f (x) = 〈x,y〉 for all x ∈ E . Thus, F ⊂ E ′always.

Let linear spaces E and F be dual with respect to the bilinear form 〈x,y〉,x ∈ E,y ∈ F , and M be a subset of E . A polar of M (with respect to this duality) isa set

M◦ = {y ∈ F | |〈x,y〉| ≤ 1,x ∈M} .Let linear spaces E and F be dual with respect to the bilinear form 〈x,y〉,

x ∈ E,y ∈ F , and let M be a subset of E . A set is said to be total in E if the lin-ear spaces M and F are dual also (with respect to 〈x,y〉).

An operator L which acts from the linear space E into the linear space F iscalled linear if it satisfies the following two conditions:

1 (Additivity) L (x+ y) = L (x)+L (y) for all x,y ∈ E ,2 (Homogeneity) L (λx) = λ (L x) for all x ∈ E , λ ∈R.

The algebraically adjoint to a linear operator L : E → F is an operator L ′ :F ′ → E ′ such that:

(L ′ϕ)(x) = ϕ(L x) for all x ∈ E,ϕ ∈ F ′.

Thus, L ′ϕ = ϕ ◦L .Suppose that in a linear space E a topology T (a system of open subsets O of

the set E) is introduced, such that E is a linear topological space (see [40]). A setMx ⊂ E is called a neighborhood of a point x ∈ E if there exist such an open setO ∈ T that x ∈O⊂Mx.

Suppose that two topologies T1 and T2 are given on a set E . The topology T1 issaid to be stronger than the topology T2 (this fact is denoted as T2 ⊂ T1) if forany O2 ∈ T2 there exists such O1 ∈ T1 that O1 ⊂ O2. If one topology is strongerthan the other, then these topologies are said to be comparable.

1 The Major Definitions, Concepts and Auxiliary Facts 3

Let L be an operator which acts from the linear topological space E into thelinear topological space F . The operator L is called continuous if for any open setOF ⊂ F there exists such an open set OE ⊂ E that L (OE ) ⊂ OF . In particular, ifF = R, then the operator L is a continuous functional. A set E∗ of linear contin-uous functionals defined on E is called conjugate to E .

A subset M ⊂ E is called bounded in the linear topological space E if for anysequence of elements xn ∈M and for any numerical sequence λn ∈ R the sequenceλnxn converges to zero.

Let E be a linear set and F ⊂ E ′ be a total subset. An important example ofa linear topological space is a space E with a weak topology σ(E,F). Denote asOε, f1, f2,..., fn , where fi ∈ F , ε > 0, a set of points x ∈ E such that | fi(x)| < ε for alli∈ {1, . . . ,n}. The topology σ(E,F) is given by the sets of neighborhoods of zero inthe following way. A neighborhood of zero is a set containing Oε , f1, f2,..., fn . Neigh-borhoods of an arbitrary point x = 0 are defined as a shift of the neighborhoodof zero on element x, in other words an arbitrary neighborhood Vx of a point x hasthe form Vx =V0+ x, where V0 is a neighborhood of zero. In such linear topologicalspace, linear (and only linear) functionals f ∈ F are continuous, i.e. E∗ = F .

Let E and F be normed spaces and L be a linear continuous operator which actsfrom E into F . If ϕ( f ) is a linear continuous functional on F (ϕ ∈ F∗), then thefunctional l(x) = ϕ(L x) is a linear continuous functional on E (l ∈ E∗). Thus, wehave constructed a map L ∗ : F∗ ϕ �→ l ∈ E∗ which is called an adjoint operator.Adjoint operators exist also when E and F are linear topological spaces (see, forexample, [97]). Remember that a subdifferential of a convex functional f : E→ R

in a point x0 ∈ E is a set ∂ f (x0)⊂ E∗ of linear continuous functionals x∗0 ∈ E∗ suchthat f (x)− f (x0)≥ x∗0(x− x0) for all x ∈ E . If E is a Banach space and a functionalf is continuous in a point x0 ∈ E then ∂ f (x0) is a non-empty convex and compactwith respect to the topology σ (E∗,E) set [15].

Under the investigations of generalized solutions of operator equations the theoryof embedding and intermediate Banach spaces plays an important role (see [86]).Remember the major concepts of this theory. Let E and F be two sets. An operatorL : E → F is called injective if L x = L y when x = y; surjective if L (E) = Fand bijective if it is injective and surjective.

A linear normed space E is said to be embedded into a linear normed space Fwith the help of an embedding operator jEF if jEF is a linear bounded injectiveoperator with a domain coinciding with the space E . Let the space E be embeddedinto a space G that in its turn is embedded into the space F using correspondingembedding operators. The space G is called intermediate between E and F if thefollowing diagram is commutative:

EjEG−−−−→ G

∥

∥

∥

⏐

⏐

�jGF

EjEF−−−−→ F

4 1 The Major Definitions, Concepts and Auxiliary Facts

The concepts of embedded and intermediate spaces have another, more simpleinterpretation. Let the space E be embedded into F . Let us consider an imagejEF(E) ⊂ F with a norm ‖y‖0 = ‖x‖E where y = jEF (x). Obviously, the spaces(E,‖ · ‖) and ( jEF(E),‖ · ‖0) are isometric. So, elements of E and jEF (E) can beidentified and we may consider that E is a subspace of F . Since the operator jEF isbounded, there exists some positive number C that does not depended on y such that‖y‖F ≤C‖y‖0 for all y ∈ jEF . Therefore, the definition of embedded spaces can bereformulated in the following way: a linear normed space E is embedded into F ifE is a subspace of F and there exists C > 0 for which ‖x‖F ≤C‖x‖E , where x is anarbitrary element from E .

The concept of an intermediate space is interpreted in the following way: a spaceG is an intermediate space between E and F if E ⊂ G ⊂ F and ‖x‖G ≤C1‖x‖E forall x∈E , and ‖x‖F ≤C2‖x‖G for all x∈G, where the constants C1,C2 do not dependon x.

A space E is said to be embedded densely into F if the set E considered as asubset of the space F is dense in F with respect to the norm ‖ ·‖F . If E is embeddeddensely into F , then a restriction to the set E of any linear continuous functionalf ∈ F∗ induces a continuous linear functional over E . Indeed, taking into consider-ation the inequality ‖x‖F ≤C‖x‖E we have

‖ f‖E∗ = sup‖x‖E≤1

| f (x)|‖x‖E

≤C sup‖x‖E≤1

| f (x)|‖x‖F

≤C sup‖x‖F≤C

| f (x)|‖x‖F

= C sup‖x‖F≤1

| f (x)|‖x‖F

=C‖ f‖F∗ .

The injectivity of the map jF∗E∗ which maps a functional f ∈ F∗ to a functionalfrom E∗ follows from the density of the embedding E into F . Thus, F∗ is embeddedinto E∗ and ‖ f‖E∗ ≤ C‖ f‖F∗ . It is easy to see that the density of the embeddingE ⊂ F implies the density of the embedding F∗ into E∗ if the space E∗ is endowedwith topology σ(E∗,E); an embedding F∗ ⊂ E∗ which is dense with respect to thenorm of E∗ may not exist. Consider an example. Let E = l1, F = c0 then E is denselyembedded into F (with C = 1). However, the set F∗ = l1 is not dense in the spaceE∗ = l∞, since the element e = (1,1, . . .) ∈ l∞ is away from l1 over the distance 1.

Let us consider some examples of embedded and intermediate Banach spaces.A Banach space E1 =C1(0,1) is embedded into E0 =C(0,1) using the operator ofnatural embedding: if x(t) ∈C1(0,1) then jE1E0(x(t)) = x(t). Indeed,

‖x‖E0 = max0≤t≤1

|x(t)| ≤ max0≤t≤1

|x(t)|+ max0≤t≤1

|x′(t)|= ‖x‖E1.

Note that C1(0,1) is embedded densely into C(0,1), since by the Weierstrass theo-rem every continuous function can be approximated uniformly with arbitrary accu-racy by algebraic polynomials which obviously are the elements of C1(0,1). More-over, C1(0,1) is embedded compactly into C(0,1) (E is embedded compactly intoF if the unit ball S1(E) in E is a relatively compact set in F , i.e. its closing in F with

1 The Major Definitions, Concepts and Auxiliary Facts 5

respect to the norm F is compact in F). Indeed, if x(t) ∈ S(θ ,1), where S(θ ,1) isthe unit ball in C1(0,1) then |x(t)|+ |x′(t)| ≤ 1 for all t ∈ [0,1]. This implies theinequalities |x(t)| ≤ 1 and |x(t)− x(τ)| ≤ |t− τ| for all t,τ ∈ [0,1]. Thus, the set offunctions x(t) is uniformly bounded and equicontinuous in C(0,1), hence accordingto the Arzela theorem this set if relatively compact in C(0,1).

The other examples of an embedding of Banach spaces are C(0,1) and Lp(0,1)(1≤ p<∞). Let us define the embedding operator j of C(0,1) into Lp(0,1) mappinga continuous function x(t) to a class x(t) ∈ Lp(0,1) consisting of functions, whichdiffer from x(t) on a zero Lebesgue measure set; then C(0,1) in embedded intoLp(0,1). Actually, j is an algebraic isomorphism, and moreover

‖ j(x)‖Lp(0,1) =

(

∫ 1

0|x(t)|pdt

)1/p

≤ max0≤t≤1

|x(t)|= ‖x‖C(0,1).

The density of C(0,1) ⊂ Lp(0,1) with respect to the metric of Lp is implied fromthe Luzin theorem.

If p > q then Lp(0,1) is embedded densely into Lq(0,1) using the identity em-bedding operator j. Indeed, when p > q the Holder inequality implies that

‖x‖Lq =

(

∫ 1

0|x(t)|qdt

)1/q

≤(

∫ 1

0|x(t)|q× p

q dt

)qp× 1

q

= ‖x‖Lp .

Since C(0,1)⊂ Lp(0,1)⊂ Lq(0,1) is an everywhere dense set in any of the spacesLq, when p> q, and Lp(0,1) is an everywhere dense set in Lq(0,1). Also, the Holderinequality implies that lp is embedded into lq when q > p; the density of the em-bedding lp into lq is implied from the fact that a linear manyfold M consisting fromsequences having only finite number of non-zero elements belongs to any lp and itis an everywhere dense set.

One more important example of embedding of the Banach spaces are the Sobolev

space and the space Lp(D), where D is a bounded region in Rn. The space W (l)

p (D)consisting of the functions x(t), t = (t1, t2, . . . , tn) ∈ D ⊂ R

n having pth-ordersummable generalized partial derivatives in D up to lth order inclusive is embed-ded into Lq(D) using the operator of natural embedding. In addition,

‖x‖Lp(D) =

(

∫

D|x(t)|pdt

)1/p

≤(

∫

D|x(t)|pdt

)1/p

+n

∑k1,k2,...,kl=1

(

∫

D

∣

∣

∣

∣

∂ lx∂ tk1∂ tk2 . . .∂ tkl

∣

∣

∣

∣

p

dt

)1/p

= ‖x‖W

(l)p (D)

.

The fact that the embedding W (l)p (D) into Lp(D) is dense follows from the fact

that the set C(l)(D) consisting of all lth-order continuously differentiable functions

in D is everywhere dense both in W (l)p (D) and in Lp(D) [29].

Chapter 2The Simplest Schemes of Generalized Solutionof Linear Operator Equation

Let E,F be Banach spaces and L be a linear operator with an everywhere densedomain D(L )⊂ E , which acts from E into F . Let us consider an operator equation

L u = f , u ∈D(L ), f ∈ F (2.1)

and an adjoint equation

L ∗ϕ = l, ϕ ∈ D(L ∗), l ∈ E∗, (2.2)

where E∗ and F∗ are conjugate Banach spaces to E and F , respectively, L ∗ is anadjoint operator to L . Suppose that the range R(L ) ⊂ F of L is an everywheredense set in F and (2.1) is uniquely solvable over R(L ), i.e. the null space Ker(L )of L consists only of the zero element θ : Ker(L ) = θ . Thus, L sets a one-to-onemapping between D(L ) and R(L ). Note that the continuity of L is not supposed.

The aim of this chapter is to give a “meaningful” definition of the solution of (2.1)when f /∈ R(L ).

2.1 Strong Generalized Solution

Let us introduce one more norm on the linear set D(L ) in the space E . Since L :E→ F is a linear injective operator with a domain D(L )⊂ E , the function

D(L ) � u→ ‖L u‖F ∈ R

has all properties of a norm on D(L ). Hence, D(L ) with this norm turns into anormed space, which may be incomplete. Let E be a completion of this normedspace.

The fact that ‖u‖E = ‖L u‖F for all u ∈ D(L ) allows to extend L from D(L )onto E . Indeed, if u is an arbitrary element from E , then the density of D(L ) inE implies that there is such a sequence ui ∈ D(L ) that ui → u in E as i→ ∞.


7

8 2 The Simplest Schemes ...

Since ui is a convergent sequence in E and hence it is a Cauchy sequence in E , and‖ui−u j‖E = ‖L ui−L u j‖F then L ui is a Cauchy sequence in F . However, F isa complete normed space. Thus, there is such an element f in F that L ui→ f inF as i→ ∞. Determine a value of the operator L on the element u in the followingway: L u = f , where L : E → F is an extended operator defined on the entirespace E . Note that the value L u is defined correctly, i.e. the element f = L u ∈ Fdoes not depend on the selection of the sequence ui. Thus, the operator L : E→ F ,D(L ) = E is an extension of L on the whole space E .

Definition 2.1. A strong generalized solution of (2.1) is such an element u ∈ E thatequality (2.1) holds for the extended operator L .

Remark 2.1. If E = F is a Hilbert space and L is a symmetric operator then theoperator L is called a self-adjoint extension of operator by Friedrichs.

As mentioned before, the concept of a strong generalized solution u arises whena right-hand side f of (2.1) does not belong to the range R(L ) of L . In this case,the ordinary (classical) solution does not exist. The word “strong” means that thetopology of the space E is normed.

Let us study some properties of L . It follows from the linearity of L that theoperator L is linear also. Let us prove that L is an injective operator. Indeed,if u ∈ E is such an element that L u = 0, then selecting a sequence ui ∈ D(L )converging to u in E as i→ ∞ we have that L ui→ L u = 0 in F as i→ ∞. The laststatement can be rewritten as ‖L ui‖F → 0 or ‖ui‖E → 0. Hence, ‖u‖E = 0. Thus,the injectivity of the operator L is proven. In addition, the equality ‖L u‖F = ‖u‖E ,which holds for an arbitrary u ∈ D(L ), clearly holds for all u ∈ E (taking intoaccount the replacement of L by L ). From ‖L u‖F = ‖u‖E , where u∈ E , it followsthat the operator L is continuous and coercive.

The properties of the operator L can be proven in another way. Indeed, the op-erator L is a one-to-one map between D(L ) and R(L ). In addition, if (D(L ) is anormed space with the norm ‖u‖E and R(L ) is a normed space with the norm ‖ f‖F )then the completion of D(L ) coincides with E and the completion R(L ) coincideswith F (remember that R(L ) is a dense subset of F). On the other hand, grantingthe equality ‖L u‖F = ‖u‖E , which holds for all u ∈ D(L ), we have that the oper-ator L is an isometry between the normed spaces D(L ) and R(L ). Hence, theircompletions are isometrical. This isometry defines the completion L of the opera-tor L . Thus, the operator L sets an isometry between E and F . This implies theabove-mentioned properties of L . The foregoing implies the following theorem.

Theorem 2.1. For any f ∈ F there exists a unique strong generalized solutionof (2.1) in the sense of Definition 2.1.

If f ∈ R(L ), then a strong generalized solution u turns into a classic solution. Itis also clear that the classic solution is strong, and it is classic if u ∈ D(L ).

Let us clarify the relations between the spaces E and E. Since D(L ) is a denselinear subset of E (of course, in the sense of the norm of the space E), then theset E may be obtained by completing D(L ) with respect to the norm ‖u‖E . Thus,the spaces E and E may be considered as completions of the same linear set D(L )

2.2 Strong Near-Solution 9

with respect to the two different norms: ‖u‖E and ‖u‖E . Unfortunately, in generalcase, elements of the spaces E and E are incomparable. It is explained by the factthat, on one hand, the operator L : E → F can be unbounded and, from the otherhand, it can be non-coercive, even though it is a linear injective operator. This meansthat in general case the norms ‖u‖E and ‖L u‖F = ‖u‖E can induce incomparabletopologies on D(L ).

When L : E → F is a linear continuous operator the case is more simple. Thenthe topology induced on D(L ) by the norm ‖u‖E is weaker than the topology ofthe space E .1

Consider another possibility. Let an operator L : E → F be coercive, i.e. thereexists such a constant c > 0 that

‖u‖E ≤ c‖L u‖F = c‖u‖E (2.3)

for all u ∈ D(L ). In this case, the norms ‖u‖E and ‖L u‖F = ‖u‖E are comparableover D(L ) (the topology of the space E is stronger than the topology of the spaceE on D(L )) and there is a relation between elements of E and E .

Theorem 2.2. Let L be a closable coercive operator. Then there exists a densecontinuous embedding E ⊂ E.

Proof. Since the spaces E and E are the completions of the linear set D(L ) withrespect to two norms and (2.3) holds, then in order to prove the theorem, it is enoughto check the condition:

(π) if ui ∈ D(L ) and ui→ u in E , ui→ 0 in E , then u = 0.

However, this condition can be rewritten in the following way:

(π) if ui ∈ D(L ) and L ui→ f in F , ui→ 0 in E , then f = 0.

The last condition is clear, since the operator L is closable. Thus, we ascertained that E ⊂ E , i.e. an arbitrary strong generalized solution

of (2.1) is an element of the space E .

2.2 Strong Near-Solution

Suppose that the right-hand side of (2.1), i.e. the element f , does not belong to therange R(L ) of an operator L . Since R(L ) is everywhere dense in F and (2.1)

1 Note that studying of a closable operator L : E → F can be reduced (at least theoretically) tostudying of a linear continuous operator L1 defined on the same set D(L ), but with respect toanother norm. Indeed, introducing in D(L ) a graph norm

‖u‖Γ = ‖u‖E +‖L u‖F ,

with respect to which the linear set D(L ) is Banach, we have that the operator L1 : D(L )→ F islinear and continuous (L1u = L u, u ∈D(L ) = D(L1)).


is uniquely solvable, then there exists a sequence fn ∈ R(L ) such that fn → f asn→ ∞, and a sequence un = L −1( fn) convergent to some element u ∈ E in E .

Definition 2.2. A sequence of elements un ∈D(L ) is called a strong near-solutionof the operator equation (2.1), if fn = L un → f as n→ ∞ in the metric of thespace F . An element u ∈ E is called the strong limit element of the near-solution.

The concept of a “near-solution” is justified by the following arguments. In manyimportant practical cases, it is impossible or almost impossible to determine theright-hand side f of (2.1) absolutely exactly; therefore, we have to consider itsε-approximation, i.e. and element f ∈ R(L ) such that ρ( f , fε ) = || f − fε || < ε.In this case, there exists an element uε = L −1( fε ) from the domain of the operatorL , which can be considered as “ε - approximation” of the solution of (2.1), i.e. theright-hand (2.1) is closely approximated by its image L uε = fε . If the elementsuε “become stabilize” as ε → 0, i.e. if they converge in some topology (in E) tothe fixed element u /∈ D(L ), then it is naturally to consider the element uε as an“ε -solution” or “near-solution”. Note that in many cases the “accuracy” of a solu-tion uε is defined by the closeness of its image L uε = fε to the element f , i.e. by anorm of the space E .

The definitions of a strong generalized solution and a near-solution of (2.1) implythat these concepts are equivalent, i.e. an element u ∈ E is a strong generalizedsolution of the operator equation (2.1) iff it is a strong limit element of a near-solution.

2.3 Weak Generalized Solution

Let us consider a definition of a generalized solution of an operator equation in alinear topological space with a topology which is not necessarily induced by a norm.

As before, suppose that L : E → F is a linear injective operator, which actsbetween Banach spaces E,F with everywhere dense domain and range in E and F ,respectively. In addition, suppose that D(L ∗) is a total subset of F∗ in a duality(F,F∗), and R(L ∗) is a total subset of E∗ in a duality (E,E∗). Note that the totalityproperty of R(L ∗) may be replaced by one of the following conditions:

(a)The space E is reflexive (if the space F is reflexive also, then the set D(L ∗) isstrongly dense in F∗).

(b)The operator L is continuous, i.e. D(L ) = E;

In condition (a), the totality of R(L ∗) follows from [40], and in case (b) itfollows from the formulae

R(L ∗)◦ ∩D(L ) = Ker(L ), (2.4)

where R(L ∗)◦ ⊂ E is a polar of the set R(L ∗) ⊂ E∗ in a duality (E,E∗). Let usprove (2.4) for an arbitrary linear operator. Since R(L ∗) is a linear set, then

R(L ∗)◦ ∩D(L ) = {u ∈ E : u ∈ D(L ), l(u) = 0,∀l ∈ R(L ∗)}= {u ∈ E : u ∈ D(L ), ϕ(L u) = 0,∀ϕ ∈D(L ∗)}

2.3 Weak Generalized Solution 11

Since D(L ∗) is a total linear subspace, then

R(L ∗)◦ ∩D(L ) = {u ∈ E : u ∈ D(L ), L u = 0}= Ker(L ).

Therefore, formulae (2.4) is proved. From (2.4) it is follows that

(R(L ∗)◦ ∩D(L ))◦ = (Ker(L ))◦.

If L is a continuous injective operator, then D(L ) = E , Ker(L ) =∅. Therefore,

(R(L ∗))◦◦ = (Ker(L ))◦ = E∗.

So, a bipolar of the set R(L ∗), i.e. a weak closure R(L ∗) coincides with E∗; hence,R(L ∗) is total in E∗.

Finally, we see that the set of functionals R(L ∗)⊂ E∗ is a total linear manyfoldwith respect to the duality (E∗,E); the linear subspaces F and D(L ∗) are in dualityalso.

Denote by ˜E a completion of a space E with respect to a topology σ(E,R(L ∗)).Since the sets E and R(L ∗) are in duality, then the space ˜E is a Hausdorff locallyconvex topological vector space. Each of the functionals l ∈ E∗ which has the forml = L ∗ϕ , where ϕ ∈D(L ∗), allows a unique extension by continuity on the wholespace ˜E, which we will denote as ˜l. A conjugate space to ˜E is a space consisting ofvarious functionals ˜l, where l = L ∗ϕ , ϕ ∈ D(L ∗).

Let us consider an arbitrary continuous linear functional ϕ ∈D(L ∗). Then (2.1)implies that

ϕ(L u) = ϕ( f ), l(u) = (L ∗ϕ)(u) = ϕ( f ). (2.5)

Definition 2.3. A weak generalized solution of the operator equation (2.1) is anelement u ∈ ˜E , which satisfies the relation

˜l(u) = ϕ( f ) for all ϕ ∈ D(L ∗), (2.6)

where l = L ∗ϕ .

A weak generalized solution u ∈ ˜E , as a strong generalized solution of (2.1) alsoarises when the right-hand side of (2.1), i.e. the element f , does not belong to therange R(L ) of the operator L and a classic solution does not exist.

Relations (2.5) imply that any classic solution is a weak solution also. On theother hand, if f ∈ R(L ), then a weak generalized solution u turns into a classic one.Indeed, let f ∈ R(L ) and u ∈ ˜E be a weak generalized solution. Therefore, for allϕ ∈ D(L ∗) we have

˜l(u) = ϕ( f ), l = L ∗ϕ .

Moreover, there exists such an element u1 ∈ D(L ) that L u1 = f . This element u1

is a weak generalized solution, i.e.

˜l(u1) = ϕ( f ), l = L ∗ϕ


for all ϕ ∈ D(L ∗). Thus, for all ϕ ∈ D(L ∗) the equality ˜l(u1) = ˜l(u) holds, wherel = L ∗ϕ . Since the set of all functionals ˜l, where l = L ∗ϕ , ϕ ∈D(L ∗), coincideswith the space ˜E∗, then u = u1 in ˜E , and therefore in E .

Analogously, if a weak generalized solution u belongs to D(L ), then it is a clas-sic solution.

2.4 Weak Near-Solution

Analogous to a strong near-solution let us introduce a weak near-solution.

Definition 2.4. A sequence un ⊂ D(L ) is called a weak near-solution of the op-erator equation (2.1) if fn = L un→ f as n→ ∞ with respect to the metric of thespace F and un = L −1( fn)→ u ∈ ˜E as n→ ∞ with respect to the weak topologyσ(˜E,R(L ∗)); an element u ∈ ˜E is called a Limit element!weak.

As it will be proved below, the effect of stabilizing of a sequence of elements un

in the space ˜E is a corollary of the convergence of fn to f ; so, there exists an analogybetween string and weak near-solutions.

Consider the relation between a weak generalized solution and a weak near-solution. Let us prove that u is a weak generalized solution of the operator equa-tion (2.1) iff it is a weak limit element of a near solution. Indeed, let u be a limitelement of a near-solution, then u ∈ ˜E and there exists a sequence { fn} ⊂ R(L )such that fn tends to f as n → ∞ with respect to the metric of F , and un =L −1( fn) ∈ D(L ) tends to u as n→ ∞ in the topology σ(˜E,R(L ∗)). Therefore,for all ϕ ∈D(L ∗)⊂ F∗ we have that ϕ(L un) = ϕ( fn), hence

l(un) = (L ∗ϕ)(un) = ϕ( fn)→ ϕ( f )

as n→ ∞, where l = L ∗ϕ ∈ R(L ∗).In addition, since l ∈ R(L ∗), then l(un) = ˜l(un)→ ˜l(u) as n→ ∞. Thus, we

have that˜l(u) = ϕ( f )

for all ϕ ∈D(L ∗) such that l = L ∗ϕ , i.e. u is a weak generalized solution of (2.1).Conversely, let us suppose that u is a weak solution of (2.1), i.e.

˜l(u) = ϕ( f ) for all ϕ ∈ D(L ∗),

where l = L ∗ϕ .Let { fn} be an arbitrary sequence from R(L ) convergent to f as n→ ∞ with

respect to the norm of F . Denote L −1( fn) as un. Then for an arbitrary functionalϕ ∈ D(L ∗) such that l = L ∗ϕ we have

l(un) = (L ∗ϕ)(un) = ϕ(L xn) = ϕ( fn)→ ϕ( f )

as n→ ∞.

2.5 Existence and Uniqueness of a Weak Generalized Solution of a Linear Operator Equation 13

Thus, for any functional ∈ R(L ∗) we have that l(un) → ϕ( f ) = ˜l(u) asn → ∞. Therefore, the sequence un converges to u with respect to the topologyσ(E,R(L ∗)), hence u is a limit element of a near-solution {un}.

2.5 Existence and Uniqueness of a Weak Generalized Solutionof a Linear Operator Equation

In this section, we prove the theorem on existence and uniqueness of a weak gener-alized solution of the operator equation (2.1) on the assumptions stated above, i.e.if L is a linear operator with dense domain D(L ) and dense range R(L ), (2.1) isuniquely solvable, and the sets D(L ∗) and R(L ∗) are total in the spaces F∗ and E∗with respect to the corresponding weak topologies.

Let us start with the relatively simple problem of uniqueness. Suppose that theoperator equation (2.1) in addition to a weak generalized solution u∈ ˜E has anotherweak generalized solution u ∈ ˜E (u �= u), then

˜l(u) = ϕ( f ) = ˜l(u)

for all ϕ ∈D(L ∗), l = L ∗ϕ .Since the set of the functionals ˜l coincides with the conjugate space ˜E∗, then

u = u, and we have a contradiction. Thus, the operator equation (2.1) may not havemore than one weak generalized solution.

Now, let us consider the problem of existence of a weak generalized solution.Suppose that the right-hand side of the operator equation (2.1), i.e. the elementf does not belong to the range R(L ) of the operator L . Since (2.1) is denselysolvable, then there exists such a sequence of elements fn from R(L ) that fn → fas n→ ∞ with respect to the norm F . Let us prove that the sequence un = L −1( fn)is a weak near-solution, and its limit element u belongs to ˜E. For this purpose letus consider the inverse operator u = L −1( f ), which acts from the vector spaceR(L ) into E . Denote by T the topology induced in R(L ) ⊂ F by the norm ofthe Banach space F , and denote by (R(L ),T ), (E,σ(E,R(L ∗))) the vector spacesR(L ) and E endowed with the topologies T and σ(E,R(L ∗)), respectively. Letus prove that the inverse operator B = L −1 is a continuous linear operator, whichacts from the normed space (R(L ),T ) into the Hausdorff topological vector space(E,σ(E,R(L ∗))). Since the set

W (l1, . . . , ln;ε) = {u : u ∈ E, l1(u)< ε, . . . , ln(u)< ε},

where ε ∈ R, li ∈ R(L ∗), i ∈ {1,2, . . . ,n}, form a fundamental system of neigh-borhoods of zero in (E,σ(E,R(L ∗))), it is enough to prove that the followingpreimages B−1[W (l1, . . . , ln;ε)] = L [W (l1, . . . , ln;ε)] are neighborhoods of zero in(R(L ),T ).


Indeed,

L [W (l1, . . . , ln;ε)] = {L u : l1(u)< ε, . . . , ln(u)< ε}= {L u : ϕ1

(

L u)

< ε, . . . ,ϕn(

L u)

< ε},

where li(u) = (L ∗ϕi)(u) = ϕi(L u), ϕi ∈ D(L ∗), i ∈ {1,2 . . . ,n}. Therefore,

L [W (l1, . . . , ln;ε)] = { f ∈ R(L ) : ϕ1( f )< ε, . . . ,ϕn( f )< ε}= WR(L )(ϕ1, . . . ,ϕn;ε),

where WR(L )(ϕ1, . . . ,ϕn;ε) is a neighborhood that belongs to a fundamental sys-tem of neighborhoods of zero in the vector space R(L ) endowed with the topol-ogy σ(R(L ),D(L ∗)). Since the normed topology T is stronger than the weaktopology σ(R(L ),D(L ∗)), then the set WR(L )(ϕ1, . . . ,ϕn;ε) is a neighborhood ofzero with respect to the topology T . Thus, the operator B = L −1 : (R(L ),T )→(E,σ(E,R(L ∗))) is continuous.

Since the space ˜E is complete, F,E are the Hausdorff topological vector spaces,and every continuous linear map B of the space (R(L ),T ) into E is uniquelyextendable to a continuous linear map ˜B from F into ˜E [7], then the sequence{un = L −1( fn) = ˜B( fn)} converges to some element u ∈ ˜E , which is a limit el-ement of the near-solution {un = L −1( fn)}. As it was shown above, in this case uis a weak generalized solution of (2.1). Thus, the existence of a weak generalizedsolution of (2.1) is proved.

2.6 Relation Between Weak and Strong Solutions of a LinearOperator Equation

Let us establish the relation between the solvability in sense of Definitions 2.1and 2.3.

Theorem 2.3. The space E is densely embedded into the space ˜E.

Proof. Let some network {uα}α∈A ,uα ∈E converges to 0 with respect to the topol-ogy of the space E . Then L uα → 0 in F , hence ϕ (L uα)→ 0 for any ϕ ∈ F∗.Thus, l(uα )→ 0 for all l ∈ R(L ∗). Therefore, the topology ˜E is weaker than thetopology E . It remained only to prove that if uα → u with respect to the topology ofthe space E and uα → 0 with respect to the topology ˜E , then u = 0 (condition π)).Taking into account the fact that uα → u is a convergent sequence, we have that

l(uα) = L ∗ϕ(uα) = ϕ(L uα)→ ϕ(L u)

for all l ∈ R(L ∗). In addition, the fact that uα → 0 implies that l(uα)→ 0 for alll ∈ R(L ∗) also. Thus, we have that ϕ(L u) = 0 for any ϕ ∈ D(L ∗). Since the set

2.6 Relation Between Weak and Strong Solutions of a Linear Operator Equation 15

D(L ∗) is total and the operator L is injective, then u = 0. Thus, the embeddingE ⊂ ˜E is proved.

The fact that the embedding is dense follows from the fact that the spaces E ⊂ ˜Eare obtained as a result of completing of the set D(L ), i.e. D(L ) is a dense set bothin E and ˜E . Theorem 2.4. Definitions 2.1 and 2.3 are equivalent.

Proof. Let u ∈ E be a strong generalized solution of the equation L u = f . Takinginto account the fact that the set R(L ) is dense in F , we have that there exists sucha sequence fn ∈ R(L ) that converges to f , or, in other words, there exists such anelement un ∈D(L ), that un→ u in E . By virtue of Theorem 2.3 the elements u∈ Ebelongs to the space ˜E , and in addition un→ u in ˜E. Now, it is easy to see that, fromone hand, for all l = L ∗ϕ ∈ R(L )

l(un) = L ∗ϕ(un) = ϕ(L un)→ ϕ( f ),

and, from the other hand, – l(un)→ l(u) as n→ ∞. Thus, u – is a weak generalizedsolution.

Let us prove that the solution u ∈ ˜E in the sense of Definition 2.3 is a solution inthe sense of Definition 2.1 (and vice versa). Indeed, there exists a solution u∗ ∈ Eof the equation L u = f . It is clear that L ∗ϕ(u) = ϕ( f ) = ϕ(L u∗) for all ϕ ∈D(L ∗). Hence u = Ou∗, where O is an operator of embedding of the space E intothe space ˜E .

Finally, let us point out that the concept of a generalized solution of the operatorequation L u = y is very different from various concepts u∗ of such equations (forexample, from the concept of a quasi-solution introduce by V. K. Ivanov), which aredescribed in [47] and [112], as far as L u = y for the generalized solution u always,where L is a natural extension of the operator L , whereas the equality L u∗ = yfor the generalized solutions u∗ holds not always.

Chapter 3A Priori Estimates for Linear ContinuousOperators

In this chapter, we will study a linear continuous operator L , which acts from anormed space E (D f = E) into a normed space F . We will suppose that the L isinjective and has a dense range in F .

Strong and weak solutions considered in the previous chapter belong to spaces Eand ˜E , but constructive description of the spaces E and ˜E is a very difficult problemfor various operators L which are important from the practical point of view. So, itis necessary to establish the existence of a dense embedding of E or ˜E into anotherwell-studied Banach or locally convex linear topological space H. In this chapter, wedescribe such spaces H for some integral, differential and abstract Hilbert–Schmidtoperators in Hilbert space. In addition, we will study the properties of generalizedsolutions in H.

3.1 A Priori Inequalities

Let us consider the case when the space E in embedded into a Banach space H. Thisembedding implies that c1‖u‖H ≤ ‖u‖E for all u ∈ E . Hence,

c1‖u‖H ≤ ‖L u‖F ≤ c2‖u‖E , ∀u ∈ E, (3.1)

where c1,c2 are positive constants.Such estimations are common in applications. They are called a priori estima-

tions [40, 62]. In addition (3.1), the following a priori estimates hold

c1‖u‖E ≤ ‖L u‖F ≤ c2‖u‖E, ∀u ∈ E,

c1‖u‖H ≤ ‖L u‖F ≤ c2‖u‖E , ∀u ∈ E,

where L , as in Chap. 2, is an extension L onto the entire space E by continuity,i.e. L : E → F .


17

18 3 A Priori Estimates for Linear Continuous Operators

Note that inequalities (3.1) themselves do not guarantee the embedding E ⊂ H.They only allow to compare topologies induced in E by the norms ‖ · ‖E and ‖ · ‖H.

Further, let us prove that estimations (3.1) may be a basis for constructing thetheory of generalized solvability of an operator equation

L u = f (3.2)

As in Chap. 2 we shall consider the following concepts.

Definition 3.5. A strong generalized solution of (3.2) is such an element u ∈ E thatL u = f .

3.2 A Generalized Solution of an Operator Equationin Banach Spaces

The previous section implies that inequalities (3.1) are the necessary conditions forconstruction of the theory of generalized solvability of linear operator equations ina Banach space H. Let us prove that inequalities (3.1) are sufficient conditions forthe solvability of L u = f also (in some generalized sense) L u = f . Note that vari-ous approaches to construction of the theory of generalized solutions of differentialequations are relevant to this scheme also (see [5]).

Let us suppose that the linear operator L , (D(L ) = E , R(L ) = F) satisfiesinequalities (3.1), where u ∈ E , c1,c2 > 0, H is a completion of the space E withrespect to the norm ‖u‖H . It is clear that the right-hand sides of inequalities (3.1)imply the continuity of the operator L , and the left-hand sides imply its injectivity.In addition, by virtue of the density of the embedding E ⊂ H the set H∗ (conjugateto H) is total in the conjugate space E∗ and, hence, the spaces E and H∗ are dual toeach other.

Lemma 3.1. An operator equation

L ∗ϕ = l (3.3)

is solvable on a subset H∗ of the space E∗.

Proof. Let us consider the operator ˜L : H → F (D( ˜L ) = E), defined in the fol-lowing way: ˜L u = L u,u ∈ E . Then the left-hand side of inequalities (3.1) impliesthe correct solvability of the operator ˜L . Let us consider also the adjoint opera-tor ˜L ∗ : F∗ → H∗, D( ˜L ∗) ⊂ D(L ∗) = F∗. It is clear that if ϕ ∈ D( ˜L ∗), then˜L ∗ϕ |E = L ∗ϕ , where ˜L ∗ϕ|E is a restriction of the functional ˜L ∗ϕ ∈ H∗ from

the set H onto the set E . As well-known, the correct solvability of the operator ˜L

implies the solvability of the operator ˜L ∗ everywhere [40]; hence, taking into ac-count the facts above, we have the solvability of the operator L ∗ over the set H∗ (asa subspace of E∗).

3.2 A Generalized Solution of an Operator Equation in Banach Spaces 19

Remark 3.2. If the operator L satisfies inequalities (3.1), then

H∗ ⊂ R(L ∗)⊂ E∗.

Definition 3.6. A generalized solution of (3.2) with a right-hand side f ∈ F is suchan element u ∈H, that the equality

˜L ∗ϕ(u) = ϕ( f ), (3.4)

holds for any ϕ ∈ D( ˜L ∗).

It is clear that equality (3.4) is equivalent to

L ∗ϕ(u) = ϕ( f ), ∀ϕ ∈ F∗,L ∗ϕ ∈H∗.

Theorem 3.1. For any right-hand side f ∈ F there exists a unique solution u ∈ Hof (3.2) in the sense of Definition 3.6.

Proof. Let us choose a sequence fp ∈R(L ) such that fp→ f in the space F . Hence,if up ∈ E is a solution of the equation L u = fp, then taking into account (3.1) andthe fact that the sequence { fp} is Cauchy, we have

‖up1−up2‖H ≤ c−11 ‖L up1−L up2‖F = c−1

1 ‖ fp1− fp2‖F → 0, p1, p2→ ∞.

Thus, there exist such u∗ ∈ H that up→ u∗ in H. Further, we have

L ∗ϕ(up) = ϕ(L up) = ϕ( fp), ϕ ∈ F∗.

Passing to the limit in the last equality as p→ ∞, we have

L ∗ϕ(u∗) = ϕ( f ), ϕ ∈ F∗,L ∗ϕ ∈ H∗.

Thus, u∗ is a solution of (3.2) in the sense of Definition 3.6.Since H∗ ⊂ R(L ∗), then the equality

l(u∗) = L ∗ϕ(u∗) = 0, ∀ϕ ∈ F∗,L ∗ϕ ∈H∗

implies that u∗ = 0, and hence the solution is unique. Definition 3.7. A generalized solution of problem (3.2) with a right-hand side f ∈ Fis such an element u ∈ H that there exists a sequence ui ∈ E , which satisfies theconditions

‖ui−u‖H → 0, ‖L ui− f‖F → 0, i→ ∞.

Theorem 3.2. Definitions 3.6 and 3.7 are equivalent.


Proof. Let u be a solution of the equation L u = f in the sense of Definition 3.6, i.e.L ∗ϕ(u)=ϕ( f ). Repeating reasonings which were used for proving of Theorem 3.1we conclude that u = u∗, and hence ‖up−u‖H → 0. From the other hand, ‖L up−f‖F = ‖ fp− f‖F → 0. Thus, u is a solution of (3.2) in the sense of Definition 3.7.

Let us prove the inverse statement. Let u be a solution of (3.2) in the sense ofDefinition 3.7. Then

L ∗ϕ(u) = L ∗ϕ(ui)+L ∗ϕ(u−ui) = ϕ(L ui)+L ∗ϕ(u−ui)

= ϕ(L ui− f )+ϕ( f )+L ∗ϕ(u−ui),

for all ϕ ∈ F∗,L ∗ϕ ∈H∗.Let us estimate the first and third term in the right-hand side of the last equality

|ϕ(L ui− f )| ≤ ‖ϕ‖F∗‖L ui− f‖F → 0,

|L ∗ϕ(u−ui)| ≤ ‖L ∗ϕ‖H∗‖u−ui‖H → 0, i→ ∞.

Hence,L ∗ϕ(u) = ϕ( f ), ϕ ∈ F∗,L ∗ϕ ∈H∗,

i.e. u is a solution of (3.2) in the sense of Definition 3.6. Remark 3.3. It is clear that solution in the sense of Definitions 3.6 and 3.7 coincidewith a classic solution u ∈ E if f ∈ R(L ). Also, it is easy to prove that the classicalsolution is a generalized solution, and if the generalized solution u belongs to D(L ),it is a classic solution.

Theorem 3.3. If the space E is embedded into H, then Definitions 3.5, 3.6 and 3.7are equivalent.

Proof. Let us prove that Definition 3.5 is equivalent to Definition 3.7. Let u ∈ H bea solution of (3.2) in the sense of Definition 3.7. As it was mentioned above, for anyright-hand f ∈ F there exists a unique solution u∗ ∈ E in the sense of Definition 3.5.Let us prove that u∗ = u considering u∗ to be the element of H (by virtue of theembedding E ⊂ H). Indeed,

‖u−u∗‖H ≤ ‖u−ui‖H +‖ui−u∗‖H ≤ ‖u−ui‖H + c−11 ‖ui−u∗‖E =

= ‖u−ui‖H + c−11 ‖L ui− L u∗‖F → 0, i→ ∞,

where ui ∈ E is a sequence that converges to the solution u ∈ H.Thus, u is a solution of (3.2) in the sense of Definition 3.5.And vice versa, let u be a solution of (3.2) in the sense of Definition 3.5. Let us

choose an arbitrary sequence ui ∈ E such that ‖ui−u‖E→ 0. Then, by virtue of theembedding E ⊂ H we have

‖u− ui‖H → 0, ‖ui−u‖E = ‖L ui− L u‖F = ‖L ui− f‖F → 0, i→ 0.

3.2 A Generalized Solution of an Operator Equation in Banach Spaces 21

Remark 3.4. The theorem implies that there exist a constant c > 0 such that

‖u‖H ≤ c‖ f‖F , ∀ f ∈ F,(c > 0),

where u is a solution of (3.2) with a right-hand side f in the sense of Definitions 3.5–3.7.

Remark 3.5. The embedding E ⊂ H follows either from the density of D( ˜L ∗) inthe space F∗ with respect to the weak topology σ(F∗,F), either from the fact thatthe operator ˜L is closable.

In many cases, inequalities (3.1) for the direct operator L immediately implysimilar inequalities for the adjoint operator

c1‖ϕ‖G ≤ ‖L ∗ϕ‖E∗ ≤ c2‖ϕ‖F∗ , ∀ϕ ∈ F∗, (3.5)

where G is a completion of the set F∗ with respect to some norm. Consider this casefor reflexive Banach spaces E,F . In this case, L ∗∗ =L and similarly to Lemma 3.1we have that the operator equation (3.2) is solvable over G∗ ⊂ F . In addition, ana-logues of Theorems 3.1–3.3 for solvability of the adjoint equation (3.3) hold.

Theorem 3.4. There is such a constant c > 0 that for any f ∈ G∗ ⊂ F and for anyl ∈H∗ ⊂ E∗ the following inequalities are satisfied:

‖u‖E ≤ c‖ f‖G∗ , (3.6)

‖ϕ‖F∗ ≤ c‖l‖H∗ , (3.7)

where u ∈ E,ϕ ∈ F∗ are the solutions of the equation L u = f and L ∗ϕ = l.

Proof. Let us prove inequality (3.6) (inequality (3.7) can be proved in a similarway). Since (3.3) is solvable (in the sense of analogues of Definitions 3.6 and 3.7for the adjoint operator) for any l ∈ E∗, then for any u ∈ E : L u ∈G∗ the followingequality holds (the second conjugate space is identified with the original space)

L u(ϕ) = u(l), (3.8)

where ϕ ∈ G is a solution of (3.3) with the right-hand side l ∈ E∗. Hence,

|u(l)| ≤ ‖L u‖G∗ ×‖ϕ‖G

or∣

∣

∣

∣

(

u‖L u‖G∗

)

(l)

∣

∣

∣

∣

≤ ‖ϕ‖G.

Thus, the set of functionals{

u‖L u‖G∗

: L (u) ∈G∗,u ∈ E

}

⊂ E∗∗ = E


is bounded in any point l ∈ E∗, and therefore, by virtue of Banach–SteinhausTheorem, it is bounded with respect to the norm of the space E∗∗= E , and thereforeinequality (3.6) is proved. Theorem 3.5. There exists such a constant c > 0 that for any f ∈ F and for anyl ∈ E∗ the following inequalities are satisfied:

‖u‖H ≤ c‖ f‖F , (3.9)

‖ϕ‖G ≤ c‖l‖E∗, (3.10)

where u ∈H and ϕ ∈ G are the solutions of the equations L u = f and L ∗ϕ = l inthe sense of Definition 3.6 and 3.7.

Proof. Reasoning from inequality (3.8), we have

|ϕ(L u)|= |L u(ϕ)| ≤ ‖u‖E×‖l‖E∗.

Applying inequality (3.6) to the right-hand side of inequality, we have

|ϕ(L u)| ≤ c‖L u‖G∗ ×‖l‖E∗

or

‖ϕ‖G = ‖ϕ‖G∗∗ = supL u∈G∗

|ϕ(L u)|‖L u‖G∗

≤ c‖l‖E∗.

Taking into account the fact that G∗ ⊂ R(L ), we obtain inequality (3.10). Inequal-ity (3.9) is proved in a similar way. Remark 3.6. Since ‖ f‖F = ‖L u‖F = ‖u‖E and inequality (3.9) holds for any f ∈F ,it may seem that (3.9) guarantees the embedding E ⊂ H. But it is not true. Indeed,in the space E there exists an element u∗ such that ‖u∗‖E = ‖ f‖F . But until theembedding is not proved E ⊂ H, it is impossible to compare u ∈ H as a solutionof (3.2) in the sense of Definitions 3.6 and 3.7, and u∗ ∈ E as a solution of (3.2) inthe sense of Definition 3.5.

3.3 A Generalized Solution in Locally Convex LinearTopological Spaces

Let us introduce several more definitions of a generalized solution. As before, let ussuppose that L is a linear continuous operator. Let us select in the space E∗ a totallinear set M ⊂ R(L ∗)⊂ E∗. Let ˜M be a completion of the set E with respect to thetopology σ(E,M). By virtue of the Banach theorem on a weakly continuous linearfunctional, the functional l = L ∗ϕ ∈ M admits a unique extension by continuityonto the entire space ˜M.

3.3 A Generalized Solution in Locally Convex Linear Topological Spaces 23

Definition 3.8. A generalized solution of (3.2) is an element u ∈ ˜M satisfying therelation

L ∗ϕ(u) = ϕ( f ), ∀ϕ ∈ F∗,L ∗ϕ ∈M.

Definition 3.9. A generalized solution of (3.2) is an element u ∈ ˜M for which thereexists a sequence ui ∈ E such that

ui→ u in topology ˜M, ‖L ui− f‖F → 0, i→ ∞.

If M = R(L ∗), Definitions 3.8 and 3.9 turn into Definitions 2.3 and 2.2 of a weaksolution and a near-solution, respectively.

It is easy to prove the following theorem.

Theorem 3.6. Definitions 3.8 and 3.9 are equivalent and for any element f ∈ Fthere exists a generalized solution u ∈ ˜M of (3.2) in the sense of Definitions 3.8and 3.9.

Remark 3.7. Definition 3.9 implies that if u ∈ ˜M is a generalized solution of (3.2),then the point (u, f ) is an adherent point of graphs Γ (L ) of the operator L , i.e. anadherent point of the set

{(u, f ) : L u = f ,u ∈ E} ⊂ ˜M×F

with respect to the topology ˜M×F.Moreover, the point (u, f ) belongs to a sequential closure of the graph Γ (L )

with respect to the topology ˜M× F and hence u ∈ ˜Ms, where ˜Ms is a sequentialclosure of the set E with respect to the topology ˜M.

Remark 3.8. Taking into account Remark 3.7, we can define a generalized solutionin sequentially complete spaces ˜Ms.

In addition, it is easy to prove that a classic solution (3.2) is a generalized solutionin the sense of Definitions 3.8 and 3.9. If f ∈ R(L ) or a generalized solutionbelongs to D(L ), then the generalized solution turn into the classic solution.

Since M ⊂ R(L ∗), then there exists such a set MF ⊂ F∗, that M = L ∗(MF ).

Theorem 3.7. An embedding of the space EL into the space ˜M exists iff MF is atotal linear subset of F∗. In this case, Definitions 3.8, 3.9, and 3.5 are equivalent.

Proof. Similar to Theorem 2.3, it can be proved that in set E the topology inducedby the norm ‖·‖EL

is stronger that the topology σ(E,M). The condition π is consid-ered similarly. We have that ϕ(L u) = 0 for any ϕ ∈MF . The fact that the operatorL sets an isometric isomorphism between the spaces EL and F implies that thecondition u = 0 is equivalent to the totality of the set MF . The equivalence of thedefinitions is proved similar to Theorem 2.4. Remark 3.9. Theorem 3.7 implies that the set ˜M can be defined as a completionof the set EL with respect to the topology σ(E,M) (under the conditions of thetheorem).


Remark 3.10. The condition of totality of MF is of fundamental importance anddoes not holds always.

Let us consider an example of a linear continuous injective operator which mapsnot total sets into total sets. Indeed, the operator A : �2 → �2, which acts on thevector x = (ξ1,ξ2, . . .) by the rule

A x =

(

ξ1

20 ,ξ2

21 ,ξ3

22 , . . .

)

.

is linear, continuous (and even totally continuous), and injective. Also, it maps thevector system

g1 = (1,2,0,0,0, . . .),

g2 = (0,2,4,0,0, . . .),

g3 = (0,0,4,8,0, . . .),

. . .

into the system

e1 = (1,1,0,0,0, . . .),

e2 = (0,1,1,0,0, . . .),

e3 = (0,0,1,1,0, . . .),

. . .

The vector system {gi} is orthogonal to the vector

x∗ =(

1,−12,

14,−1

8, . . .

)

and, hence, it is not total, but from the other hand, the totality of the system {ei} in�2 is obvious.

In addition, if L satisfies a priori inequalities (3.1), then the following theoremholds.

Theorem 3.8. The Banach space H is embedded into the space ˜H∗. Definitions3.8, 3.9 (for H∗), and 3.6, 3.7 are equivalent.

Proof. It is clear that the topology induced by the norm ‖ · ‖H is stronger than thetopology σ(E,H∗). Let us test the condition π . Let a sequence un be convergentto u with respect to the topology of the space H and un → 0 with respect to thetopology σ(E,H∗). Then Hahn–Banach Theorem implies that the norm ‖ · ‖H canbe represented as

‖un−u‖H = supl∈H∗

|l(un−u)|‖l‖H∗

.

3.4 Relation Between Generalized Solutions in Banach and Locally Convex Spaces 25

Hence, l(un− u)→ 0 as n→ ∞; therefore l(un)→ l(u) for all l ∈ H∗. On the otherhand, l(un)→ 0 for all l ∈H∗. Thus, l(u) = 0 for all l ∈H∗, and therefore, u= 0.

3.4 Relation Between Generalized Solutions in Banachand Locally Convex Spaces

Note that there is some kind of analogy between the schemes of the construction ofgeneralized solutions in the sense of Definitions 3.6 and 3.7 and the schemes of theconstruction of generalized solutions in the sense of Definitions 3.8 and 3.9.

Just as we construct a Hausdorff locally convex linear topological space ˜M bya total set M ⊂ R(L ∗) ⊂ E∗, we can construct a Banach space M, which definesa generalized solution in the sense of Definitions 3.6 and 3.7, by the total set M ⊂R(L ∗)⊂ E∗. Namely, let M be a completion of the set E with respect to the norm

‖u‖M = supL ∗ϕ∈M

|L ∗ϕ(u)|‖ϕ‖F∗

. (3.11)

Norm (3.11) can be rewritten as

‖u‖M = supϕ∈MF

|ϕ(L u)|‖ϕ‖F∗

. (3.12)

From the other hand, if M = R(L ∗), then the norm ‖u‖M coincides with thenorm of the space EL and, hence, M = EL (an analogue of the space ˜EL ). Indeed,

‖u‖M = supL ∗ϕ∈M

|L ∗ϕ(u)|‖ϕ‖F∗

= supϕ∈F∗

|ϕ(L u)|‖ϕ‖F∗

= ‖L u‖F .

This equality holds because, by virtue of Hahn–Banach Theorem, for any elementL u∈F there exists such a functional ϕ ∈F∗ with unit norm that ϕ(L u)= ‖L u‖F .

In addition, we have the following lemma.

Lemma 3.2. If M ⊂ R(L ∗) and MF is a total subset of the space F∗, then the spaceEL is embedded into the space M.

Proof. The totality of the set MF and the injectivity of the operator L imply thetotality of the set M. It is easy to see that the norm ‖ · ‖EL

is stronger that the norm‖ · ‖M. It remains only to test the condition π . Let a sequence un ∈ E be convergentto u∈ EL with respect to the norm ‖u‖EL

and un be convergent to zero with respectto the norm ‖u‖M, then, from the one hand, ϕ(L un)→ ϕ(L u), and from the otherhand, ϕ(L un)→ 0 for any ϕ ∈MF . Hence, ϕ(L u) = 0 for all ϕ ∈MF . The totalityof MF and the injectivity of L imply that u = 0. Remark 3.11. Theorem implies that the space M can be constructed by completingthe set EL with respect to (3.11).


Thus, we can obtain an a priori estimate by any total set M ⊂ R(L ∗) (and if MF

is total then the embedding EL ⊂ M also)

c1‖u‖M ≤ ‖L u‖F ≤ c2‖u‖E, ∀u ∈ E,

This embedding implies the statements on solvability of (3.2) (in the sense ofanalogues of Definitions 3.5–3.7) which are similar to Theorems 3.1–3.3.

It is also easy to see that a topology induced by the norm ‖ ·‖M on the space E isstronger than a topology induced by a topology of the space ˜M.

Thus, evolving the idea on relation between generalized solutions in Banach andlocally convex linear topological spaces, we can describe the results of Sect. 3.2in the style of Sect. 3.3 (proceeding from a total linear subset of the set R(L ∗)),and the results of Sect. 3.3 – in the style of Sect. 3.2 (using analogues of a prioriinequalities, i.e. assuming that a Hausdorff locally convex topology defined on theset E is weaker than the norm ‖ · ‖EL

).

Lemma 3.3. The following equality holds: R(L ∗) = (EL )∗, where R(L ∗), (EL )∗are subsets of the space E∗.

Proof. Let L : EL → F be a completion of the operator L by continuity, then theoperators L ∗ and L ∗ set an isomorphism between the linear sets

L ∗ : F∗ ←→ (EL )∗, L ∗ : F∗ ←→ R(L ∗)

Let us prove that the sets R(L ∗) and (EL )∗ coincide each other as sets of thespace E∗. Let O : E→ EL be a linear continuous operator, which defines a canonicembedding of the space E into EL . For any functional l ∈ (EL )∗ there exists suchan element ϕ ∈ F∗, that

O∗l(u) = l(Ou) = L ∗ϕ(Ou) = ϕ(L (Ou)) = ϕ(L u) = L ∗ϕ(u), ∀u ∈ E.

Hence, R(O∗) = R(L ∗). Q.E.D. Remark 3.12. Identifying R(L ∗) and (EL )∗, we can interpret the set R(L ∗) as aBanach space REL

(L ∗) with the norm

∥

∥l∥

∥

REL(L ∗) = ‖ϕ‖F∗ ,

where L ∗ϕ = l.Indeed, using this identification, we have

∥

∥l∥

∥

REL(L ∗) = ‖l‖(EL )∗ = ‖L ∗ϕ‖(EL )∗ = sup

u∈EL

|L ∗ϕ(u)|‖u‖EL

= supf∈F

|ϕ( f )|‖ f‖F

= ‖ϕ‖F∗ .

Thus, the spaces REL(L ∗) and (EL )∗ are isometrically isomorphous.

In conclusion of the theoretical part we note that many aspects of our analysishave the topological character. Therefore, we can study (3.2) in locally convex linear

3.4 Relation Between Generalized Solutions in Banach and Locally Convex Spaces 27

topological spaces (and, may be, in just topological) spaces E and F also. In thiscase, instead of EL we may consider a completion of E with respect to the topologyinduced by the system of semi-norms

pα ,EL(u) = pα ,F(L u), α ∈ A,

where {pα ,F}α∈A is a system of semi-norms, which induces the topology of thespace F , and instead of estimations (3.1) we have a chain of dense embeddings

E ⊂ EL ⊂ H, ∀u ∈ E,

where H is a completion of the set E with respect to some locally convex topologywhich is weaker than the norm of the space EL .

Chapter 4Applications of the Theory of GeneralizedSolvability of Linear Equations

4.1 Application to the Equations with Hilbert–Schmidt Operator

Let L2(−π,π) be the Hilbert space of measurable, square integrable, complex val-ued functions with the standard inner product (·, ·)0 and {ek}∞

k=−∞ be an orthonormalbasis consisting of eigenvectors of an self-adjoint Hilbert–Schmidt operator

L u = f , L : L2(−π,π)→ L2(−π,π). (4.1)

Then

L u =∞

∑k=−∞

λk(u,ek)0ek,∞

∑k=−∞

λ 2k <+∞.

Let us denote by E the vector space of all infinitely differentiable numerical func-tions over (−π,π). Let us consider on the set E a countable system of semi-norm

pm( f ) = supt∈[−π ,π ]

| f (m)(t)|,

where m ≥ 0 is an integer number, f (m) is the derivative of the function f of orderm. Thus, the set E turns into a metrizable topological vector space.

Let us select in the space E a closed subset E= which consists of functions satis-fying the additional condition

f (m)(−π) = f (m)(π), ∀m ∈ N∪{0}.

Denote by E ∗= the conjugate space of E= endowed with a weak-* topologyσ(E ∗=,E=). EL means a completion of the space L2(−π,π) with respect to the norm

‖u‖EL= ‖L u‖L2(−π,π) =

∞

∑k=−∞

λ 2k |(u,ek)0|2.


29

30 4 Applications of the Theory ...

Lemma 4.1. Let the basis ek be a trigonometric system of functions ek = eikt , k ∈ Z

and the eigenvalues satisfy the estimation

|λk|> cks ,

for fixed constants c> 0, s≥ 1. Then the Banach space EL is densely embedded intothe space of distributions E ∗= with the help of the operator of canonical embedding j(for an arbitrary function x(t) ∈ EL we have ( j(x))( f ) =

∫ π−π f (t)x(t)dt, where

j(x) ∈ E ∗= and f ∈ E=).

Proof. To prove the lemma, it is enough to check whether the following statementsare true:

1. The topology τ0 induced by the norm ‖·‖ELon the set L2(−π,π) is stronger than

the topology τ1 induced by the weak-* topology σ(E ∗=,E=) on the set L2(−π,π).2. For the topologies τ0,τ1 the following condition is satisfied:

π) if the sequence {un}, where un ∈ L2(−π,π), is a Cauchy sequence in the topologyτ0 and it converges to zero in the topology τ1, then the sequence {un} converges to zeroin the topology τ0 also.

Let us verify Condition 1. Let {un} be a sequence of elements of the spaceL2(−π,π) which converges to zero in the topology τ0, i.e.

‖un‖EL=

∞

∑k=−∞

λ 2k |(un,ek)0|2→ 0, n→ ∞.

Let us prove that un converges to zero in the topology τ1 also:∫ π

−πun(t)ϕ(t)dt = (un,ϕ)0 −−−→

n→∞0, ∀ϕ ∈ E=.

Indeed, expanding ϕ into a Fourier series and using the continuity of the innerproduct (u,v)0, we have

|(un,ϕ)0| =∣

∣

∣

(

un,∞

∑k=−∞

(ϕ,ek)0ek

)

0

∣

∣

∣=∣

∣

∣

∞

∑k=−∞

(ϕ,ek)0 (un,ek)0

∣

∣

∣

≤∞

∑k=−∞

|(ϕ,ek)0|× |(un,ek)0|.

Further, we have

|(un,ϕ)0| ≤∞

∑k=−∞

|(ϕ,ek)0|λk

×λk|(un,ek)0|

≤(

∞

∑k=−∞

|(ϕ,ek)0|2λ 2

k

)1/2

×(

∞

∑k=−∞

λ 2k |(un,ek)0|2

)1/2

=

(

∞

∑k=−∞

|(ϕ,ek)0|2λ 2

k

)1/2

×‖un‖EL.

4.1 Application to the Equations with Hilbert–Schmidt Operator 31

If for a fixed ϕ∞

∑k=−∞

|(ϕ,ek)0|2λ 2

k

<C <+∞, (4.2)

then

0≤ |(un,ϕ)0| ≤√

C×‖un‖EL−−−→n→∞

0.

Let us prove (4.2). Integrating by parts, it is easy to see that the coefficients

ak =(

ϕ(t),eikt)

0

of the Fourier series ϕ ∈ E= satisfy the following relations

ak =∫ π

−πϕ(t)eikt dt =− 1

ik

∫ π

−πϕ ′(t)eikt dt =− 1

ikbk, (4.3)

where bk are the Fourier coefficients of the function ϕ ′(t). It is well-known that|bk| → 0 as k→ ∞, and hence, |kak| → 0.

Using the analogous formulae several times, it is easy to prove that

|k|p|ak| −−−→k→∞

0, ∀p ∈ N.

Hence,∞

∑k=−∞

k2p|ak|2 <+∞, ∀p ∈ N.

Since |λk|> c/ks,

∞

∑k=−∞

|(ϕ,ek)0|2λ 2

k

<1c2

∞

∑k=−∞

k2s|ak|2 <C <+∞.

Thus, Condition 1 is verified.Let us verify the condition π). Let a sequence {un} ⊂ L2(−π,π) be a Cauchy

sequence with respect to the norm ‖ · ‖EL, and in the topology τ1 it converges to

zero, i.e.

L un→ f ∗ in the space L2(−π,π) and

(un,ϕ)0 −−−→n→∞

0, ∀ϕ ∈ E=.

Since L is a Hermitian operator,

(L un,ϕ)0 = (un,ψ)0, ψ = L ϕ. (4.4)

Let us prove that ψ = L ϕ is an infinitely differentiable function. Having differen-tiated L ϕ formally and taking into account (4.3), we have

ψ ′(t) = (L ϕ)′(t) =∞

∑k=−∞

λkake′k(t) =∞

∑k=−∞

λk−bk

ike′k(t) =−

∞

∑k=−∞

λkbkek(t).


The correctness of differentiating follows from convergence of the series

∞

∑k=−∞

λkbkek(t).

Indeed, using the Hilbert–Schmidt condition

( ∞

∑k=−∞

λ 2k

)1/2<+∞

and the Bessel inequality, we have

∣

∣

∣

∞

∑k=−∞

λkbkek(t)∣

∣

∣≤∞

∑k=−∞

|λkbk| ≤( ∞

∑k=−∞

λ 2k

)1/2×( ∞

∑k=−∞

|bk|2)1/2

<+∞.

Thus, the function ψ(t) is differentiable. Repeating these reasonings, we can provethat ψ(t) is an infinitely differentiable function.

On the other hand, since the eigenvectors of the basis {ek} belong to the spaceE=, then

ψ = L ϕ =∞

∑k=−∞

λk(ϕ ,ek)0ek ∈ E=.

Thus, proceeding to the limit in (4.4) as n→ ∞, we have

( f ∗,ϕ)0 = 0, ∀ϕ ∈ E=.

Taking into account the fact that the set E= is dense in the space L2(−π ,π), weconclude that f ∗ = 0, so the condition π) holds. �

The embedding EL ⊂ E ∗= implies the existence and uniqueness of a generalizedsolution.

Theorem 4.1. It follows from Lemma 4.1 that there exists a unique generalized so-lution of (4.1) in the sense of Definitions 3.8, 3.9 (M = E=).

Note that the theorem proved above can be interpreted in the sense of Sect. 3.3.Indeed, the space E ∗= is induced by the set M = E= ⊂ (L2(−π ,π))∗. It is justnecessary to verify the embedding E= ⊂ R(L ∗) or, taking into account the self-adjointness of the operator, the embedding E= ⊂ R(L ). To prove the latter inclu-sion, we should suppose that in R(L ) a complete system of functions exists a priori(in the example considered above this system consisted of the functions eikt ). Se-lecting other systems we can get other results. For example, for the integral operator

L u =∫ 1

0K(t,s)u(t)dt, (4.5)

which acts in the real Hilbert space L2(0,1), we can prove the following theorem.


Theorem 4.2. If

(1)There exist functions fn ∈ L2(0,1) such that

L ∗ fn =

∫ 1

0K(t,s) fn(s)ds =

√2sinπnt,

(2)There exist fixed constants c > 0, p such that

‖ fn‖L2(0,1) ≤ cnp, ∀n = 0,1, . . . ,

then (4.5) has a unique solution in the sense of Definitions 3.8 and 3.9 (M =E=(0,1)).

Remark 4.13. By virtue of the Mercer Theorem it is possible to construct integralHilbert–Schmidt operators with the kernel

K(t,s) =∞

∑n=−∞

λneinte−ins, |λk|> cks ,

which satisfy the conditions of Lemma 4.1. In the similar way we can constructkernels satisfying the conditions of Theorem 4.2. An example of such operator is

L ϕ =

∫ 1

0K(x, t)ϕ(t)dt,

where

K(x, t) =

{

(1− x)t, 0≤ t ≤ x,x(1− t), x≤ t ≤ 1.

The eigenvalues of this kernel are

λ1 = π2,λ2 = (2π)2, . . . ,λn = (nπ)2, . . . ,

and the corresponding eigenfunctions are

ϕ1 =√

2sinπx,ϕ2 =√

2sin2πx, . . . ,ϕn =√

2sinnπx, . . . .

Note that in general it is not a rule that the generalized solution of (4.1) is a gen-eralized function even from D∗(−π,π) at least, where D∗(−π ,π) is the conjugatespace of D(−π,π), which is the space of finite and infinitely differentiable functionson (−π,π) with a standard topology. Indeed, consider the integral operator

L u =∫ π

−πK(t,s)u(t)dt,

with the kernel

K(t,s) = K(|t− s|) = K(τ),

K(τ) ={

0, τ ≤ 0,τ, τ > 0.


This operator satisfies the Hilbert–Schmidt condition∫ π

−π

∫ π

−πK2(t,s)dtds <+∞.

Let us prove that this operator is injective. We have

f = L u =∫ s

−πK(|t− s|)u(t)dt +

∫ π

sK(|t− s|)u(t)dt

=

∫ s

−πK(s− t)u(t)dt+

∫ π

sK(t− s)u(t)dt

=∫ π

sK(t− s)u(t)dt.

Differentiating the function f , we have f ′′ =−u, whence it follows that the operatorL is injective.

On the other hand, it is easy to verify, that a solution of (4.1) with the right-handside

f (s) =

{

s(lns−1)+ 1, t ∈ (0,π ],0, t ∈ [−π ,0]

is the function

u(t) =

{

1/t, t �= 0, t ∈ [−π ,π ],0, t = 0,

which is not locally integrable, and hence it does not belong to the space D∗(−π,π).Let us show that in general the space S(−π,π) which consists of measurable

functions on (−π,π) does not cover the space EL . Indeed, considering (4.1) withthe first-order integral Fredholm operator with a kernel K(t,s) which is square inte-grable, we have that the sequence of functions

{

un(t) = ei(−1)n[n/2]t}+∞

n=1

does not converge to zero with respect to the metric of space S(−π,π), i.e. it doesnot converge to zero with respect to the Lebesgue measure. On the other hand, thesequence un converges to zero in the space EL . The sequence {un(t)}+∞

n=1 formsan orthonormal basis in the space L2(−π,π), and hence it converges weakly in thespace L2(−π ,π). Since the function ks(t) = K(t,s) is square integrable for almostall s, then

fn = L un =∫ π

−πK(t,s)un(t)dt =

(

ks,un)

0→ 0, n→ ∞.

for almost all s ∈ (−π,π).


Further, we have

| fn(s)|2 =

∣

∣

∣

∣

∫ π

−πK(t,s)un(t)dt

∣

∣

∣

∣

2

≤(

∫ π

−π|K(t,s)|2dt

)

×(

∫ π

−π|un(t)|2dt

)

=

∫ π

−π|K(t,s)|2dt.

Since K(t,s) is a square summable function on the set (−π,π)× (−π,π), then byvirtue of the Fubini theorem the function

g(s) =∫ π

−π|K(t,s)|2dt,

is a summable function on (−π,π). Thus, the sequence | fn|2 converges to zero al-most everywhere, and | fn(s)|2 ≤ g(s), where g(s) is a summable function. Then byvirtue of Lebesgue theorem

∫ π

−π| fn(s)|2ds→ 0, n→ ∞.

i.e. un→ 0 in EL . Thus, the embedding EL ⊂ S(−π,π) does not exist and hencethere exists a solution of (4.1), which does not belong to the space S(−π,π).

Note that the development of the sufficiently general theory of solvability of (4.1)(including the proof of the embedding EL ⊂H, where H is a known space) is ratherdifficult. Indeed, let M be an arbitrary total linear subset of the space L2(−π,π). Letus consider the class M of operators of the form (4.1) which satisfy the conditionM ⊂ R(L ∗). Then it follows from Sect. 3.3 that it is reasonable to try to provethe embedding EL into ˜M for all L ∈M . However, it can be shown that in Mthere exist linear total subsets M+, M−, such that for some operator LM ∈M theembedding ELM ⊂ ˜M+ exists, but the embedding ELM ⊂ ˜M− does not exist. (Notethat, nevertheless, the space ELM induces on L2(−π,π) a stronger topology than thetopology in ˜M−). Thus, to prove the embedding of EL into ˜M it is necessary to useinherent properties of the set M.

Let us show how to select the sets M−,M− and the operator LM . The densityof the set M in the space L2(−π,π) implies the fact that in the space L2(−π,π)it is possible to select an orthonormal basis {tk}∞

k=1 from the vectors of M. Let usconsider the operator LM : L2(−π,π)→ L2(−π,π)

LMu =∞

∑k=1

λk(u, tk)0tk,∞

∑k=1

λ 2k <+∞,

which is an injective linear and continuous Hilbert–Schmidt operator. Extending theexample from Remark 3.10, let us consider the system of vectors

g1 =t1λ1

+t2λ2

,

g2 =t2λ2

+t3λ3

,


g3 =t3λ3

+t4λ4

,

. . . .

The operator LM maps this system to the following system

s1 = t1 + t2,

s2 = t2 + t3,

s3 = t3 + t4,

. . . .

The system of vectors {gi} is orthogonal to the vector

g∗ =∞

∑k=1

(−1)k+1λktk

and hence it is not total (and the linear span of the system G = l.h.{gi} is not densein L2(−π,π)), and the totality of the system {si} in the space L2 is clear. Let M− =l.s.{si} be a linear span of the system of vectors si. So, it is obvious that M− ⊂M, M− is dense in L2(−π,π) and L −1

M (M−) = G. Theorem 3.7 implies that theembedding ELM ⊂ ˜M− does not exist.

The set M+ can be selected in the following way. Let M+ = l.s.{ti}, then theset L −1

M (M+) = M+ is total and hence, by virtue of Theorem 3.7, there exists theembedding ELM ⊂ ˜M+.

4.2 Generalized Solutions of an Infinite System of LinearAlgebraic Equations

Let us consider several examples of generalized solutions of operator equations witha bounded linear operator L , which acts in the Hilbert space

�2 ={

x = (x1, . . . ,xk, . . .),∞

∑k=1

|xk|2 < ∞}

with the inner product

(x,y) =∞

∑k=1

xkyk, y = (y1, . . . ,yk, . . .).

4.2 Generalized Solutions of an Infinite System of Linear Algebraic Equations 37

As well-known [114], every such operator L is defined by an infinite matrix, whichwe will denote by the same letter L (and we will identify it with the operator L ):

L =

⎛

⎜

⎜

⎜

⎜

⎝

a11 a12 . . . a1n . . .a21 a22 . . . a2n . . .. . . . . . . . .

an1 an2 . . . ann . . .. . . . . . . . .

⎞

⎟

⎟

⎟

⎟

⎠

(4.6)

The element f = L u is represented as the product of the matrix L on the column-vector u from �2 : f = L u. If the matrix L satisfies the Hilbert–Schmidt condition

∞

∑i, j=1

|ai j|2 < ∞, (4.7)

then L is a compact operator, which acts in the space �2 [114]. The domain D(L )of a bounded linear operator L : �2 → �2 coincides with the space �2. In order toapply our theory of generalized solutions we have to clarify under which conditionson L its range R(L ) is everywhere dense in �2 and (2.1) is uniquely solvable (i.e thekernel Ker(L ) of the operator L consists of only zero element Ker(L ) = {θ}).For this purpose, let us put ai = (ai1,ai2, . . . ,ain, . . .); so, inequality (4.7) impliesai ∈ �2, i = 1,2, . . ., therefore, each row ai of the matrix L can be considered as theelement of �2. Then

f = L u, f = ( f1, . . . , fn, . . .),u = (u1, . . . ,un, . . .),

fi =∞

∑j=1

ai ju j = (ai,u),(i = 1,2, . . .).

It is easy to see, that Ker(L ) = {θ} iff the system of the elements

ℜ = (a1,a2, . . . ,an, . . .)

is total in �2 (or, saying in terms of Hilbert spaces theory, it is complete or closed):if (ai,u) = 0 for any i ∈N, then u = θ . To study whether R(L ) is dense in �2 let usintroduce the following definition.

Definition 4.10. We will say that the infinite matrix (4.6) which satisfies theHilbert–Schmidt condition (4.7) is a matrix with sparse rows if any row ai of thematrix L does not belong to closed linear subspace

Li = L(a1, . . . ,ai−1,ai+1, . . .),

induced by the other vectors (a1, . . . ,ai−1,ai+1, . . .) of the system L with respect tothe metric �2 : ai /∈ Li for all i ∈ N.

Let us denote by ek(k ∈N) the unit vectors in the space �2:

e1 = (1,0,0, . . .),e2 = (0,1,0, . . .), . . . ,ek = (0, . . . ,0,1,0, . . .), . . .


and let us put B= {ek}∞k=1. Let us show that the system of unit vectors B is contained

in the range R(L ) of the operator L iff matrix (4.6) has sparse rows. Indeed, ifai /∈ Li for all i ∈ N, then Li �= �2 and there exists an element ui �= θ , which isorthogonal to the closed subspace Li : ui⊥Li. It is easy to see that ci = (ai,ui) �=0, because otherwise fi = L (ui) = ((a1,u1), . . . ,(an,ui), . . .) = (0,0, . . .), but thiscontradicts the fact that the operator L is injective. Supposing that ui = ui/ci, wehave L (ui) = ei. Thus, ei ∈ R(L ) for all i ∈ N and B⊂ R(L ). On the other hand,if B ⊂ R(L ), then for any natural number i there exists such an element ui thatei = L ui. The latter inequality means that (ai,ui) = 1 and (ak,ui) = 0 for all k �=i, and hence (g,ui) = 0 for all g ∈ L(a1, . . . ,ai−1,ai+1, . . .) = Li, i.i. ui⊥Li. Thisimplies that ai /∈ Li, therefore L is a matrix with sparse rows. It is easy to see, thatevery injective bounded linear operator L in �2 which is defined by a matrix Lwith sparse rows has an everywhere dense range R(L ). Indeed, for such operatorwe have B ⊂ R(L ); since R(L ) is a linear manyfold, then the vector space L(B)induced by unit vectors is contained in R(L ) also, and since L(B) is everywheredense in �2, then the completion of R(L ) coincides with �2. The concept of a matrixwith sparse rows is an extension of classical concept of nonsingular matrix in caseof infinite matrices, since a finite matrix is non-singular iff it has sparse rows. Thereare many examples of matrices with sparse rows: diagonal matrices L = {ai j}∞

i, j=1,where ai j = 0 for all i �= j, aii �= 0, triangular matrices L = {ai j}∞

i, j=1, where ai j = 0for i > j (or i < j) with the non-zero diagonal elements, unitary matrices an so on.

The problem of solving an operator equation L u = f in �2 with a bounded lin-ear operator L is equivalent to the problem of solving an infinite system of linearalgebraic equations

∑ai ju j = bi, u = (u1, . . . ,un, . . .),b = (b1, . . . ,bn, . . .) ∈ �2 i = 1,2, . . . (4.8)

In order to find a generalized solution of (4.8) let us consider the space s of allnumerical sequences with the metric

ρ(x,y) =∞

∑n=1

12n

|xn− yn|1+ |xn− yn| ,

where x = (x1, . . . ,xn, . . .), y = (y1, . . . ,yn, . . .) ∈ s. As already known [114], sis a complete linear metric space, the conjugate space s∗ consists of functionalsϕ(x) = ∑n

k=1 ϕkxk, and the convergence in the metric of s is equivalent to coordi-natewise convergence. This implies that the convergence of the sequence {xn} ⊂ swith respect to the weak topology σ(s,s∗) is equivalent to the coordinatewise con-vergence also. Therefore, in s the weak convergence and the convergence in themetric ρ are equivalent. Remember [40] that a sequence {xn} is a weakly Cauchysequence if for any ϕ ∈ E∗ there exists a finite limit limϕ(xn). The space is weaklysequentially complete if every weakly Cauchy sequence weakly converges to an el-

ement of E . If {xn = (x(n)1 , . . . ,x(n)i , . . .)} is a weakly Cauchy sequence in s, then

selecting ϕ as ϕi(x) = xi, we have that the coordinate sequence ϕi(xn) = x(n)i is


convergent, and hence, the sequence {xn} is coordinatewise convergent. Whence itfollows that the sequence {xn} is weakly convergent to an element of s, and hence,s is a weakly sequentially complete space.

It is easy to see that the Hilbert space �2 is densely embedded into the spaces : �2 ⊂ s [40]. The analysis of Definitions 1 and 2 of a generalized solution showsthat it belongs to weak sequential closure ES of a vector space E in a Hausdorfflocally convex topological vector space E (i.e., in a completion of E with re-spect to the topology σ(E,R(L ∗))). If the range R(L ∗) of the adjoint operatorL ∗ : �2 → �2 contains the unit vectors ek, then s∗ = L(B) ⊂ R(L ); therefore, thetopology σ(�2,R(L )) is stronger than the topology σ(�2,L(B)) = σ(�2,s∗). Letu be an arbitrary element of weak sequential closure �s

2 and {un} be a sequence from�2, which converges to the element u with respect to the topology σ(E,R(L ∗)). Thisimplies that the sequence {un} is a Cauchy sequence in the topology σ(�2,R(L ∗)),and hence it is a Cauchy sequence in the topology σ(�2,s∗), and hence in the topol-ogy σ(s,s∗) also, i.e. the sequence {un} is a weakly Cauchy sequence in s. Sincethe space s is weakly sequentially complete, then {un} converges in s to some ele-ment u; in addition, the space �2 is embedded into a completion of s in the topologyσ(s,s∗), and the latter topology is Hausdorff, therefore u = u ∈ s. Thus, any gener-alized solution of system (4.8), for which the matrix L and the transposed matrixL ∗ have sparse rows, belongs to the space s. This fact is non-trivial for variousclasses of matrices with sparse rows (e.g., for the triangular matrixes). It shows thata generalized solution is an element of s, which is a limit of an near-solution {un}in the metric of this space.

Finally, consider an example of generalized solving of system (4.8). Let aninfinite matrix L = ||ai j|| is diagonal: ai j = 0 for all i �= j, aii = λi �= 0 and∑∞

i=1 |λi|2 < ∞; in this case the matrix L induces in �2 an injective Hermitianoperator L = L ∗ and the matrices L and L ∗ have sparse rows. Let us putann = λn =

1n , then

L u =(

u1,u2

2, . . . ,

un

n, . . .)

, u = (u1, . . . ,un, . . .) ∈ �2. (4.9)

It is easy to see that y = (b1,b2, . . . ,bn, . . .) = (1, 12 , . . . ,

1n , . . .) /∈ R(L ), therefore,

system (4.8) does not have a classical solution, but the element u=(1,1, . . . ,1, . . .)∈s is a generalized solution; this element is a limit of the near-solution

un = (1,1, . . . ,1,0,0, . . .) = L −1(yn)

in s, where yn = (1, 12 , . . . ,

1n ,0,0, . . .). Let us consider an infinite system of linear

algebraic equations

a11x1 +a12x2 + ...+a1nxn + ...= b1,a21x1 +a22x2 + ...+a2nxn + ...= b2,

.........................................an1x1 +an2x2 + ...+annxn + ...= bn,

(4.10)


A numerical sequence x = (x1,x2, . . . ,xn, . . .) is called a classical solution ofsystem (4.10), if after substitution of these values into the left-hand side of equali-ties (4.10) we obtain convergent (in the usual sense) numerical series and all theseequalities are satisfied. Denote by L the infinite matrix L = ‖ai j‖i, j=1,∞ generatedby the coefficients if system (4.10).

The infinite matrix L may be considered as an linear operator mapping somevector space E , consisting of sequences, into a similar space F containing the ele-ment b, i.e. L : E → F . We shall call the matrix operator L an operator definingsystem (4.10). Thus, applying the matrix operator L defining system (4.10) to theelement x we have

y = L (x) =

(

∞

∑i=1

a1kxk,∞

∑i=1

a2kxk, ...,∞

∑i=1

ankxk, ...

)

(4.11)

The solution of an operator equation (4.11) is an element x from E such thatL (x) = b ∈ F . Passing from the system of linear algebraic equations to an oper-ator equation allows to specify the concept of the solution of system (4.10); nowby the solution of the operator equation (4.11) we will means an element x fromthe vector space of sequences E for which L x = b ∈ E . Comparing this conceptwith the concept of the classical solution x of system (4.10), it would seem that thissequence does not belong to the space E , since the solution of the operator (4.11)corresponding to the element b=(b1,b2, . . . ,bn, . . .) does not exist whereas the clas-sical solution exists. However, the classical solution of system (4.10) is a partial caseof the solution of equation (4.11) if both E and F are the vector space S consistingof all numerical sequences.

Let us rewrite system (4.10) in the following way:

xi =∞

∑i=1

cikxk +bi (4.12)

where cik = δik− aik, δik is the Kroneker symbol.Let us introduce the following notations: C = ‖cik‖i,k=1,∞ is an infinite matrix,

which induces an operator C. If F ⊂ E then Cx = (I−L )x, where I is a unit oper-ator in E . Affine operator T which acts from E to F such that T (x) = (I−L )x+ballows to rewrite (4.11) in the following way:

x = Tx = (I−L )x+b (4.13)

Therefore, solving system (4.12) is reduced to searching of a fixed point T .It is known that [30] an infinite system like (4.12) is called regular, if

∞

∑k=1

|cik|< 1 i = 1,2, ...,n (4.14)

and completely regular if

∞

∑k=1

|cik|< 1−θ < 1 i = 1,2, ...,n (4.15)


Let us prove that for regular systems the operators L and T are continuous in theBanach space m consisting of all bounded sequences with the norm ‖x‖= sup

k∈[1,∞)

|xk|,(x1, ...,xk, ...) ∈ m. Indeed, if x ∈ m then for a regular system

‖L x‖ = ‖y‖= supi∈[1,∞)

|yi|= supi∈[1,∞)

∞

∑k=1

|aikxk| ≤ supi∈[1,∞)

∞

∑k=1

|aik| supk∈[1,∞)

|xk|

≤(

supi∈[1,∞)

∞

∑k=1

|aik|)

‖x‖ (4.16)

=

(

supi∈[1,∞)

∞

∑k=1

|cik|+ |cii−1|=)

‖x‖ (4.17)

≤(

supi∈[1,∞)

∞

∑k=1

|cik|+1

)

‖x‖ ≤ 2‖x‖ (4.18)

So, the operator L is a bounded linear operator which acts in m and its norm doesnot exceed 2. By formula (4.12)

‖Tx‖ = ‖y‖= ‖(I−L )x+b‖ ≤ ‖L x‖+‖x‖+‖b‖ (4.19)

≤ 2‖x‖+‖x‖+‖b‖≤ 3‖x‖+‖b‖ (4.20)

whence,

‖y‖= supk∈[1,∞)

|yi| ≤ 3‖x‖+‖b‖< ∞,

therefore, T x ∈ m.When the operator T is completely regular in the space m, it is a contraction

operator, as

ρ(

T x,T x′)

= ‖Tx−Tx′‖= ‖∞

∑k=1

cikxk−∞

∑k=1

cikx′k‖

= ‖∞

∑k=1

cik(

xk− x′k)‖ ≤

∞

∑k=1

|cik|‖x− x′‖ ≤ ‖x− x′‖∞

∑k=1

|cik|

≤ (1−θ ))ρ(

x,x′)

= qρ(

x,x′)

,

where q = 1−θ < 1. Therefore, a completely regular infinite system of linear alge-braic equations in the space m has a unique solution and we can find this solution bythe reduction method proposed by V.L. Kantorovich [30]. For solving finite systemsof linear algebraic equations which occur in this method we will use the combinedmethod described in Chap. 5.

Moreover, existence and uniqueness of a solution of a completely regular systemand Banach inverse operator theorem imply that the operator L always has abounded linear inverse operator L −1, which acts in the space m.


The solving of a regular systems is a more complex problem. In this case, theoperator T is not a contraction operator and the concept of generalized solutionmust be considered.

Consider an arbitrary regular infinite system of linear algebraic equations with abounded linear operator L , which acts in the space m. Denote by L1 a restrictionof the operator L on the space c0 of all sequences convergent to zero with thenorm ‖x‖ = sup

k∈[1,∞)

|xk|. Since c0 ⊂ m, then the operator L1 maps the space c0 into

m and it is a bounded linear operator. P.S.Bondarenko proved that the operator L1

is injective, as a regular system may have only one solution converging to zero [6].Denote by R(L ) and R(L1) the ranges of the operators L and L1, respectively,by F = R(L ) and F1 = R(L1) the closures of the ranges R(L ) and R(L1) in thespace m. Then R(L1) ⊂ R(L ), F1 ⊂ F , and F , F1 are closed Banach subspacesof m. Let us show that c0 ⊂ F . Indeed, let b = (b1,b2, ...,bn, ...) be an arbitraryelement of the space c0 and bn = (b1,b2, ...,bn,0, ...). When the right-hand side ofa regular system has only finite number of nonzero element, this regular systemhas a bounded solution [30], therefore there exists such an element xn ∈ m thatL (xn) = bn ∈ R(L ). Since bn→ b as n→∞ in the space m, then c0 ⊂ R(L ) = F .

Assume that the matrix L defining system (4.10) is symmetrical. In this casethe matrix C defining system (4.12) and all columns of the matrix C are ele-ments of the space l1, as ∑∞

i=1 |cik| < 1. Since l1 ⊂ c0, the columns of the ma-trix C are elements of the space c0. By virtue of condition (4.13), all columnsof the matrix L are elements of the space c0. Denote by e1 = (1,0,0, ...,), e2 =(0,1,0, ...,), ..., en = (0,0, , ...,1,0, ...) the orts of the space c0. Formula (4.11) im-plies that y1 = L1 (e1) = L (e1) = (a11,a21, ...,ak1, ...), y2 = L1 (e2) = L (e2) =(a12,a22, ...,ak2, ...), ..., yk = L1 (ek) = L (ek) = (a1k,a2k, ...,akk, ...), ... belong tothe space c0. Hence, ∀u∈ c0: u = ∑∞

i=1 αiek and L1 (u) =L (u) = ∑∞i=1 αiL1 (ei) =

limn→∞ ∑ni=1 αiL1 (ei)∈ c0. Thus, the operator L1 maps elements u∈ c0 to elements

L1 (u) ∈ c0, i.e. the operator L1 acts into c0, so that F1 = R(L1)⊂ c0.Thus, we conclude the following:

(1)If b ∈ R(L1) then there exist a unique classical solution x ∈ c0.(2)If b ∈ R(L1), but b �∈ R(L1) then a classical solution does not exist, but the

operator equation L1 (x) = b, where x ∈ c0, has a unique generalized solution xin some space E , which is a completion of c0 with respect to the norm ‖x‖∗ =‖L1 (x)‖c0 .

(3)If b �∈ R(L1) then there are no either classical either generalized solution of theequation L1 (x) = b.

The most interesting case is that in which R(L1) is everywhere dense in c0 (withrespect to the norm or in a weak topology). Then R(L1) = c0, as R(L1) ⊂ c0. Inthis case, (4.10) always has a generalized solution if L is a symmetrical matrix, andsystem (4.12) is regular. If the operator L1 is injective, then B = (e1,e2, ...,en, ...)⊂R(L1) iff the matrix L1 has sparse rows in �2. Since all rows of the matrix L of a


regular system belong to l1 and l1⊂ �2, then all rows of the matrix L are elements of�2. From the other hand, the condition B⊂ R(L1) implies that R(L1) = c0, hence,R(L1) = c0 if the matrix L1 has sparse rows.

Let us consider a generalized solution x of system (4.10) when b = (b1,b2, ...,bn, ...) ∈ c0 and a classic solution in c0 or m does not exist. In other words, let us tryto give a constructive description of a space containing the generalized solution x ofa system L1 (x) = b, where x and b belong to c0. In addition, this description mustnot depend on the kind of the operator L .

The space c0 is densely embedded into s. Definitions 1 and 2 imply that the gen-eralized solutions belong to weak sequential closure Es of the vector space E in aHausdorff locally convex topological vector space E , which is a completion of c0

in topology σ (E,R(L ∗1 )). Let us show that the range R(L ∗

1 ) of the adjoint oper-ator L ∗

1 : l1→ l1 contains the orts ek = (0, ...,0,1,0, ...), k = 1,2, .... By definitionof the adjoint operator f = L ∗

1 (g), where f ,g ∈ c∗0 = l1, f (x) = g(L1 (x)) ,x =(x1,x2, ...,xn, ...) ∈ c0. Put y = L1 (x). Then formula (4.11) implies that

y =

(

∞

∑k=1

a1kxk,∞

∑k=1

a2kxk, ...,∞

∑k=1

aikxk, ...

)

.

Thus,

f (x) =∞

∑i=1

giyi =∞

∑i=1

gi

(

∞

∑k=1

aikxk

)

=∞

∑i=1

(

∞

∑ki=1

aikgixk

)

=∞

∑i=1

∞

∑k=1

aikgixk =∞

∑k=1

(

∞

∑i=1

aikgixk

)

=∞

∑k=1

(

∞

∑i=1

akigi

)

xk

since the matrix L is symmetrical. Swapping the indexes i and k, we obtain

f (x) =∞

∑k=1

fixi =∞

∑i=1

(

∞

∑k=1

aikgk

)

xi,

therefore,

f (x) =∞

∑i=1

aikgk

This implies that the adjoint operator L ∗1 : l1 → l1 is a matrix operator, and its

inducing matrix L ∗1 coincides with the matrix L . Therefore, the matrix L has

sparse rows in the space �2 and hence R(L ∗1 ) contains the set of all orts in �2.

Therefore, s∗ = L (B) ⊂ R(L1) and hence the topology σ (c0,R(L1)) is strongerthan σ (c0,L (B)) = σ (c0,s∗). Let u is an arbitrary element of weak sequential clo-sure cs

0 and {un}∞n=1 is a sequence from c0, which converges to u in the topology

σ (c0,R(L ∗1 )). Therefore, the sequence {un}∞

n=1 is a Cauchy sequence in the topol-ogy σ (c0,R(L ∗

1 )) and hence it is a Cauchy sequence in the topology σ (c0,s∗)and, therefore, in the topology σ (s,s∗). Hence, the sequence {un}∞

n=1 is a Cauchy


sequence in the space s. Since the space s is sequentially complete, then the sequence{un}∞

n=1 converges to s to some element u. In addition, the space c0 is embedded tothe completion of the space s in the topology σ (s,s∗) and this topology is Hausdorff.Therefore, u = u ∈ s. Thus, any generalized solution of system (4.10) belongs to s,if inducing matrix L is symmetrical and has sparse rows, and b ∈ c0. Hence, whilethe operator L1 acts in the space c0 and the generalized solution u does not belongto c0, the generalized solution is just an element of more wide space s and coincideswith a classical solution of system (4.10).

Thus, a classical solution of the system L x = b and a classical solution of theoperator equation L (x) = b, where the operator L : E→ F does not coincide eachother. Indeed, if x is the classical solution of the operator equation, then x ∈ E andx is the classical solution of the system L x = b. However, if x is the generalizedsolution of the system L x = b, then x �∈ E and x is not a classical solution of theoperator equation L (x) = b, but x may be a classical solution of the system L x= b.

The investigation of a infinite system of linear algebraic equations confirms thatthe main notion of functional analysis is an operator, but not a space [43].

4.3 Application to Volterra Integral Equation of the First Kind

Consider a Volterra integral operator of the first kind

L u =∫ t

0K(t,s)u(s),ds, L : L2(0,1)→ L2(0,1), (4.21)

where L2(0,1) is a space of measurable, square-integrable, real functions.Let us denote by D the bounded region {(t,s) |0≤ s≤ t ≤ 1}.

Lemma 4.2. Let a kernel K(t,s) be bounded in D and have a square-integrable inD partial derivative (in the Sobolev sense) ∂K/∂ t with respect to the variable t, thefunction ks(t) = K(t,s) be absolutely continuous with respect to t, s≤ t ≤ 1, and thefunction K(t, t) be a square-integrable on [0,1], then the function

f = L u =∫ t

0K(t,s)u(s)ds

is absolutely continuous on [0,1] and the following formulae is true:

d fdt

= K(t, t)u(t)+∫ t

0

∂K(t,s)∂ t

u(s)ds.

Proof. At first, let us prove that the function f is absolutely continuous. Let us selecton [0,1] the points

0≤ a1 < b1 ≤ a2 < b2 ≤ . . .≤ an < bn ≤ 1.

4.3 Application to Volterra Integral Equation of the First Kind 45

Then

n

∑i=1| f (bi)− f (ai)| =

n

∑i=1

∣

∣

∣

∣

∫ bi

0K(bi,s)u(s)ds−

∫ ai

0K(ai,s)u(s)ds

∣

∣

∣

∣

≤n

∑i=1

∣

∣

∣

∣

∫ bi

ai

K(bi,s)u(s)ds

∣

∣

∣

∣

+n

∑i=1

∣

∣

∣

∣

∫ ai

0

(

K(bi,s)−K(ai,s))

u(s)ds

∣

∣

∣

∣

≤ max(t,s)∈D

|K(t,s)|×n

∑i=1

∣

∣

∣

∣

∫ bi

ai

u(s)ds

∣

∣

∣

∣

+n

∑i=1

∣

∣

∣

∣

∫ ai

0

(∫ bi

ai

∂K(t,s)∂ t

dt)

u(s)ds

∣

∣

∣

∣

≤ Cn

∑i=1

∫ bi

ai

|u(s)|ds+n

∑i=1

∣

∣

∣

∣

∫ bi

ai

(∫ ai

0

∂K(t,s)∂ t

u(s)ds)

dt

∣

∣

∣

∣

≤n

∑i=1

∫ bi

ai

(

C|u(t)|+∫ 1

0

∣

∣

∣

∣

∂K(t,s)∂ t

u(s)

∣

∣

∣

∣

ds

)

dt.

By the conditions of the lemma, we have that the function

z(t) =C|u(t)|+∫ 1

0

∣

∣

∣

∣

∂K(t,s)∂ t

u(s)

∣

∣

∣

∣

ds

is integrable on [0,1], and the absolute continuity of the Lebesgue integral impliesthe absolute continuity of the function f .

Thus, the function f (t) has an integrable derivative almost everywhere on [0,1].On the other hand

∫ t

0

(

K(τ,τ)u(τ)+∫ τ

0

∂K(τ,s)∂ t

u(s)ds

)

dτ

=∫ t

0K(τ,τ)u(τ)dτ +

∫ t

0

∫ t

s

∂K(τ,s)∂ t

dτu(s)ds

=∫ t

0K(τ,τ)u(τ)dτ +

∫ t

0(K(t,s)−K(s,s))u(s)ds = f (t).

Hence, the lemma is proved. �Remark 4.14. Usually, the formula from Lemma 4.2 is cited under the strongerrestrictions on K(t,s).

Remark 4.15. Lemma 4.2 directly implies that the function f ′(t) is square inte-grable.

Lemma 4.3. Let the conditions of Lemma 4.2 be satisfied and K(t, t)≥ ε > 0, thenthe range of operator (4.21) coincides with the set W 1

2,0(0,1) of absolutely continu-ous functions that are equal to zero at the point t = 0 and have a square-integrablederivative.


Proof. The fact that f ′(t) is square-integrable was probed in Remark 4.15. It isalso easy to see that f (0) = 0. Let us prove that for any function f ∈W 1

2,0 thereexists u∈ L2(0,1) such that L u = f . To do this, let us consider the Volterra integralequation of the second type

K(t, t)u(t)+∫ t

0

∂K(t,s)∂ t

u(s)ds = f ′(t), (4.22)

or

u(t)+∫ t

0

1K(t, t)

∂K(t,s)∂ t

u(s)ds =f ′(t)

K(t, t).

As well-known, this equation has a solution in L2(0,1) for any right-hand side. Thus,for any function f there exists u∈ L2(0,1) , which is a solution of (4.22). Integratingthe equality (4.22) from 0 to t and taking into account Lemma 4.2, we have that anyfunction f ∈W 1

2,0 belongs to the range of the operator (4.21). �Remark 4.16. In particular, the lemma implies that the set R(L ) is dense in L2(0,1).

Remark 4.17. Similarly, we can prove the following statement. Let us suppose thatthe following conditions are satisfied:

(a)A kernel K(t,s) is bounded and absolutely continuous with respect to s for anyfixed value t.

(b)∂K/∂ s is square-integrable in D.(c)K(t, t)≥ ε > 0.(d)The function K(t, t) is square-integrable in [0,1].

Then R(L ∗) = W 12,1(0,1) and since R(L ∗) is dense in L2(0,1); the operator L

is injective, where W 12,1(0,1) is a set of absolutely continuous functions on [0,1],

which are equal to zero at the point t = 1 and have square-integrable derivative on(0,1).

Lemma 4.4. Let the conditions of Remark 4.17 be satisfied. Then the spaceREL

(L ∗) (in terms of Lemma 3.3) is isomorphic to the space W 12,1 (with respect to

the structures of topological vector spaces) in the norm

‖ f‖W 12,1

=

(

∫ 1

0

(

∂ f∂ t

)2

dt

)1/2

.

Proof. It is sufficiently to prove that the norm ‖ · ‖W12,1

is equivalent to the norm

‖ · ‖REL(L ∗). Indeed, the result of Lemma 4.2, the boundedness of the kernel K(t,s)

and the square-integrability of ∂K/∂ s imply that

‖ f‖2W 1

2,1=∫ 1

0

(

∂L ∗ϕ∂ s

)2

ds =∫ 1

0

(

K(s,s)ϕ(s)+∫ s

0

∂K(t,s)∂ s

ϕ(t)dt

)2

ds

≤ 2∫ 1

0(K(s,s)ϕ(s))2 ds+2

∫ 1

0

(

∫ s

0

∂K∂ s

ϕ(t)dt

)2

ds

4.4 Application to the Statistics of Random Processes 47

≤ 2∫ 1

0K2(s,s)ds

∫ 1

0ϕ2ds+2

∫ 1

0

(∫ s

0

(

∂K∂ s

)2

dt)

ds×∫ 1

0

(∫ s

0ϕ2(t)dt

)

ds

≤ C‖ϕ‖2L2(0,1)

=C‖ f‖2REL

(L ∗).

Thus, we have proved that the norm ‖ · ‖REL(L ∗) is stronger than the norm ‖ · ‖W1

2,1.

In addition, the spaces REL(L ∗), W 1

2,1 are complete and hence by virtue of theBanach theorem on the inverse operator, these norms are equivalent. �Lemma 4.5. Let the following conditions be satisfied:

(a)K(t,s) is a measurable and bounded function in D.(b)The function K(t,s) is absolutely continuous with respect to each variable t,s

and have square-integrable partial derivatives ∂K/∂ t and ∂K/∂ s.(c)The function K(t, t)≥ ε > 0 is square-integrable on [0,1].

Then the space EL is densely embedded into the space H, which is homeomorphicto the negative space W−1

2,1 , where the space W−12,1 is constructed by the pair W 1

2,1 ⊂L2(0,1).

Proof. Lemmas 3.3 and 4.4 imply that the spaces W 12,1 and (EL )∗ are linear and

topologically isomorphic. Therefore, the spaces (W 12,1)∗ =W−1

2,1 and (EL )∗∗ are lin-ear and topologically isomorphic also. Taking into account the fact that the embed-ding EL ⊂ E∗∗L is dense, we prove the lemma. �Theorem 4.3. If the conditions of the previous lemma are satisfied there exists aunique generalized solution of the Volterra equation in the sense of Definitions 3.6and 3.7 (H is homeomorphic to W−1

2,1 ).

Remark 4.18. The conditions such that the kernel is absolute continuous with re-spect to t and its partial derivative ∂K/∂ t is square-integrable are necessary only toprove that the range R(L ) is densely embedded into L2(0,1). If it is known a priori,then these conditions may be discarded.

4.4 Application to the Statistics of Random Processes

Let x(t) be a random process with continuous time, whose trajectory is observedon the segment [a,b], r(t,s) be a correlation function of this process. By Mercertheorem a correlation function can be represented as a uniformly convergent series

r(t,s) =∞

∑k=1

ϕk(s)ϕk(t)λk

, (4.23)


if the eigenvalues λk and the eigenvectors ϕk of the integral operator

Aϕ =

∫ b

ar(t,s)ϕ(s)ds

are known.If λk > 0, then the function r(t,s) defined by the formulae (4.23) is positive-

definite, and by Khinchin’s theorem r(t,s) can be considered as a correlation func-tion of some random (may be Gaussian) ) process x(t).

In the theory of testing of statistical hypotheses on random processes with con-tinuous time and in the theory of estimation of unknown coefficients in a linearregression scheme, there is an important problem of solving Fredholm equation ofthe first type

∫ b

ar(t,s)ϕ(s)ds = a(t)

for the right-hand sides a(t) ∈ L2(a,b) [23, 96]. In particular, if the process x(t) hasa constant, but unknown mathematical expectation Mx(t) = m, then the best linearestimation m∗ of this mathematical expectation is defined by the following formula:

m∗ =∫ b

af (t)x(t),dt, f ∈ L2(a,b)

and it exists if the function f ∈ L2(a,b) is a solution of the integral equation

∫ b

ar(s, t) f (t)dt = 1. (4.24)

Since (4.24) rarely has a classic solution, it is naturally to study a generalized so-lution of this equation and to search for the best linear estimation of the mathe-matical expectation as a functional from the space of generalized functions (seeTheorems 4.1, 4.2).

4.5 Application to Parabolic Differential Equationin a Connected Region

A priori estimates and generalized solutions are the effective methods of qualitativeanalysis of partial differential equations (e.g. see [5, 62, 63]). As an example let usconsider the application of the method of a priori estimates to the issue of existenceand uniqueness of generalized solutions of parabolic equations.

4.5 Application to Parabolic Differential Equation in a Connected Region 49

4.5.1 Problem Definition

Let Ω ⊂ Rn be a bounded region with a regular border ∂Ω . Let us consider in the

cylinder Q = (0,T )×Ω the following parabolic equation

ut +A u = f (t,ξ ), (4.25)

where A is an elliptic differential expression

A u =−n

∑i, j=1

(

ai juξ j

)

ξi+

n

∑i=1

(aiu)ξi+au.

Let us study the following boundary value problem: we need to find the functionu(t,ξ ), which satisfies (4.25) in Q and the following conditions

u|t=0 = 0, u|ξ∈∂Ω = 0. (4.26)

Let D⊂C1(Q) be a linear manyfold of functions, which satisfy conditions (4.26),D+ ⊂C1

(

Q)

be a linear manyfold of functions, which satisfy the conditions

v|t=T = 0, v|ξ∈∂Ω = 0.

To study boundary value problem (4.25), (4.26) let us consider the followingHilbert spaces. Let W be a Hilbert space, which is a completion of D with respectto the norm

‖u‖2W =

∫

Q

(

u2t +

n

∑i=1

u2ξi

)

dξ dt, (4.27)

W+ be a Hilbert space, which is a completion of D+ with respect to the norm (4.27),and H be a Hilbert space, which is a completion of D with respect to the norm

‖u‖2H =

∫

Q

(

u2 +n

∑i=1

u2ξi

)

dξ dt. (4.28)

Note that the space H is a completion of D+ with respect to the norm (4.28). Inaddition, there are dense and compact embeddings

W ⊂ H ⊂ L2(Q), W+ ⊂ H ⊂ L2(Q).

Passing to the conjugate spaces, we have the following chains of dense and compactembeddings

L2(Q)⊂ H− ⊂W−, L2(Q)⊂ H− ⊂W−+ .

Denote by 〈·, ·〉W,W− an extension of the inner product (·, ·)L2(Q) by continuityon W ×W−. The next denotations have similar sense 〈·, ·〉W−+ ,W+

, 〈·, ·〉H− ,H and

〈·, ·〉H,H− .


Let us suppose that the following conditions are satisfied:

(1) Functions ai j = a ji belong to the space C(

Ω)

and satisfy the uniform ellipticitycondition

n

∑i, j=1

ai j (ξ )λiλ j ≥ αn

∑i=1

λ 2i , λi ∈ R, ξ ∈Ω ,

where α is a positive constant.(2) Functions a ∈C(Ω), ai ∈C1(Ω) satisfy the inequalities

n

∑i=1

∂ai

∂ξi(ξ )+ a(ξ )≥ 0; a(ξ )≥ 0

for all ξ ∈Ω .

4.5.2 Properties of Operators Associated with a Boundary ValueProblem

Let the function u ∈ C2(

Q)

satisfy (4.25), (4.26) with a smooth right-hand sidef , then the Gauss–Ostrogradsky formula implies, that for any function v ∈ D+ thefunction u satisfies the identity

l(u,v) =∫

Q

(

utv+n

∑i, j=1

ai juξ jvξi−

n

∑i=1

aiuvξi+auv

)

dξ dt = ( f ,v)L2(Q) (4.29)

or the identity

l(u,v) =∫

Q

(

−uvt +n

∑i, j=1

ai juξ jvξi−

n

∑i=1

aiuvξi+auv

)

dξ dt = ( f ,v)L2(Q) . (4.30)

Also, the integral identities (4.29), (4.30) are meaningful for the pair of functions(u,v) ∈W ×H, (u,v) ∈ H×W+, respectively (of course, all of the derivatives areregarded as generalized).

For the bilinear forms

W ×H � (u,v) �→ l(u,v), H×W+ � (u,v) �→ l(u,v)

using the integral Cauchy–Bunyakovsky inequality it is easy to get the estimate

|l(u,v)| ≤ c‖u‖W‖v‖H,∣

∣l(u,v)∣

∣≤ c‖u‖H ‖v‖W+.

Thus, there exists such linear continuous operators

L : W → H−, L ∗ : H→W−, L : H →W−+ , L ∗ : W+→ H−,


that

l(u,v) = 〈L u,v〉H− ,H = 〈u,L ∗v〉W,W− ∀(u,v) ∈W ×H,

l(u,v) = 〈L u,v〉W−+ ,W+=⟨

u,L ∗v⟩

H,H− ∀(u,v) ∈ H×W+.

Obviously,L ⊂ L , L ∗ ⊃ L ∗,

i.e. the operator L is a completion of the operator L by continuity and the operatorL ∗ is a contraction of the operator L ∗.

Let us obtain a priori estimates in negative norms for these operators.

Lemma 4.6. The following inequalities hold:

‖u‖L2(Q) ≤ c1∥

∥L u∥

∥

W−+≤ c2 ‖u‖H , u ∈ H, (4.31)

‖u‖H ≤ c1 ‖L u‖H− ≤ c2 ‖u‖W , u ∈W, (4.32)

‖v‖L2(Q) ≤ c1 ‖L ∗v‖W− ≤ c2 ‖v‖H , v ∈ H, (4.33)

‖v‖H ≤ c1∥

∥L ∗v∥

∥

H− ≤ c2 ‖v‖W+, v ∈W+. (4.34)

Proof. It is sufficient to prove only left-hand sides of the double inequalities (4.31)–(4.34). Let us prove inequality (4.31). Let us put for an element u ∈ D

v(t,ξ ) =∫ T

te−Nτ u(τ,ξ )dτ,

where N is a sufficiently large positive number. Its exact value we will determinelater. It is clear that v ∈W+ and u = −eNtvt . Let us consider the value of the func-tional L u on the element v

⟨

L u,v⟩

W−+ ,W+=∫

QeNt(

v2t −

n

∑i, j=1

ai jvξ jvξit +

n

∑i=1

aivt vξi−avvt

)

dξ dt.

Using the formula of integration by parts and taking into account the boundaryconditions and the coefficient conditions, we have

−∫

QeNt

n

∑i, j=1

ai jvξ jvξit dξ dt = −1

2

∫

Ω

n

∑i, j=1

eNt ai jvξ jvξi

∣

∣

∣

t=T

t=0dξ

+N2

∫

QeNt

n

∑i, j=1

ai jvξ jvξi

dξ dt

≥ αN2

∫

QeNt

n

∑i=1

v2ξi

dξ dt,

−∫

QeNt avvt dξ dt = −1

2

∫

Ωk

eNtav2∣

∣

t=Tt=0 dξ +

N2

∫

QeNt av2 dξ dt ≥ 0,

∫

QeNt

n

∑i=1

aivtvξidξ dt ≥ −c0

∫

QeNt

n

∑i=1|vt |×

∣

∣vξi

∣

∣dξ dt,

where c0 = maxi=1,n

supξ∈Ω|ai (ξ )|<+∞.


Gathering all these estimates together gives the following inequality:

⟨

L u,v⟩

W−+ ,W+≥∫

QeNt(

v2t +

αN

2

n

∑i=1

v2ξi− c0

n

∑i=1|vt |×

∣

∣vξi

∣

∣

)

dξ dt

=

∫

QeNt(1

2v2

t +αN4

n

∑i=1

v2ξi

)

dξ dt

+∫

QeNt

n

∑i=1

( 12n

v2t +

αN4

v2ξi− c0 |vt |×

∣

∣vξi

∣

∣

)

dξ dt.

Using the elementary inequalities c0 |vt |×∣

∣vξi

∣

∣≤ 12n v2

t +n2 c2

0v2ξi

, we have

⟨

L u,v⟩

W−+ ,W+≥∫

QeNt(1

2v2

t +αN4

n

∑i=1

v2ξi

)

dξ dt+∫

QeNt

n

∑i=1

(αN4− n

2c2

0

)

v2ξi

dξ dt;

hence, for a given N > 2nα c2

0, we have

⟨

L u,v⟩

W−+ ,W+≥ c‖v‖2

W+.

Applying the Schwarz inequality to the expression⟨

L u,v⟩

W−+ ,W+and taking into

account the estimates ‖u‖L2(Q) ≤ eNT ‖v‖W+, we get the inequality (4.31) for smooth

functions u∈D. For reasons of the density, the inequalities (4.31) hold on the entirespace H.

To prove (4.32), let us consider the value of the functional L u on an elemente−Ntu, where u ∈W and N is sufficiently large positive number. We have

⟨

L u,e−Ntu⟩

H−,H =

∫

Qe−Nt

(

uut +n

∑i, j=1

ai juξ juξi−

n

∑i=1

aiuuξi+au2

)

dξ dt

≥ N2

∫

Qe−Nt u2dξ dt+

∫

Qe−Ntα

n

∑i=1

u2ξi

dξ dt

−∫

Qe−Nt

n

∑i=1

aiuξiudξ dt.

Further,

⟨

L u,e−Ntu⟩

H−,H ≥∫

Qe−Nt

(N4

u2 +α2

n

∑i=1

u2ξi

)

dξ dt

+

∫

Qe−Nt

n

∑i=1

( N4n

u2 +α2

u2ξi− c0 |u|

∣

∣uξi

∣

∣

)

dξ dt.


Using the inequality c0 |u| ·∣

∣uξi

∣

∣≤ 12α c2

0u2 + α2 u2

ξi, we have

⟨

L u,e−Ntu⟩

H−,H ≥∫

Qe−Nt

(N4

u2 +α2

n

∑i=1

u2ξi

)

dξ dt

+

∫

Qe−Nt

n

∑i=1

( N4n− 1

2αc2

0

)

u2 dξ dt;

hence, for a given N > 2nα c2

0, we obtain

⟨

L u,e−Nt u⟩

H− ,H ≥ c‖u‖2H .

Applying the Schwarz inequality to⟨

L u,e−Ntu⟩

H− ,H , we get the esti-mate (4.32).

Estimates (4.33) and (4.34) can be proved similarly. In the first case, we have toestimate below the form

⟨

∫ t

0eMτ v(τ,ξ )dτ,L ∗v

⟩

W,W−,

where v∈D+, M > 0. It is clear that u(t,ξ ) =∫ t

0 eMτ v(τ,ξ )dτ ∈W and v= e−Mtut .Let us consider the value of the functional L ∗v on an element u

〈u,L ∗v〉W,W− =∫

Qe−Mt

(

u2t +

n

∑i, j=1

ai juξ juξit −

n

∑i=1

aiuutξi+auut

)

dξ dt.

Using the formula of integration by parts and taking into account the boundary con-ditions and the coefficient conditions, we have

∫

Qe−Mt

n

∑i, j=1

ai juξ juξit dξ dt =

12

∫

Ω

n

∑i, j=1

e−Mtai juξ juξi

∣

∣

∣

t=T

t=0dξ

+M2

∫

Qe−Mt

n

∑i, j=1

ai juξ juξi

dξ dt

≥ αM2

∫

Qe−Mt

n

∑i=1

u2ξi

dξ dt,

∫

Qe−Mtauut dξ dt−

∫

Qe−Mt

n

∑i=1

aiuutξidξ dt =

∫

Qe−Mt

( n

∑i=1

∂ai

∂ξi+a)

uut dξ dt

+∫

Qe−Mt

n

∑i=1

aiuξiut dξ dt

≥−c0

∫

Qe−Mt

n

∑i=1

|ut |∣

∣uξi

∣

∣dξ dt.


Gathering all these estimates together gives the following inequality:

〈u,L ∗v〉W,W− ≥∫

Qe−Mt

(

u2t +

αM

2

n

∑i=1

u2ξi−

n

∑i=1

c0 |ut |∣

∣uξi

∣

∣

)

dξ dt

=

∫

Qe−Mt

(12

u2t +

αM4

n

∑i=1

u2ξi

)

dξ dt

+∫

Qe−Mt

n

∑i=1

( 12n

u2t +

αM4

u2ξi− c0 |ut |

∣

∣uξi

∣

∣

)

dξ dt.

Using the inequality c0 |ut | ·∣

∣uξi

∣

∣≤ 12n u2

t +n2 c2

0u2ξi

, we have

〈u,L ∗v〉W,W− ≥∫

Qe−Mt

(12

u2t +

αM4

n

∑i=1

u2ξi

)

dξ dt

+∫

Qe−Mt

n

∑i=1

(αM4− n

2c2

0

)

u2ξi

dξ dt,

hence, for a given M > 2nα c2

0 we have 〈u,L ∗v〉W,W− ≥ c‖u‖2W . Applying the

Schwarz inequality to 〈u,L ∗v〉W,W− and taking into account ‖v‖L2(Q) ≤ c‖u‖W ,we obtain the inequality (4.33) for smooth functions v ∈ D+. For reasons of thedensity, the inequalities (4.33) hold on the entire space H.

To prove (4.34) let us consider the value of the functional L ∗v ∈ H− on theelement eMt v, where v ∈W+ and M is a sufficiently large positive number. �

4.5.3 Generalized Solvability of the Boundary Value Problem

Using the inequalities proved above we will prove that the operator equations L u=f , L ∗v= g and L u = f , L ∗v = g are correctly and densely solvable. It is naturallyto call the solutions of these equations the generalized solutions of the boundaryvalue problem (4.25), (4.26) and of the adjoint problem.

Theorem 4.4. For any function f ∈ L2(Q) there exists a unique solution u ∈W ofthe operator equation L u = f , and the following estimate holds

‖u‖W ≤ c‖ f‖L2(Q) .

Proof. Let us consider f ∈ L2(Q). By virtue of inequalities (4.33) the followingestimate holds

( f ,v)L2(Q) ≤ ‖ f‖L2(Q) ‖v‖L2(Q) ≤ c‖L ∗v‖W− ,v ∈ H.

Therefore, the expression l(w) = ( f ,v)L2(Q) (w = L ∗v) specifies a linear boundedfunctional on R(L ∗) ⊂W−, moreover ‖l‖ ≤ c‖ f‖L2(Q). Let us extend the func-tional l linearly and with preservation of norm to a linear continuous functional


l ∈ (W−)∗. By the Riesz representation theorem for a linear continuous functionalin W− there exists such an element u ∈W that l (w) = 〈u,w〉W,W− and ‖u‖W =

∥

∥l∥

∥.Since for an arbitrary element v ∈ H

( f ,v)L2(Q) = l (L ∗v) = 〈u,L ∗v〉W,W− = 〈L u,v〉H− ,H ,

then L u = f . The uniqueness of the solution follows from the inequality (4.32) andthe embedding W ⊂ H. �Corollary 4.1. Problem (4.25), (4.26) for any right-hand side f ∈ L2(Q) has aunique weak solution u ∈W , i.e. an element u ∈W for an arbitrary function v ∈ Hthat satisfies the integral identity

∫

Q

(

utv+n

∑i, j=1

ai juξ jvξi−

n

∑i=1

aiuvξi+auv

)

dξ dt = ( f ,v)L2(Q) .

Similar to Theorem 4.4 we can prove the following theorems.

Theorem 4.5. For any element f ∈ H− there exists a unique solution u ∈ H of theoperator equation L u = f , and the following estimate holds

‖u‖H ≤ c‖ f‖H− .

Corollary 4.2. For any right-hand side f ∈H− the boundary value problems (4.25)and (4.26) has a unique weak solution u ∈ H, i.e. an element that for any functionv ∈W+ satisfies the integral identity

∫

Q

(

−uvt +n

∑i, j=1

ai juξ jvξi−

n

∑i=1

aiuvξi+auv

)

dξ dt = 〈 f ,v〉H− ,H .

Theorem 4.6. For an arbitrary function g ∈ L2(Q) there exists a unique solutionv ∈W+ of the operator equation L ∗v = g, and the following estimate holds

‖v‖W+≤ c‖g‖L2(Q) .

Theorem 4.7. For any element g ∈ H− there exists a unique solution v ∈ H of theoperator equation L ∗v = g, and the following estimate holds

‖v‖H ≤ c‖g‖H− .

To prove the solvability of problems (4.25) and (4.26) for the right-hand sideswhich are more singular, let us extend the class of generalized solutions.

Definition 4.11. A function u ∈ L2(Q) is called a generalized solution of the oper-ator equation L u = f ( f ∈W−+ ) if it satisfies the identity

(u,g)L2(Q) = 〈 f ,v〉W−+ ,W+∀g ∈ L2(Q),

where v ∈W+ is a solution of the operator equation L ∗v = g.


Remark 4.19. The generalized solutions in L2(Q) are nothing else than ultra-weaksolutions in the method of transposition of isomorphism defined by J.-L.Lions.

The naturalness of this definition is based on the following properties of a gener-alized solution.

1. A classic solution of the equation L u = f is a generalized solution. Indeed,since H ⊂ L2(Q), then such an element u ∈ H, that L u = f ( f ∈W−+ ), satisfiesthe identity

〈 f ,v〉W−+ ,W+=⟨

L u,v⟩

W−+ ,W+=⟨

u,L ∗v⟩

H,H− =(

u,L ∗v)

L2(Q)

for all v ∈W+ : L ∗v ∈ L2(Q), i.e. u is a generalized solution.2. If a generalized solution u ∈ L2(Q) of the equation L u = f belongs to the space

H, than it is a classic solution. Indeed, let u ∈ H be a generalized solution. Thenthe following equality holds

〈 f ,v〉W−+ ,W+=(

u,L ∗v)

L2(Q)=⟨

u,L ∗v⟩

H,H− =⟨

L u,v⟩

W−+ ,W+

for all v ∈W+ : L ∗v ∈ L2(Q).The density of the set

{

v ∈W+ : L ∗v ∈ L2(Q)}

in the space W+ implies thatL u = f .

3. If u∈ L2(Q) is a generalized solution of the equation L u= f with the right-handside f ∈ R

(

L)

, than u ∈ H is a classic solution also.

Using the inequality (4.31) we can prove the following theorem

Theorem 4.8. For an arbitrary element f ∈W−+ there exists a unique generalizedsolution u ∈ L2(Q) of the operator equation L u = f , and the following estimateholds

‖u‖L2(Q) ≤ c‖ f‖W−+ .

Remark 4.20. Similarly, we can give the definition of a generalized solution for theadjoint problems in L2(Q) and prove its existence.

4.6 Application to Parabolic Differential Equationin a Disconnected Region

Let us consider once more example of application of the method of a priori estimatesto a boundary value problem for parabolic equation in a disconnected region witha contact condition. Such boundary value problems arises in the theory of heat andmass transport in heterogeneous media Ω1,Ω2, when media contact each other viaa thin three-layered inclusion γ = γ1∪ γ2∪ γ3 (see, e.g. [67]).

4.6 Application to Parabolic Differential Equation in a Disconnected Region 57

In such cases, a zone of foreign layer γ is excluded from a region, where theprocess is carried, and the influence of an inclusion is described by conditions ofconjugation Ω1,Ω2. In order to define the problem correctly, it is necessary to posethe contact conditions in every region Ω1 and Ω2 in addition to initial and classicalboundary conditions. In the case of three-layered inclusion, we obtain a problemwith heterogeneous non-ideal contact conditions [11,55–57,81] (γ1,γ3 – poorly per-meable inclusions where the coefficient of filtration is much less than in other mediaΩ1,Ω2, γ2 is a highly permeable inclusion, for example, thin tectonic break, thinlayer of highly permeable media, formed as a result of dissolution of thin salt layersand so on). Thus, we come to a boundary condition in disconnected region.

This process can be considered from the other point of view [55–57, 77, 83]. Anexcluded layer is returned to the region of solution, but coefficients of equations be-come generalized functions. In this section, we develop this approach for a parabolicsystem with heterogeneous non-ideal contact conditions of conjugation.

4.6.1 Main Definitions

Let the state of a system be described by the function u(t,ξ1,ξ2, . . . ,ξn) defined ina cylindrical region Q = (0,T )×Ω , where Ω = Ω1 ∪ γ ∪Ω2 ⊂ R

n is a boundedsimply connected domain of changes of space variables ξ = (ξ1, . . . ,ξn) with a reg-ular bound ∂Ω which is broken by a smooth hypersurface γ = Ω1 ∩Ω 2 ⊂ R

n ofdimension (n−1) onto two simply connected domains Ω1 and Ω2 (Ω1∩Ω2 = ∅).Let us introduce the following notations: Qi = (0,T )×Ωi, i = 1,2, Q3 = (0,T )×γ .

Let us consider the diffusion process that evolves in two heterogeneous regionsQ1 and Q2 Hausdorff by a three-layered inclusion Q3

∂u∂ t

+ q(ξ )u−n

∑i, j=1

∂∂ξi

(

ki j(ξ )∂u∂ξ j

)

= f (t,ξ ), (t,ξ ) ∈ Q1∪Q2, (4.35)

u|t=0 = 0, u|ξ∈∂Ω = 0, (4.36)

[(ω ,n)Rn ] = f0(t,ξ ), (t,ξ ) ∈Q3, (4.37)

[u]+R1(ω ,n)−Rn +R3(ω,n)+

Rn = 0, (t,ξ ) ∈ Q3, (4.38)

where ω = −Kgradu in Q1 ∪Q2, K = {ki j}ni, j=1 is a non-degenerated matrix,

gradu =(

uξ1, . . . ,uξn

)

, [u] is a jump of function u(t,ξ ) on Q3, i.e.

[u](t,ξ0) = u+(t,ξ0)−u−(t,ξ0), ξ0 ∈ γ,u+(t,ξ0) = lim

ξ+→ξ0

u(t,ξ+),

u−(t,ξ0) = limξ−→ξ0

u(t,ξ−), ξ+ ∈Ω2,ξ− ∈Ω1,


[(ω ,n)Rn ],(ω,n)+Rn ,(ω ,n)−

Rn are defined similarly, R1(ξ ) ≥ 0, R3(ξ ) ≥ 0 are thefunctions, continuous on γ and describe physical parameters of inclusions γ1,γ3

(R1(ξ ) + R3(ξ ) > 0), n = (nξ1, . . . ,nξn) is the normal to the surface γ , which is

outward to Ω1, q(ξ ),ki j(ξ ) have a simple disconnection on the surface Q3.Note that when f0 = 0 the conditions of conjugation (4.37), (4.38) turn into

heterogeneous conditions of conjugation of non-ideal contact type

[(ω ,n)Rn ] = 0, α[u]+ (ω,n)Rn = 0, (t,ξ ) ∈Q3,

where α = 1/(R1 +R3).According to [56, 57], let us pass from (4.35) to (4.38) to a first order system of

linear differential equations (with respect to (u,ω)), which allow to take into accountthe conditions of conjugation (4.37) and (4.38) in the equations of the system.

Let Ck(

Q1,Q2)

be a set of functions of the class Ck(Q1 ∪Q2), which allow ex-tensions keeping the smoothing from Q1 into Q1 and from Q2 into Q2.

Let us define on the set C0(Q1,Q2) the linear functionals δ−(γ),δ+(γ) in thefollowing way:

δ−(γ)(u) =∫

Q3

u−(t,ξ )dQ3, δ+(γ)(u) =∫

Q3

u+(t,ξ )dQ3, u ∈C0(Q1,Q2).

By a generalized left (right) derivative of the function f ∈ C1(Q1,Q2) we mean afunctional defined on the functions u ∈C0(Q1,Q2) by the rule

∂l f∂ξi

= f ∗ξi+[ f ]nξi

δ+(γ),(

∂r f∂ξi

= f ∗ξi+[ f ]nξi

δ−(γ))

,

where f ∗ξiis a classic derivative of the function f in Q1∪Q2.

Then for the function u(t,ξ ) ∈C2(Q1,Q2) the following equalities hold:

R3

R1 +R3gradl u =

R3

R1 +R3(u∗ξ1

, . . . ,u∗ξn)+

R3[u]nδ+(γ)R1 +R3

,

R1

R1 +R3gradr u =

R1

R1 +R3(u∗ξ1

, . . . ,u∗ξn)+

R1[u]nδ−(γ)R1 +R3

.

Taking into account the equality ω = −Kgradu in Q1 ∪Q2 and the condi-tion (4.38) on the surface Q3, we get

R3

R1 +R3gradl u =−R3K−1ω

R1 +R3− R3nδ+(γ)(R1(ω ,n)−

Rn +R3(ω ,n)+Rn)

R1 +R3,

R1

R1 +R3gradr u =−R1K−1ω

R1 +R3− R1nδ−(γ)(R1(ω ,n)−

Rn +R3(ω ,n)+Rn)

R1 +R3,

or

R3 gradl u+R1 gradr uR1 +R3

=−K−1ω− (R1nδ−(γ)+R3nδ+(γ))2 ωR1 +R3

, (t,ξ ) ∈ Q.

(4.39)


Similarly, generalized derivatives of the vector ω may be written as

R1 divl ωR1 +R3

=n

∑i=1

R1(ωi)∗ξi

R1 +R3+

R1δ+(γ)([ω ],n)Rn

R1 +R3,

R3 divr ωR1 +R3

=n

∑i=1

R3(ωi)∗ξi

R1 +R3+

R3δ−(γ)([ω ],n)Rn

R1 +R3.

Hence, taking into account (4.37) we have

R1 divl ω +R3 divr ωR1 +R3

=n

∑i=1

(ωi)∗ξi+

R1 f0δ+(γ)+R3 f0δ−(γ)R1 +R3

.

Thus, (4.35) may be rewritten as

∂u∂ t

+q(ξ )u+R1 divl ω +R3 divr ω

R1 +R3= f +

R1 f0δ+(γ)+R3 f0δ−(γ)R1 +R3

, (4.40)

where (t,ξ ) ∈Q and the derivatives are meant in a the generalized sense.Thus, instead of (4.35) and conditions (4.37) and (4.38) we obtain the first order

system of linear partial derivative equations (4.39) and (4.40).Let us consider this problem in the generalized sense.Let C1

bd(

Q1,Q2)

be a subset of C1(

Q1,Q2)

which consists of functions satisfy-

ing initial and boundary conditions (4.36). Similarly, let C1bd*

(

Q1,Q2

)

be a subset

of functions of C1(

Q1,Q2)

satisfying the adjoint conditions

v|t=T = 0, v|ξ∈∂Ω = 0. (4.41)

Let us denote by Cbd the set of pairs of functions x = (u,ω) ∈ C1bd(

Q1,Q2

)×(C0(Q1,Q2))

n which satisfy adjoint condition (4.38). Similarly, let Cbd* be a set ofpairs of functions y = (v,η) ∈C1

bd*(

Q1,Q2

)× (C0(Q1,Q2))n, satisfying on Q3 the

condition[v] = R1(η ,n)−

Rn +R3(η,n)+Rn .

In addition, let W 1,1/12 (Q) be a completion of the set C1

bd(

Q1,Q2)

with respectto the norm

‖u‖2W1,1/1

2 (Q)=

2

∑k=1

∫

Qk

u2t +

n

∑i=1

u2ξi

dQk, (4.42)

and W 1,12 (Q) be a completion of C1

(

Q)

with respect to the same Sobolevnorm (4.42).

It is clear that elements of the space W 1,1/12 (Q) may be interpreted as pairs

of functions (u1,u2) ∈W 1,12 (Q1)×W 1,1

2 (Q2) satisfying condition (4.36) on corre-

sponding parts of bound. Similarly, W 1,1/12,∗ (Q) is a completion of C1

bd*(

Q1,Q2)

with respect to the norm (4.42). Denote as W−1,1/12 (Q) and W−1,1/1

2,∗ (Q) the spaces

which are conjugate to W 1,1/12 (Q) and W 1,1/1

2,∗ (Q), respectively.


By the theorem on traces, functions from (u1,u2) ∈W 1,12 (Q1)×W 1,1

2 (Q2) leavethe traces (u−,u+) ∈ L2(Q3)×L2(Q3) on the surface Q3, and the trace operator is

continuous. That is why for all u ∈W 1,1/12 (Q) the following inequality holds:

∫

Q3

[u]2dQ3 ≤ c‖u‖2W1,1/1

2 (Q),

where c hereinafter is some positive constant. The inequality ‖[v]‖L2(Q3) ≤c‖v‖

W1,1/12,∗ (Q)

for all v ∈W 1,1/12,∗ (Q) can be proved in the same way.

Let us introduce the space X (Y ) as a completion of the set Cbd (Cbd*, respec-tively) in the norm

‖x‖2 = ‖u‖2W1,1/1

2 (Q)+‖ω‖2

Ln2(Q).

In the pair x=(u,ω)∈X the vector ω leaves a trace like R1(ω ,n)−Rn +R3(ω ,n)+

Rn

on the surface Q3, defined by the equality

R1(ω ,n)−Rn +R3(ω ,n)+

Rn =−[u].

The relations between the vector-function ω and its trace R1(ω ,n)−Rn +

R3(ω ,n)+Rn on Q3 is seen better when we consider on Cbd the norm

‖x‖20 = ‖u‖2

W1,1/12 (Q)

+‖ω‖2Ln

2(Q) +‖R1(ω ,n)−Rn +R3(ω ,n)+

Rn‖2L2(Q3)

which is equivalent to the norm of the space X . Now, let us denote as Ln2,γ(Q) the

completion of the set (C0(Q1,Q2))n with respect to the norm

‖ω‖2Ln

2,γ (Q) = ‖ω‖2Ln

2(Q) +‖R1(ω ,n)−Rn +R3(ω ,n)+

Rn‖2L2(Q3)

,

then, element of the space Ln2,γ(Q) is a set of functions ω from Ln

2(Q), whose traces

R1(ω ,n)−Rn +R3(ω ,n)+

Rn ∈ L2(Q3) are meaningful. More precisely, the set Ln2,γ(Q)

is isometric Ln2(Q)×L2(Q3), where the operator of isometry O : Ln

2,γ (Q)→ Ln2(Q)×

L2(Q3) is set as a completion by continuity of the operator

(C0(Q1,Q2))n � ω → Oω = (ω ,R1(ω ,n)−

Rn +R3(ω,n)+Rn) ∈ Ln

2(Q)×L2(Q3),

on the entire space Ln2,γ(Q).

Similarly, in the pair y = (v,η) ∈ Y the vector η leaves the trace R1(η,n)−Rn +R3(η ,n)+

Rn = [v] on Q3.Let the natural bilinear form 〈·, ·〉X×X∗ be defined on Cartesian product of the

original space and its conjugate space (for example, X and X∗) .Let us consider a system describing heat and mass transport in two heterogeneous

media with heterogeneous conditions of conjugation of non-ideal contact type:

L x = F, (4.43)


where the operator L is defined as a symbolic matrix

L =

⎛

⎜

⎜

⎝

∂∂ t

+qR1 divl +R3 divr

R1 +R3

R3 gradl +R1 gradr

R1 +R3M

⎞

⎟

⎟

⎠

, x =

(

uω

)

.

The function u(t,ξ ) describes heat and mass transport, ω = (ω1, . . . ,ωn) is a vectorof specific flux of substance. The operator L acts from X into Y ∗, the domain of Lis the set D(L ) =Cbd.

The coefficients of the system satisfy the following conditions: q(ξ ) ∈C0(Ω 1,Ω 2), q≥ 0, the coefficient matrix M = {σi j}n

i, j=1 has the form

M = K−1 +(R1nδ−(γ)+R3nδ+(γ))2

R1 +R3,

where K−1 = {ki j}ni, j=1 is an inverse matrix to the coefficient matrix K = {ki j}n

i, j=1

of the original parabolic equation (ki j(ξ )∈C0(Ω 1,Ω 2)). We suppose that the matrixK is symmetric ki j = k ji and uniformly positive defined in Ω1∪Ω2

n

∑i, j=1

ki j(ξ )λiλ j ≥ c−1n

∑i=1

λ 2i , λi ∈ R,ξ ∈Ω1∪Ω2,

where c is a positive constant which does not depend on ξ ,λi.Under (R1 divl ω+R3 divr ω)/(R1+R3) in the equations (4.43) we mean a linear

continuous functional over v ∈W 1,1/12,∗ which acts by the rule

⟨

R1 divl ω +R3 divr ωR1 +R3

,v

⟩

W−1,1/12,∗ ×W 1,1/1

2,∗=−

2

∑k=1

n

∑i=1

∫

Qk

ωi∂v∂ξi

dQk

−∫

Q3

R1(ω ,n)−Rn +R3(ω ,n)+

Rn

R1 +R3[v]dQ3.

Note, that for smooth functions this equality corresponds to the formula of integra-tion by parts.

By (R3 gradl u+R1 gradr u)/(R1 +R3) we mean a linear continuous functionalover η ∈ Ln

2,γ(Q) (or over y ∈ Y ):

⟨

R3 gradl u+R1 gradr u

R1 +R3,η⟩

(Ln2,γ )∗×Ln

2,γ

=2

∑k=1

n

∑i=1

∫

Qk

∂ u

∂ξiηi dQk

+

∫

Q3

R1(η ,n)−Rn +R3(η,n)+Rn

R1 +R3[u]dQ3.


By Mω we mean a functional over η ∈ Ln2,γ (Q):

〈Mω ,η〉(Ln2,γ )∗×Ln

2,γ=

2

∑k=1

n

∑i, j=1

∫

Qk

ki jω jηi dQk

+∫

Q3

(

R1(ω,n)−Rn +R3(ω ,n)+

Rn

)(

R1(η,n)−Rn +R3(η ,n)+Rn

)

R1 +R3dQ3.

Thus, taking into account (4.38), we have

〈L x,y〉Y ∗×Y =2

∑k=1

∫

Qk

∂u∂ t

v+quv+n

∑i, j=1

ki jω jηi dQk

+2

∑k=1

n

∑i=1

∫

Qk

∂ u∂ξi

ηi− ∂v∂ξi

ωi dQk +

∫

Q3

[u][v]R1 +R3

dQ3. (4.44)

By L + we denote an adjoint operator

L +y = G, L + : Y → X∗, y = (v,η).

Let us write a symbolic matrix which defines the operator L +:

L + =

⎛

⎜

⎜

⎝

− ∂∂ t

+q −R1 divl +R3 divr

R1 +R3

−R3 gradl +R1 gradr

R1 +R3M

⎞

⎟

⎟

⎠

,

where values of symbolic operators are defined similarly to the matrix of the op-erator L . For the moment, let take as a domain of the operator L + the setD(L +) =Cbd*. Then

〈L +y,x〉X∗×X =2

∑k=1

∫

Qk

−∂v∂ t

u+quv+n

∑i, j=1

ki jω jηi dQk

+2

∑k=1

n

∑i=1

∫

Qk

∂u∂ξi

ηi− ∂v∂ξi

ωi dQk

+∫

Q3

[v][u]R1 +R3

dQ3

= 〈y,L x〉Y×Y ∗ , (4.45)

for all x ∈D(L ), y ∈ D(L +).


4.6.2 Properties of Operators Associated with a Boundary ValueProblem

By means (4.44) and (4.45), we can prove that the operators L and L + are contin-uous in their domains. The density of the set D(L ) in X (D(L +) in Y , respectively)allows to extend L (L +, respectively) by continuity on the entire space X (Y , re-spectively). Extended operators we denote by L , L +. Thus, the following lemmais true.

Lemma 4.7. There exists such a positive constant c > 0 that for all x ∈ X ,y ∈Y thefollowing inequalities hold

‖L x‖Y∗ ≤ c‖x‖X , ‖L+y‖X∗ ≤ c‖y‖Y . (4.46)

Remark 4.21. Passing to the limit in (4.45) and taking into account (4.46), it is easyto prove that the operators L ,L + satisfy the relation

⟨

L x,y⟩

Y ∗×Y =⟨

x,L +y⟩

X×X∗ , ∀x ∈ X ,y ∈ Y,

i.e. L + is adjoint operator to L .

Let us show that a solution of the equation L x=F is connected with the classicalsolvability of the problems (4.35)–(4.38).

Theorem 4.9. Let the coefficients ki j of the operator L and the solution x =(u,ω) ∈ X of the equation

L x =

(

f + f0R3δ−(γ)+R1δ+(γ)

R1 +R3,0)

∈Y ∗, f ∈ L2(Q), f0 ∈ L2(Q3),

are smooth sufficiently to guarantee the classical solvability of the problem (4.35)–(4.38):

(1)ut ∈C(Q1 ∪Q2),uξiξ j∈C(Q1∪Q2), ki j ∈C1(Ω1∪Ω2), i, j = 1,n.

(2)There exist one-sided pointwise limits:lim

ξ1→ξ0

(Kgradu,n)Rn , limξ2→ξ0

(Kgradu,n)Rn , ξ0 ∈ γ , ξk ∈Ωk, k = 1,2.

Then there exist such f ∈C(Q1∪Q2), f0 ∈C(Q3), that the function u(t,ξ ) pointwisesatisfies relations (4.35)–(4.38), ω =−Kgradu in Q1∪Q2, and the equalities f = f ,f0 = f0 hold almost everywhere.

Proof. The conditions (4.36) hold because x = (u,ω) ∈ X and the norm of

W 1,1/12 (Q) preserves the corresponding limit values.On the other hand, for any y = (v,η) ∈ Y the following inequality holds.

〈L x,y〉Y ∗×Y = ( f ,v)L2(Q) +

(

R3 f0

R1 +R3,v−)

L2(Q3)

+

(

R1 f0

R1 +R3,v+)

L2(Q3)

.

(4.47)


At the same time, let y = (0,η) ∈ Y , then we can rewrite (4.47) as

〈L x,y〉Y ∗×Y =2

∑k=1

n

∑i=1

∫

Qk

n

∑j=1

ki jω jηi +uξiηi dQk = 0, ∀ηi ∈ L2(Q).

Hence, ω =−Kgradu in the sense of equality in L2(Q), and taking into account thesmoothing of u(t,ξ ) we can consider this equality in the pointwise sense in Q1∪Q2

also. Thus, ω ∈ (C1(Q1∪Q2))n, and (ω ,n)−

Rn ,(ω,n)+Rn make sense.

Let us substitute into (4.47) such y = (v,η) ∈ Y that v ∈ C1(Q) and v = 0 overQ3. Then [v] = 0 over Q3. Integrating by parts we have

〈L x,y〉Y ∗×Y =2

∑k=1

∫

Qk

∂u∂ t

v+quv+n

∑i=1

∂ωi

∂ξivdQk = ( f ,v)L2(Q).

Hence, by virtue of the density of the considered set of functions v(t,ξ ) in L2(Q),we have

∂u∂ t

+qu+n

∑i=1

∂ωi

∂ξi=

∂ u∂ t

+qu−n

∑i, j=1

∂∂ξi

(

ki j∂u∂ξ j

)

= f

in L2(Q). Denoting the left-hand side by f ∈C(Q1∪Q2), we have that f = f almosteverywhere in Q1∪Q2.

Let us substitute into (4.47) such y = (v,η) ∈Y that v ∈C1(Q1) and v = 0 in Q2.Integrating by parts we have

〈L x,y〉Y ∗×Y − ( f ,v)L2(Q) =

=∫

Q3

(

[u]R1 +R3

+(ω,n)−Rn

)

[v]dQ3 =−∫

Q3

R3 f0

R1 +R3[v]dQ3.

Whence it follows that

[u]R1 +R3

+(ω,n)−Rn =− R3 f0

R1 +R3, (t,ξ ) ∈ Q3.

Similarly, we can prove that

[u]R1 +R3

+(ω,n)+Rn =

R1 f0

R1 +R3,

This guarantees the holding of the conditions (4.37) and (4.38). �

Remark 4.22. The theorem remains true if f ∈ ◦W−1,1

2,∗ , where◦

W 1,12,∗ is a completion

of the set of the functions from the space C1(Q), satisfying conditions (4.41) and

vanishing on the surface Q3 in the norm (4.42), and◦

W−1,12,∗ is the conjugate space

to◦

W 1,12,∗ .


Lemma 4.8. There exists such a positive constant c > 0 that for all x = (u,ω) ∈ Xthe following inequality holds:

c−1‖u‖L2(Q) ≤ ‖L x‖Y ∗ .

Proof. Let us consider the value of functional L x ∈ Y ∗ on the element y = Ix ∈ Y ,where

v =−∫ t

Te−τu(τ,ξ )dτ, η = Kgradv.

It is clear, that y = Ix belongs to the space Y .By definition of the operator L x, we have

〈L x,y〉Y ∗×Y =2

∑k=1

(ut +qu,v)L2(Qk)+

2

∑k=1

n

∑i, j=1

(ki jω j,ηi)L2(Qk)

+2

∑k=1

n

∑i=1

(

uξi,ηi)

L2(Qk)−

2

∑k=1

n

∑i=1

(

ωi,vξi

)

L2(Qk)

+( 1

R1 +R3[u], [v]

)

L2(Q3).

Let us consider every item separately. Integrating by parts and taking into ac-count (4.36), we have

(ut +qu,v)L2(Q) =−(u,vt)L2(Q)−(

qetvt ,v)

L2(Q)

=∫

Qe−tu2dQ+

12

∫

Ωqetv2

∣

∣

t=0 dΩ +12

∫

Qqetv2dQ

≥ c−1‖u‖2L2(Q).

Let us pass to the second item.

2

∑k=1

n

∑i, j=1

∫

Qk

ki jωiη j dQk =2

∑k=1

n

∑i=1

∫

Qk

ωi

n

∑j=1

ki jη j dQk

=2

∑k=1

n

∑i=1

∫

Qk

ωi∂v∂ξi

dQk.

Let us consider the third item. Integrating by parts and taking into account thefact that the matrix {ki j}n

i, j=1 is positively defined, we have


2

∑k=1

n

∑i=1

∫

Qk

uξiηi dQk =

2

∑k=1

n

∑i, j=1

∫

Qk

uξiki jvξ j

dQk

=−2

∑k=1

n

∑i, j=1

∫

Qk

ki jetvtξi

vξ jdQk

=2

∑k=1

n

∑i, j=1

∫

Ωk

12

ki jetvξi

vξ j|t=0 dΩk +

2

∑k=1

n

∑i, j=1

∫

Qk

12

etki jvξivξ j

dQk

≥ c−12

∑k=1

n

∑i=1

∫

Qk

(

∫ t

Te−τuξi

dτ)2

dQk.

Let us estimate the last item.∫

Q3

[u][v]

R1 +R3dQ3 =−

∫

Q3

et [vt ][v]

R1 +R3dQ3

=

∫

γ

et [v]2

2(R1 +R3)|t=0 dγ +

∫

Q3

et [v]2

2(R1 +R3)dQ3

≥ 0.

Let ‖x‖X1 be a semi-norm over X :

‖x‖2X1

=∫

Qu2dQ+

2

∑k=1

n

∑i=1

∫

Qk

(

∫ t

Te−τuξi

dτ)2

dQk.

Thus, we conclude that 〈L x,y〉Y ∗×Y ≥ c−1‖x‖2X1

. Applying the Schwarz inequal-ity, we have ‖L x‖Y∗ · ‖y‖Y ≥ c−1‖x‖2

X1.

Let us show that ‖y‖Y ≤ c‖x‖X1 . Indeed, since η = Kgradv, then

‖y‖2Y = ‖vt‖2

L2(Q) +2

∑k=1

n

∑i=1

‖vξi‖2

L2(Qk)+‖η‖2

Ln2(Q)

≤ ‖vt‖2L2(Q) + c

2

∑k=1

n

∑i=1

‖vξi‖2

L2(Qk)

≤ c∫

Qu2dQ+ c

2

∑k=1

n

∑i=1

∫

Qk

(∫ t

Te−τuξi

dτ)2

dQk

= c‖x‖2X1.

Thus, we proved the inequality

‖L x‖Y ∗ ≥ c−1‖x‖X1 ≥ c−1‖u‖L2(Q), ∀x ∈ X .

�Similarly, we can prove the following lemma for an adjoint operator.


Lemma 4.9. There exists such a positive constant c > 0, that for all y = (v,η) ∈ Ythe following inequality holds

c−1‖v‖L2(Q) ≤ ‖L +y‖X∗ .

To prove this, we have to consider the operator L +y on the element x = (u,ω) =Iy, where

u =

∫ t

0eτv(τ,ξ )dτ, ω =−Kgradu.

Note that in the left-hand side of the inequalities of Lemmas 4.8 and 4.9 semi-norms of the element x and y appear, and not the norms as in the previous chapters.

Lemma 4.10. The operators L and L+ are injective.

Proof. Let us suppose that there exists such x = (u,ω) ∈ X that L x = 0 in Y ∗.Then 〈L x,y〉Y ∗×Y = 0 for all y ∈ Y , including y = Ix defined in Lemma 4.8.Applying the inequality from Lemma 4.8, we have 0 = 〈L x,y〉Y ∗×Y ≥ c−1‖x‖2

X1.

Whence it follows that, u = 0 in L2(Q); hence, in W 1,1/12 (Q) also. Then the equality

〈L x,y〉Y ∗×Y = 0 can be rewritten as

2

∑k=1

n

∑i=1

∫

Qk

ωi

( n

∑j=1

ki jη j− ∂v∂ξi

)

dQk = 0, ∀y = (v,η) ∈Y.

If y = (0,Kω) this inequality takes the form ‖ω‖2Ln

2(Q) = 0, whence it follows that

ω = 0 in Ln2(Q).

The injectivity of the operator L + can be prove similarly. �

4.6.3 Generalized Solution of a Parabolic Systemwith Discontinuous Coefficients and Solutions

Theorem 4.10. For any right-hand side F ∈ S1 = {( f ,0) | f ∈ L2(Q)} ⊂ Y ∗ thereexists the unique element x ∈ X such that L x = F in Y ∗.Proof. In view of Lemma 4.9, for any y ∈ Y we have

|〈F,y〉Y ∗×Y |= |( f ,v)L2(Q)| ≤ ‖ f‖L2(Q)‖v‖L2(Q) ≤ c‖L +y‖X∗ .

By virtue of the injectivity of the operator L + the expression 〈F,y〉Y ∗×Y can beconsidered as a linear continuous functional of μ = L +y in X∗. Applying TheHahn–Banach Theorem on on the extension of linear functionals, let us extend thefunctional from the set R(L +) on the entire space X∗. By the Riesz Representa-tion Theorem on the general form of a linear continuous functional in X∗ there


exists such an element x ∈ X that 〈x,L +y〉X×X∗ = 〈F,y〉Y ∗×Y for all y ∈ Y . Hence,〈L x,y〉Y ∗×Y = 〈F,y〉Y ∗×Y or L x = F in Y ∗. The uniqueness of the solution followsfrom the injectivity of the operator L . �Corollary 4.3. The parabolic system (4.35)–(4.38) with homogeneous conjugation

conditions of non-ideal contact type ( f0 = 0) has the unique solution u ∈W 1,1/12 (Q)

for any right-hand side f ∈ L2(Q).

Corollary 4.4. The following equality holds {g∈ L2(Q) |(g,0)∈R(L +)}= L2(Q).

In order to study the problem in the case if f0 �= 0, let us introduce the conceptof a generalized solution.

Definition 4.12. The function u ∈ L2(Q) is called a generalized solution of theequation L x = F , if there exists such a sequence xk = (uk,ωk) ∈ X that

‖u−uk‖L2(Q)→ 0, ‖F− L xk‖Y ∗ → 0, k→ ∞.

By Lemmas 4.7 and 4.8, it is easy to see that if x = (u,ω) ∈ X satisfies the equa-tion L x = F , then u is a generalized solution also. In addition, if u is a generalized

solution and F ∈ R(L ) then u ∈W 1,1/12 (Q) and there exists ω such that L x = F ,

where x = (u,ω).

Theorem 4.11. For any right-hand side F ∈ S2 = {( f ,0) | f ∈W−1,1/12,∗ (Q)} ⊂ Y ∗

there exists the unique generalized solution u∈ L2(Q) in the sense of Definition 4.12.

Proof. By the density of the set S1 in S2 in the sense of the convergence in thespace Y ∗, there exists such a sequence Fk ∈ S1 that Fk → F in Y ∗ as k→ ∞. ByTheorem 4.10 there exists such a sequence xk = (uk,ωk) ∈ X that L xk = Fk and byLemma 4.8 the sequence uk is a Cauchy with respect to the norm ‖ · ‖L2(Q). Thus,there exists such an element u∈ L2(Q) that ‖u−uk‖L2(Q)→ 0, i.e. u is a generalizedsolution in the sense of Definition 4.12.

Let us suppose that there exists one more solution u∈ L2(Q). Then the followinginequalities hold

‖u− u‖L2(Q) ≤ ‖uk− uk‖L2(Q) +o(1)≤ c‖L xk− L xk‖Y∗ +o(1) = o(1),

since L xk→ F , L xk→ F in Y ∗. �Corollary 4.5. The parabolic system (4.35)–(4.38) in a region with inclusions has

the unique generalized solution u ∈ L2(Q) for any right-hand side f ∈W−1,1/12,∗ (Q)

and f0 ∈ L2(Q3).

Corollary 4.6. There exists such a constant c > 0 that for all F ∈ S2 the in-equality ‖u‖L2(Q) ≤ c‖F‖Y ∗ holds, where u is a solution L x = F in the sense ofDefinition 4.12.

Theorem 4.12. In order for the function u ∈ L2(Q) to be a generalized solutionof the equation L x = F in the sense of Definition 4.12 it is necessary (and when


F ∈ S2 it is sufficiently too) for all y ∈ Y such that L +y = (g,0),g ∈ L2(Q) theequality (u,g)L2(Q) = 〈F,y〉Y∗×Y to be held.

Proof. Let u be a solution of the equality L x = F in the sense of Definition 4.12and xk = (uk,ωk) ∈ X be a sequence determining this solution. Then

(uk,g)L2(Q) = 〈xk,L+y〉X×X∗ = 〈L xk,y〉Y ∗×Y

for all y ∈ Y : L +y = (g,0), g ∈ L2(Q).Passing to the limit as k→ ∞, we get the desirable equality.Vice versa, let us suppose that for all y∈Y such that L +y = (g,0),g∈ L2(Q) the

equality (u,g)L2(Q) = 〈F,y〉Y ∗×Y holds. By Theorem 4.11 the equality L x = F has asolution u∗ ∈ L2(Q). Whence it follows that (u−u∗,g)L2(Q) = 0. By Corollary 4.4,we have that functions g ∈ L2(Q) run over the entire space L2(Q), i.e. u = u∗. �Remark 4.23. Similar statements also hold for an adjoint operator.

4.6.4 Approximate Method for Solving the Boundary ValueProblem for a Parabolic Equation with InhomogeneousTransmission Conditions of Non-ideal Contact Type

In this section, we consider a new approximate method for solving the boundaryvalue problem for a parabolic equation with inhomogeneous transmission condi-tions of non-ideal contact type which is an analogue of the Galerkin method, and thestability of the method is investigated. Note that in this section (contrary to previousones) a parabolic equation is investigated in a direct statement (in a disconnecteddomain without considering generalized functions in coefficients). This approachallows to compare results obtained for a parabolic equation in a disconnected do-main under different conditions of simulation of the diffusion because theoremson convergence of the numerical method allow to prove the existence of a uniquesolution.

Let us consider the diffusion process (4.35)–(4.38) in inhomogeneous media thatare in contact with each other through a thin three-layer region.

Assume that Dbd is the set of C1(Q1,Q2) consisting of the functions satisfyingconditions (4.36) and transmission conditions (4.38). Similarly, Dbd* is the set offunctions in C1(Q1,Q2) satisfying the boundary conditions

v|t=T = 0, v|ξ∈∂Ω = 0

and the transmission conditions

[v] = R1 (η ,n)−Rn +R3 (η ,n)+

Rn , [(η ,n)Rn ] = 0, (t,ξ ) ∈ Q3,

where η = Kgradv in Q1∪Q2.


Denote by W 1,1/12,0 (Q) and W 1,1/1

2,T (Q) the completions of Dbd and Dbd*,respectively, with respect to the norm

‖u‖2 =2

∑k=1

∫

Qk

u2t +

n

∑m=1

u2ξm

dQk.

The space W 0,1/12 (Q) is a completion of Dbd with respect to the norm

‖u‖2W0,1/1

2 (Q)=

2

∑k=1

∫

Qk

u2 +n

∑m=1

u2ξm

dQk. (4.48)

Note that the completion of Dbd* with respect to norm (4.48) coincides with

W 0,1/12 (Q).

Let W−1,1/12,0 (Q), W−1,1/1

2,T (Q), and W−0,1/12 (Q) are the conjugate spaces to

W 1,1/12,0 (Q),W 1,1/1

2,T (Q), and W 0,1/12 (Q) (with respect to L2(Q)). Obviously, the fol-

lowing continuous dense embeddings hold:

W 1,1/12,0 (Q)⊂W 0,1/1

2 (Q)⊂ L2(Q)⊂W−0,1/12 (Q)⊂W−1,1/1

2,0 (Q),

W 1,1/12,T (Q)⊂W 0,1/1

2 (Q)⊂ L2(Q)⊂W−0,1/12 (Q)⊂W−1,1/1

2,T (Q).

Denote the bilinear form over W−0,1/12 (Q)×W 0,1/1

2 (Q) by 〈·, ·〉, over W−1,1/12,T (Q)×

W 1,1/12,T (Q) by 〈·, ·〉T and over W 1,1/1

2,0 (Q)×W−1,1/12,0 (Q) by 〈·, ·〉0.

It is assumed as before that the coefficients (4.35) satisfy q(ξ ) ∈C0(Ω 1,Ω 2)and q � 0; and the coefficient matrix K = {kml(ξ )}n

m,l=1 is symmetric, i.e. kml(ξ ) =klm(ξ ) ∈ C0(Ω 1,Ω 2) and uniformly positive definite in Ω1∪Ω2

n

∑m,l=1

kml(ξ )λmλl � αn

∑m=1

λ 2m, λm ∈ R, ξ ∈Ω1∪Ω2,

where α is a positive constant independent of λm or ξ .By applying the integration-by-parts formula, it can easily be shown that, if u ∈

Dbd ∩C2(Q1 ∪Q2) satisfies (4.35) and (4.37) for continuous f and f0, then therelation

2

∑k=1

(ut + qu,v)L2(Qk)+

2

∑k=1

n

∑m,l=1

(

kmluξl,vξm

)

L2(Qk)+( [u]

R1 +R3, [v])

L2(Q3)

= ( f ,v)L2(Q) +( R3 f0

R1 +R3,v−)

L2(Q3)+( R1 f0

R1 +R3,v+)

L2(Q3)(4.49)

is satisfied for all v ∈W 0,1/12 (Q).


Thus, the left-hand side of (4.49) can be considered as a definition of the operator

S : W 1,1/12,0 (Q)→W−0,1/1

2 (Q). The operator S is defined for all u ∈W 1,1/12,0 (Q), and

it is easily shown that it is linear and continuous.Analogously, the left-hand side of (4.49) can be considered as a definition of the

adjoint operator S ∗ : W 0,1/12 (Q)→W−1,1/1

2,0 (Q), which is also linear and continuous.The right-hand side of (4.49) can be considered the value of the functional

F = f +R1 f0δ+(γ)+R3 f0δ−(γ)

R1 +R3∈W−0,1/1

2 (Q)

on the element v∈W 0,1/12 (Q), where δ+(γ),δ−(γ)∈W−0,1/1

2 (Q) are the Dirac deltafunctions supported on the different sides γ+ and γ− of the hypersurface γ .

The functional F ∈W−0,1/12 (Q) makes sense for arbitrary f ∈W−0,1/1

2 (Q), f0 ∈L2(Q3). Thus, taking into account (4.49), the following equation can be considered:

S u = F , F ∈W−0,1/12 (Q).

If in (4.49) the integration-by-parts formula is applied once more to the item

2

∑k=1

(ut ,v)L2(Qk)=

2

∑k=1

(u,−vt)L2(Qk), ∀v ∈W 1,1/1

2,T (Q),

the obtained relation defines a linear continuous operator S1 : W 0,1/12 (Q) →

W−1,1/12,T (Q) that is an extension of S : W 1,1/1

2,0 (Q) → W−0,1/12 (Q), and, respec-

tively, the adjoint operator S ∗1 : W 1,1/1

2,T (Q)→W−0,1/12 (Q) that is a restriction of the

operator S ∗ : W 0,1/12 (Q)→W−1,1/1

2,0 (Q). In this case, the following equation can be

considered: S1u = F , F ∈W−1,1/12,T (Q) ( f ∈W−1,1/1

2,T (Q), f0 ∈ L2(Q3)).

Lemma 4.11. For all u ∈W 0,1/12 (Q), the following inequalities are satisfied:

c−1‖u‖L2(Q) � ‖S1u‖W−1,1/1

2,T (Q)� c‖u‖

W0,1/12 (Q)

. (4.50)

Here and below, c is a sufficiently large positive constant independent of u and v.

Proof. The definition of S1 and the integral form of the Cauchy–Schwarz inequal-ity imply the right-hand side of (4.50). In order to prove the left-hand side, it isnecessary to consider the value of the functional S1u on the element

v(t,ξ ) =−∫ t

Te−τ u(τ,ξ )dτ.

It is clear that v ∈W 1,1/12,T (Q), since the norm of W 1,1/1

2,T (Q) “does not hold” the

conditions [(η ,n)Rn ] = 0, [v] = R1 (η,n)−Rn +R3 (η,n)+Rn .

Applying the formula avvt =12

(

av2)

t− 12 atv2, going to surface integrals, and tak-

ing into account the conditions on the coefficients of S1 and the Schwarz inequality,we obtain


c−1‖v‖2W1,1/1

2,T (Q)� 〈S1u,v〉T � ‖S1u‖

W−1,1/12,T (Q)

‖v‖W

1,1/12,T (Q)

.

To complete the proof, it is sufficient to take into account that ‖u‖L2(Q) �c‖v‖

W1,1/12,T (Q)

. �

Corollary 4.7. The operator S1 (and, therefore, S ) is injective.

Lemma 4.12. For all u ∈W 1,1/12,0 (Q) the following inequalities are satisfied

c−1‖u‖W

0,1/12 (Q)

� ‖S u‖W−0,1/12 (Q)

� c‖u‖W

1,1/12,0 (Q)

. (4.51)

Proof. The right-hand side of (4.51) is proved by applying the integral form of theCauchy–Schwarz inequality. In order to prove the left-hand side, it is necessary, asin Lemma 4.11, to consider the value of S u on the element v = e−tu. �Remark 4.24. 1. The analogous of inequalities (4.50) and (4.51) for the adjointoperators S ∗ and S ∗

1 can be proved in a similar way

c−1‖v‖L2(Q) � ‖S ∗v‖W−1,1/1

2,0 (Q)� c‖v‖

W0,1/12 (Q)

∀v ∈W 0,1/12 (Q), (4.52)

c−1‖v‖W0,1/1

2 (Q)� ‖S ∗

1 v‖W−0,1/1

2 (Q)� c‖v‖

W1,1/12,T (Q)

∀v ∈W 1,1/12,T (Q). (4.53)

2. On the basis of the proved inequalities (4.50), (4.51), and (4.52), (4.53), wecan prove the following statements:

(1) For any right-hand side of F ∈ L2(Q) (for example, f ∈ L2(Q), f0 = 0) there

exists a unique element u ∈W 1,1/12,0 (Q) such that S u = F .

(2) For any right-hand side of F ∈W−0,1/12 (Q) (for example, f ∈W−0,1/1

2 (Q),

f0 = L2(Q3)) there exists a unique element u ∈W 0,1/12 (Q) such that S1u = F .

(3) For any right-hand side of F ∈W−1,1/12,T (Q) (for example, f ∈W−1,1/1

2,T (Q),f0 ∈ L2(Q3)) there exists a unique element u ∈ L2(Q) such that the equality

(S ∗v,u)L2(Q) = 〈F,v〉T (4.54)

is valid for an arbitrary function v ∈ W 1,1/12,T (Q) satisfying the condition S ∗v ∈

(L2(Q))∗ = L2(Q).3. Similar consideration for the adjoint operator imply that the set

{

S ∗v|v ∈W 1,1/1

2,T (Q)}

covers the entire space L2(Q).

Assume that φi(ξ ) ∈ C1(Ω 1,Ω 2), i ∈ N. Assume also that, for all i ∈ N, thecondition φi

∣

∣

ξ∈∂Ω= 0 is satisfied and the set of functions

{ϕ(t)φi(ξ ) | i ∈ N,ϕ(t) ∈C([0,T ]),ϕ(T ) = 0}

forms a total set in W 0,1/12 (Q).


Assume that f ∈L2(Q) and f0 ∈L2(Q3). An approximate solution to the equationS u = F is sought in the form

us(t,ξ ) =s

∑i=1

gsi (t)φi(ξ ),

where the functions gsi (t) are the solutions to the Cauchy problem for the following

system of ordinary differential equations with constant coefficients:

s

∑i=1

2

∑k=1

(

dgsi

dt(φi,φ j)L2(Ωk)

+gsi (qφi,φ j)L2(Ωk)

+gsi

n

∑m,l=1

(

kml(φi)ξl,(φ j)ξm

)

L2(Ωk)

)

+s

∑i=1

gsi

(

[φi]

R1 +R3, [φ j]

)

L2(γ)=

2

∑k=1

( f ,φ j)L2(Ωk)+

(

R3 f0

R1 +R3,φ−j

)

L2(γ)

+

(

R1 f0

R1 +R3,φ+

j

)

L2(γ), gs

m(0) = 0, m = 1,s, j = 1,s. (4.55)

Due to the well-known solvability theorems for systems of ordinary differentialequations with constant coefficients, the solution to the Cauchy problem for sys-tem (4.55) exists and gs

i (t) ∈W 12 (0,T ), where W 1

2 (0,T ) is a Sobolev space.Consider the set of functions u(t,ξ ) ∈ Dbd whose derivatives ut(t,ξ ) regarded

as functions of ξ belong to the space C1(Ω 1,Ω 2). Assume that H is a completionof this space with respect to the norm

‖u‖2H =

2

∑k=1

∫

Qk

u2t +

n

∑m=1

u2tξm

dQk.

It is easily seen that us ∈H and H ⊂W 1,1/12,0 (Q), and this inclusion is continuous and

dense.

Lemma 4.13. For all u ∈H the following inequalities are satisfied:

c−1‖u‖2W1,1/1

2,0 (Q)�⟨

S u,e−tut⟩

� c‖u‖2H.

Proof. Obviously, e−tut ∈ W 0,1/12 (Q). The right-hand side of the inequality is

proved by applying to 〈S u,e−tut〉 the integral Cauchy–Schwarz inequality, theFriedrichs inequality, and the trace theorem.

Consider the proof of the left-hand side of the inequality. Transform the expres-sion 〈S u,e−tut〉. Applying the formula auut =

12

(

au2)

t − 12 atu2, going to surface

integrals, and taking into account the condition u|t=0 = 0, we obtain

2

∑k=1

(

qu,e−tut)

L2(Qk)=

2

∑k=1

12

∫

Ωk

qe−T u2∣

∣

t=T dΩk +2

∑k=1

12

∫

Qk

qe−tu2dQk � 0.


Similarly, we obtain the inequalities

2

∑k=1

n

∑m,l=1

(

kmluξl,e−tutξm

)

L2(Qk)=

2

∑k=1

n

∑m,l=1

12

∫

Ωk

kmle−T uξl

uξm

∣

∣

t=T dΩk

+2

∑k=1

n

∑m,l=1

12

∫

Qk

kmle−t uξl

uξmdQk � c−1

2

∑k=1

n

∑m=1

∫

Qk

u2ξm

dQk,

(

[u]R1 +R3

,e−t [ut ]

)

L2(Q3)

=

∫

γ

e−T [u]2|t=T

2(R1 +R3)dγ +

∫

Q3

e−t [u]2

2(R1 +R3)dQ3 � 0.

This proves the left-hand side of the inequality in lemma. �Assume that W 1,0

2,0 (Q) is the completion of the set of functions f ∈ C1(Q1,Q2)

satisfying f (0,ξ ) = 0 (ξ ∈Ω ) with respect to the norm

‖ f‖2W 1,0

2,0 (Q)=

2

∑k=1

∫

Qk

( ft )2dQk.

Assume that W 12,0(Q3) is the completion of the set of functions f ∈C(Q3) satisfying

f (0,ξ ) = 0 (ξ ∈ γ) and having continuous t-derivatives with respect to the norm

‖ f‖2W1

2,0(Q3)=∫

Q3

( ft )2dQ3. (4.56)

Lemma 4.14. Assume that f ∈ W 1,02,0 (Q), f0 ∈ W 1

2,0(Q3). Then, the followinginequality is satisfied:

‖us‖H � c‖ f‖W 1,0

2,0 (Q)+ c‖ f0‖W 1

2,0(Q3). (4.57)

Proof. It is easy to show that, if the conditions of the theorem are satisfied, thesolutions gs

i (t) to system (4.55) belong to the Sobolev space W 22 (0,T ).

Differentiating each equality in system (4.55) with respect to t, multiplying itby e−t(gs

j)t , summing over j from 1 to s, and integrating with respect to t from 0to T , we obtain

A =2

∑k=1

(

(us)tt +q(us)t ,e−t(us)t

)

L2(Qk)+

2

∑k=1

n

∑m,l=1

(

kml(us)tξl,e−t(us)tξm

)

L2(Qk)

+

(

[(us)t ]

R1 +R3,e−t [(us)t ]

)

L2(Q3)

=2

∑k=1

(

ft ,e−t(us)t

)

L2(Qk)

+

(

R3( f0)t

R1 +R3,e−t(us)

−t

)

L2(Q3)

+

(

R1( f0)t

R1 +R3,e−t(us)

+t

)

L2(Q3)

.


Applying the integral Cauchy–Schwarz inequality to the right-hand side, we have

A � ‖ ft‖L2(Q)‖(us)t‖L2(Q) + c‖( f0)t‖L2(Q3)‖(us)−t ‖L2(Q3)

+c‖( f0)t‖L2(Q3)‖(us)+t ‖L2(Q3) � c1

(‖ ft‖L2(Q) +‖( f0)t‖L2(Q3)

)‖us‖H . (4.58)

On the other hand, let us prove that A� c−1‖us‖2H . Applying the formula aututt =

12

(

au2t

)

t − 12 atu2

t and going to surface integrals, we obtain

2

∑k=1

(

(us)tt ,e−t(us)t

)

L2(Qk)=

2

∑k=1

12

∫

Ωk

e−t(us)2t

∣

∣

t=Tt=0 dΩk +

2

∑k=1

12

∫

Qk

e−t(us)2t dQk.

Thus,

A �−12

2

∑k=1

∫

Ωk

(us)2t

∣

∣

t=0dΩk + c−12

∑k=1

∫

Qk

(us)2t +

n

∑m=1

(us)2tξm

dQk.

Multiplying each inequality of system (4.55) by (gsj)t , summing it over j from 1

to s, setting t = 0, and taking into account the lemma conditions and the equalitygs

m(0) = 0, m = 1,s, we obtain

2

∑k=1

∫

Ωk

(us)2t

∣

∣

t=0dΩk = 0.

Thus, the inequality A � c−1‖us‖2H has been proved. Taking into account (4.58)

completes the proof of the lemma. �Corollary 4.8. Assume that the conditions of the lemma are satisfied. Then inequal-ity (4.57) implies that the sequence us is bounded in the Hilbert space H and, there-fore, there exists a weakly converging subsequence usk

w→ u∗ in H, such that thesequences (usk)t , (usk)ξm

, and (usk)ξmt converge weakly in L2(Q) and the sequencesu+sk

, u−sk, (usk)

+t , and (usk)

−t converge weakly in L2(Q3) to the corresponding value

of the generalized Sobolev derivative of the function u∗ ∈ H.

Lemma 4.15. Assume that f ∈W 1,02,0 (Q), f0 ∈W 1

2,0(Q3). Then, for an arbitrary func-

tion v ∈W 0,1/12 (Q), the following equality is satisfied:

〈S u∗,v〉= 〈F,v〉,

where u∗ ∈ H is the function defined on Corollary 4.8.

Proof. Multiplying each equality in system (4.55) as s = sk by an arbitrary functionϕ j(t) ∈C([0,T ]): ϕ j(T ) = 0, summing the result over j from 1 to p (p = 1,sk) andintegrating it with respect to t from 0 to T , we obtain

⟨

S usk ,p

∑j=1

ϕ jφ j

⟩

=⟨

F,p

∑j=1

ϕ jφ j

⟩

, p = 1,sk.


Pass to the limit as k→ ∞ and assume that vp = ∑pj=1 ϕ jφ j . Due to Corollary 4.8,

we obtain⟨

S u∗,vp⟩

=⟨

F,vp⟩

.

Since the system

{ϕ(t)φi(ξ )|i ∈ N, ϕ(t) ∈C([0,T ]), ϕ(T ) = 0}

is total in W 0,1/12 (Q), a closure of its span coincides with the entire space W 0,1/1

2 (Q),which implies the lemma. �Corollary 4.9. For arbitrary functions f ∈W 1,0

2,0 (Q) and f0 ∈W 12,0(Q3) there exists

a unique solution u∗ ∈ H to the equation S u = F .

Theorem 4.13. Assume that f ∈W 1,02,0 (Q) and f0 ∈W 1

2,0(Q3). Then the sequence ofapproximations us converges to the solution to the equation S u = F in the norm of

W 1,1/12,0 (Q) and S us→ F in W−0,1/1

2 (Q) as s→ ∞.

Proof. Multiplying each equality of system (4.55) at s = sk by e−t(gskj )t , summing

then over j from 1 to sk, and integrating with respect to t from 0 to T ,we obtain

⟨

S usk ,e−t(usk)t

⟩

=⟨

F,e−t(usk)t⟩

.

Taking into account the properties of the subsequence sk (see Corollary 4.8), wepass to the limit as k→ ∞ to obtain

limk→∞

⟨

S usk ,e−t(usk)t

⟩

=⟨

F,e−t(u∗)t⟩

. (4.59)

Taking into account Lemma 4.13, we have

c−1‖usk−u∗‖2W

1,1/12,0 (Q)

�⟨

S (usk −u∗),e−t(usk −u∗)t⟩

=⟨

S usk ,e−t(usk)t

⟩− ⟨S u∗,e−t(usk)t⟩− ⟨S usk −S u∗,e−t(u∗)t

⟩

.

Since usk converges weakly to u∗ in the corresponding spaces (Corollary 4.8), weconclude that

⟨

S usk −S u∗,e−t(u∗)t⟩

tends to zero as k→∞ and⟨

S u∗,e−t(usk)t⟩

tends to 〈S u∗,e−t(u∗)t〉. Therefore, due to (4.59) and Lemma 4.15, we obtain

limk→∞

c−1‖usk −u∗‖2W1,1/1

2,0 (Q)�⟨

F,e−t(u∗)t⟩− ⟨S u∗,e−t(u∗)t

⟩

= 0.

Note that the assumption on the existence of a weak accumulation point of the se-quence us that is different from u∗ in H contradicts the uniqueness of the solution tothe equation S u = F (Corollary 4.7). Therefore, there is no necessity to choose the

subsequence sk, i.e., the entire sequence us converges to u∗ in W 1,1/12,0 (Q) as s→ ∞.

Since the operator S is continuous, we have S us→ F in W−0,1/12 (Q). �


Consider the set of functions f ∈ C(Q3) satisfying f (T,ξ ) = 0 and having acontinuous derivative with respect to t and denote by W 1

2,T (Q3) its completion withrespect to norm (4.56).

Lemma 4.16. For the arbitrary right-hand sides f ∈ L2(Q) and f0 ∈W 12,T (Q3), the

following inequality is satisfied

‖us‖W 1,1/12,0 (Q)

� c‖ f‖L2(Q) + c‖ f0‖W12,T (Q3)

.

Proof. Since us ∈ H, Lemma 4.13 implies that

c−1‖us‖2W

1,1/12,0 (Q)

�⟨

S us,e−t(us)t

⟩

. (4.60)

On the other hand, if both sides of each equality (4.55) are multiplied by e−t(gsj)t ,

summed over j from 1 to s, and integrated with respect to t from 0 to T , we obtain

⟨

S us,e−t(us)t

⟩

=(

f ,e−t (us)t)

L2(Q)

+( R3 f0

R1 +R3,e−t(us)

−t

)

L2(Q3)+( R1 f0

R1 +R3,e−t(us)

+t

)

L2(Q3).

Taking into account the conditions f0(T,ξ ) = 0, u−s (0,ξ ) = 0, and u+s (0,ξ ) = 0 andapplying the integration-by-parts formula to the last two terms, we obtain

⟨

S us,e−t(us)t

⟩

=(

f ,e−t(us)t)

L2(Q)−(R3( f0e−t)t

R1 +R3,u−s)

L2(Q3)

−(R1( f0e−t)t

R1 +R3,u+s)

L2(Q3)

� ‖ f‖L2(Q) ‖(us)t‖L2(Q) + c∥

∥( f0e−t)t∥

∥

L2(Q3)

∥

∥u−s∥

∥

L2(Q3)

+ c∥

∥( f0e−t)t∥

∥

L2(Q3)

∥

∥u+s∥

∥

L2(Q3)

� ‖ f‖L2(Q) ‖us‖W 1,1/12,0 (Q)

+ c1‖( f0)t‖L2(Q3)‖us‖W 1,1/1

2,0 (Q).

Taking into account the last equality and (4.60), we have

c−1 ‖us‖W 1,1/12,0 (Q)

� ‖ f‖L2(Q) + c1 ‖( f0)t‖L2(Q3),

which implies the lemma. �Theorem 4.14. For the arbitrary right-hand sides f ∈ L2(Q) and f0 ∈W 1

2,0(Q3),

there exists a unique solution u∗ ∈W 1,1/12,0 (Q) to the equation S u = F, the sequence

us converges to u∗ in the norm of W 1,1/12,0 (Q), and S us→ F in W−0,1/1

2 (Q).

Proof. The set W 1,02,0 (Q) is dense in L2(Q). Therefore, there exists a sequence of

functions f m ∈W 1,02,0 (Q) converging to f ∈ L2(Q) in L2(Q) as m→ ∞.


Assume that ums is a sequence of approximate solutions to system (4.55) with the

right-hand side given by f m and f0. According to Theorem 4.13, ums converges to

the solution um of the corresponding equation S u = Fm in W 1,1/12,0 (Q) as s→ ∞.

Let us prove that the sequence of functions um is a Cauchy sequence in W 1,1/12,0 (Q).

Indeed, applying the inequality of Lemma 4.16 to the term ‖uns −um

s ‖W 1,1/12,0 (Q)

, we

obtain

‖un−um‖W

1,1/12,0 (Q)

� ‖un−uns‖W1,1/1

2,0 (Q)+‖un

s −ums ‖W 1,1/1

2,0 (Q)+‖um

s −um‖W

1,1/12,0 (Q)

� c‖ f n− f m‖L2(Q) +o(1).

Passing to the limit as s→∞, m,n→∞, we conclude that um is a Cauchy sequence.

Since W 1,1/12,0 (Q) is complete, there exists an element u∗ ∈W 1,1/1

2,0 (Q) such that um→u∗ in W 1,1/1

2,0 (Q) as m→ ∞.

We prove that us→ u∗ in W 1,1/12,0 (Q) as s→ ∞. To this end,

‖us−u∗‖W1,1/1

2,0 (Q)� ‖us−um

s ‖W 1,1/12,0 (Q)

+‖ums −um‖

W1,1/12,0 (Q)

+‖um−u∗‖W 1,1/1

2,0 (Q).

Theorem 4.13 implies that ‖ums −um‖

W 1,1/12,0 (Q)

→ 0 as s → ∞. Moreover, by

Lemma 4.16

‖us−ums ‖W 1,1/1

2,0 (Q)� c‖ f − f m‖L2(Q) .

Therefore,

lims→∞‖us−u∗‖

W 1,1/12,0 (Q)

� c‖ f − f m‖L2(Q) +‖um−u∗‖W 1,1/1

2,0 (Q).

Passing to the limit as m→ ∞, we conclude that us→ u∗ in W 1,1/12,0 (Q) as s→ ∞.

Prove that u∗ is the solution to the equation S u = F . Indeed, since S um = Fm,we obtain

‖S u∗ −F‖W−0,1/12 (Q)

� ‖S (u∗ −um)‖W−0,1/12 (Q)

+‖Fm−F‖W−0,1/12 (Q)

� c‖u∗−um‖W 1,1/1

2,0 (Q)+‖Fm−F‖

W−0,1/12 (Q)

.

To complete the proof, it is sufficient to pass to the limit as m→ ∞. �If f and f0 are generalized functions, system (4.55) makes no sense, since the

integrals on the right-hand side of (4.55) may not exist. One way to overcome thisproblem is to replace these integrals by the corresponding bilinear forms, but sys-tem (4.55) then has to be considered in terms of the theory of generalized functions.


Another way is to replace the right-hand sides f and f0 with poor smoothness bysimilar but smoother functions. Below, we consider this approach and analyze theconvergence of the iteration procedures proposed.

Assume that the right-hand side F of the equation S1u = F is an element of the

negative space W−0,1/12 (Q) ( f ∈W−0,1/1

2 (Q), f0 ∈ L2(Q3)). Since L2(Q) is dense

in W−0,1/12 (Q), we choose a sequence of functions f m ∈ L2(Q) converging to f in

W−0,1/12 (Q). Similarly, assume that f m

0 is a sequence of functions from W 12,0(Q3)

converging to f0 in L2(Q3). Obviously, with such a choice, the sequence of the

right-hand sides Fm converges to F in W−0,1/12 (Q).

Consider a sequence of approximations us,m(t,ξ ) constructed according to therule

us,m(t,ξ ) =s

∑i=1

gsi,m(t)φi(ξ ),

where the functions gsi,m(t) are solutions to system (4.55) whose right-hand side is

given by f m and f m0 .

Theorem 4.15. Assume that εm is an arbitrary sequence of positive numbers con-verging to zero. Then, for an arbitrary positive integer s(m) satisfying the condition

∥

∥S us(m),m−Fm∥

∥

W−0,1/12 (Q)

< εm

(such an s(m) necessarily exists) the sequence us(m),m converges to the solution of

the equation S1u = F in the norm of W 0,1/12 (Q) as m→ ∞.

Proof. According to Theorem 4.14, the sequence us,m converges to the solution

um ∈W 1,1/12,0 (Q) of the equation S u = Fm in the norm of W 1,1/1

2,0 (Q) as s→ ∞, and‖S us,m−Fm‖

W−0,1/12 (Q)

−−−→s→∞

0.

We prove that um is a Cauchy sequence in W 0,1/12 (Q). Indeed, applying inequal-

ity (4.51) gives

‖um− un‖W0,1/1

2 (Q)� c‖S um−S un‖

W−0,1/12 (Q)

= c‖Fm−Fn‖W−0,1/1

2 (Q)−−−−→m,n→∞

0.

Thus, there exists a function u∗ ∈W 0,1/12 (Q) such that um → u∗ in W 0,1/1

2 (Q) asm→ ∞.

Prove that u∗ is the solution to the equation S1u = F . Indeed, since S1um =S um = Fm, we obtain

‖S1u∗ −F‖W−1,1/12,T (Q)

� ‖S1u∗ −S1um‖W−1,1/12,T (Q)

+‖Fm−F‖W−1,1/12,T (Q)

� c‖u∗−um‖W

0,1/12 (Q)

+ c‖Fm−F‖W−0,1/12 (Q)

−−−→m→∞

0.


Applying inequality (4.51), we have∥

∥us(m),m−u∗∥

∥

W0,1/12 (Q)

�∥

∥us(m),m−um∥

∥

W0,1/12 (Q)

+‖um−u∗‖W

0,1/12 (Q)

� c∥

∥S us(m),m−S um∥

∥

W−0,1/12 (Q)

+‖um−u∗‖W 0,1/1

2 (Q)

� cεm +‖um−u∗‖W

0,1/12 (Q)

.

Since the right-hand side of the last inequality tends to zero as m→ ∞, the theo-rem is proved. �

Now assume that the right-hand side F of the equation S1u = F belongs to

W−1,1/12,T (Q) (e.g. f ∈W−1,1/1

2,T (Q), f0 ∈ L2(Q3)).

Let f m ∈ L2(Q) be a sequence of functions converging to f in W−1,1/12,T (Q) and

f m0 ∈W 1

2,0(Q3) be a sequence converging to f0 in L2(Q3). The sequence of approxi-mations us,m satisfies the original conditions.

Theorem 4.16. Assume that εm is an arbitrary sequence of positive numbers con-verging to zero. Then, for an arbitrary positive integer s(m):

∥

∥S us(m),m−Fm∥

∥

W−1,1/12,T (Q)

< εm

(such an s(m) necessarily exists) the sequence us(m),m converges in the norm ofL2(Q) as m→ ∞ to the solution of the equation S1u = F in the sense of equal-ity (4.54).

Proof. By analogy with Theorem 4.15, we prove that a sequence um ∈W 1,1/12,0 (Q)

converges to a function u∗ ∈ L2(Q) in L2(Q). Prove that u∗ is the solution to theequation S1u = F in the sense of (4.54). Indeed,

〈S um,v〉= 〈S um,v〉T = 〈Fm,v〉T , ∀v ∈W 1,1/12,T (Q)

or(um,S ∗v)L2(Q) = 〈um,S ∗v〉0 = 〈S um,v〉= 〈Fm,v〉T

for all v ∈W 1,1/12,T (Q) such that S ∗v ∈ L2(Q).

Passing to the limit as m → ∞, we see that u∗ ∈ L2(Q) is the solution to theequation S1u = F .

The rest of the proof is similar to Theorem 4.15. �

4.7 On the Unique Solvability of Wave Systems

We study equations of the form

L u≡ A(utt)+B2(ut)+C(u) = f , (4.61)

where A, B, and C are second-order differential operators.

4.7 On the Unique Solvability of Wave Systems 81

Equations of the form (4.61) arise in applications, for example, when analyzingthe dynamics of plane motions of an incompressible viscous fluid, linear waves ona helical flow, small vibrations of an ideal non-rotating stratified liquid moving asa whole at a constant velocity in the direction perpendicular to the stratificationdirection, etc. [18, 20, 21]. Equation (4.61) generalizes the well-known Sobolev–Gal’perin equation (B ≡ 0), which has numerous applications [19, 27, 59, 93, 111].Initial-boundary value problems for (4.61) were studied from various viewpointsin [20, 21, 60, 62].

The unique solvability was analyzed in [5,41,45] for specific types of (4.61) forthe case in which the right-hand side is a distribution of a finite-order; approximatemethods were also constructed there, and some optimization problems were consid-ered. However, these results deal only with elliptic operators A, B, and C withoutfirst-order lower terms and specify the action of the operator only in a single pair ofspaces. In the book, we eliminate most of the restrictions imposed on the operatorsA, B, and C, obtain a “scale” of solvability theorems, and generalize some resultsobtained earlier.

4.7.1 Basic Notation and Statement of the Operator Equation

In the cylindrical domain (t,x) ∈ Q = (0,T )×Ω , we consider (4.61), where Ω ⊂R

n is a bounded connected domain with regular boundary ∂Ω and A is a second-order operator in the spatial variables:

A(u)≡−n

∑i, j=1

∂∂xi

(

ai j(x)∂u∂x j

)

+n

∑i=1

ai(x)∂ u∂xi

+a(x)u.

The operators B and C are given by similar differential expressions; moreover, ai j =a ji, bi j = b ji, and ci j = c ji.

We require that the function u(t,x) satisfies the homogeneous initial and bound-ary conditions (it is known that the case of inhomogeneous conditions can be re-duced to the homogeneous one by an appropriate change of the right-hand sideof (4.61))

u|t=0 = ut |t=0 = 0, u|x∈∂Ω =∂u

∂ μB|x∈∂Ω = 0, (4.62)

where μB = Bn is the conormal to the surface ∂Ω , B ={

bi j(x)}n

i, j=1 is the coeffi-cient matrix of the operator B, and n is the outward normal to the surface ∂Ω .

One can also study other “physical” boundary conditions corresponding to thesecond and third boundary value problem [21]. To avoid cumbersome expression,we carry out all considerations only for condition (4.62). All considerations canreadily be modified for the other cases.


Let H10 , W 1

0 , and V 10 be the completions of the set L0 of functions infinitely dif-

ferentiable in Q and satisfying the conditions

u|t=0 = ut |t=0 = . . .= 0, u|x∈∂Ω =∂ u

∂ μB|x∈∂Ω = 0,

with respect to the norms

‖u‖2H1

0=

n

∑i=1

∫

Qu2

txidQ+

n

∑i, j=1

∫

Qu2

txix jdQ,

‖u‖2W1

0=

n

∑i=1

∫

Qu2

txidQ+

n

∑i, j=1

∫

Qu2

xix jdQ,

‖u‖2V 1

0= ‖u‖2

W10+

n

∑i, j=1

∫

Ωu2

xix j|t=T dΩ . (4.63)

Let H1T , W 1

T , and V 1T be the completions of the set LT of functions infinitely dif-

ferentiable in Q and satisfying the adjoint conditions

v|t=T = vt |t=T = . . .= 0, v|x∈∂Ω =∂v

∂ μB|x∈∂Ω = 0,

with respect to the norms of the spaces ‖ · ‖H10, ‖ · ‖W1

0and ‖v‖2

V 1T= ‖v‖2

W1T+

∑ni, j=1 ‖v2

xix j|t=0‖2

L2(Ω), respectively.

Further, we need the definition of the spaces Hk0 ,H

kT ,W

k0 , . . . for arbitrary integer

k.1 By Hk0 we denote the completion of the set L0 in the norm ‖ · ‖Hk

0, where the

norm of the space Hk0 is defined by induction as:

‖ut‖Hk−10

= ‖u‖Hk0, ‖u‖Hk−1

0= ‖u‖Hk

0, u =

∫ t

0u(τ,x)dτ

for all k ∈ Z. The norm H10 is given by (4.63).

The remaining spaces are defined in a similar way (for the spaces HkT , W k

T , andV k

T with the subscript T the notation v should be understood as v =∫ t

T v(τ,x)dτ).

Lemma 4.17. There are dense continuous embeddings Hk0 ⊂V k

0 ⊂W k0 ⊂ Hk−1

0 andHk

T ⊂V kT ⊂W k

T ⊂ Hk−1T for arbitrary k ∈ Z. Moreover, W 0

0 ⊂ L2(Q), W 0T ⊂ L2(Q).

Proof. Let us prove the lemma, say, for the embedding V 2T ⊂W 2

T . (The remainingcases can be considered in a similar way).

Since ‖v‖W2T� ‖v‖V2

Tfor functions v ∈ LT , we need to verify the following con-

dition to prove the embedding V 2T ⊂W 2

T : for any sequence vn ∈ LT such that vn→ v0

in V 2T and vn→ 0 in W 2

T one has v0 = 0 [40].

1 We use the following notation of the spaces Hk0 ,V

k0 , . . .: the superscript indicates the number of

time derivatives, and the subscript indicates the type of initial conditions.


Let vn ∈ LT be such a sequence (i.e. vn → v0 in V 2T and vn → 0 in W 2

T ). Sincevn ∈ LT , we have (vn)t(T,x) = 0. Therefore, by using the Cauchy–Schwarz integralinequality, we obtain

|(vn)txi(0,x)|2 =∣

∣

∣

∫ T

0(vn)ttxi (τ,x)dτ

∣

∣

∣

2� T

∫ T

0(vn)

2ttxi

dτ.

By integrating this relation over the domain Ω , we obtain∫

Ω(vn)

2txi(0,x)dΩ � c

∫

Q(vn)

2ttxi

dQ � c‖vn‖2W2

T→ 0 (4.64)

as n→ ∞, here in throughout the following, c is a positive constant.The convergence of vn in the space V 2

T implies that (vn)txix j(0,x) is a Cauchysequence in L2(Ω). Let it converge to φi j(x) ∈ L2(Ω). By integrating by parts, weobtain

(

(vn)txi |t=0,uxj

)

L2(Ω)=−((vn)txix j |t=0,u

)

L2(Ω),

for all u∈C∞0 (Ω) (the set of compactly supported smooth functions in Ω ). By taking

into account (4.64) and by passing to the limit as n→∞, we obtain (φi j,u)L2(Ω) = 0.Since the function u is arbitrary, it follows that φi j = 0, i.e., ‖vn‖V 2

T→ 0 as n→ ∞.

Therefore, v0 = 0 and the embedding operator V 2T ⊂W 2

T is injective and continuous.Since the set LT is dense in both spaces V 2

T and W 2T , it follows that the embedding

V 2T ⊂W 2

T is dense. The proof of the lemma is complete. �By (Hk

0 )∗,(Hk

T )∗,(W k

0 )∗, . . . we denote the corresponding conjugate spaces. There

are natural bilinear forms defined on the pairs of primal and conjugate spaces; forexample, 〈·, ·〉Hk

0stands for the bilinear form on (Hk

0 )∗ ×Hk

0 .We set

〈Au,v〉Q =n

∑i, j=1

(ai juxi ,vx j )L2(Q) +n

∑i=1

(aiuxi ,v)L2(Q) + (au,v)L2(Q),∀u,v ∈W 0,12 (Q),

〈Au,v〉Ω =n

∑i, j=1

(ai juxi ,vx j )L2(Ω) +n

∑i=1

(aiuxi ,v)L2(Ω) + (au,v)L2(Ω),∀u,v ∈W 12 (Ω),

where W 0,12 (Q) is the completion of the set C∞(Q) in the norm ‖u‖2

W0,12 (Q)

=

∑ni=1 ‖uxi‖2

L2(Q) and W 12 (Ω) is the completion of the set C∞(Ω ) in the norm

‖u‖2W1

2 (Ω)= ∑n

i=1 ‖uxi‖2L2(Ω).

Consider the relation

〈Au,vtt 〉Q− (Bu,B∗vt)L2(Q) + 〈Cu,v〉Q = 〈 f ,v〉V 2T, (4.65)

where

B∗(v)≡−n

∑i, j=1

∂∂xi

(

bi j(x)∂v∂x j

)

−n

∑i=1

∂ (bi(x)v)∂ xi

+b(x)v. (4.66)


Relation (4.65) makes sense for arbitrary functions u∈H00 , v∈V 2

T , and f ∈ (V 2T )∗.

The coefficients of the operator are subjected to the following conditions:

(A) ai j,ci j,ai,ci,a,b,c ∈C(Ω ) and bi j, bi ∈C1(Ω ).

Let c∗ be a positive constant majorizing the corresponding norms of the coefficientsof the operator L .

The left-hand side in (4.65) specifies a linear operator L0 : H00 → (V 2

T )∗ and

the linear adjoint operator L ∗0 : V 2

T → (H00 )∗. Obviously, L0 is an extension of the

classical operator L (4.61).

4.7.2 A Priori Inequalities: Main Case

By c f we denote the positive constant in the Friedrichs inequality

‖u‖2L2(Q) � c f

n

∑i=1

‖uxi‖2L2(Q), ∀u ∈ ◦W 0,1

2 (Q), (4.67)

where◦

W 0,12 (Q) is the subspace of W 0,1

2 (Q) formed by functions vanishing on(0,T )× ∂Ω .

By using the Cauchy–Schwarz inequality in integral form and the Friedrichs in-equality, one can readily prove the estimate

‖Bu‖L2(Q) � c‖u‖W0,2

2 (Q), ∀u ∈ ◦W 0,2

2 (Q), (4.68)

where◦

W 0,22 (Q) is the completion of the set C∞(Q) of functions satisfying the bound-

ary conditions u = ∂u/∂ μB = 0 for x ∈ ∂Ω with respect to the norm ‖u‖2 =

∑ni=1 ‖uxix j‖2

L2(Q).

Lemma 4.18. Let condition (A) be satisfied. Then there exists a positive constantc > 0 such that the inequalities

‖L0u‖(V2T )∗ � c‖u‖H0

0, ‖L ∗

0 v‖(H00 )∗ � c‖v‖V2

T

are valid for arbitrary functions u ∈H00 and v ∈V 2

T .

Proof. One should apply the Cauchy–Schwarz inequality in integral form and theFriedrichs inequality (4.67) to the left-hand side of (4.65). �

Therefore, L0 and L ∗0 are continuous operators.

Remark 4.25. It follows from the definition of the operator L0 that R(L0)⊂ (W 2T )∗,

where R(L0) is the range of the operator L0, i.e.

L0u �∈ (V 2T )∗ \ (W 2

T )∗


for an arbitrary function u ∈H00 . We also have the estimates

‖L0u‖(V2T )∗ � c1‖L0u‖(W2

T )∗ � c2‖u‖H0

0, ‖L ∗

0 v‖(H00 )∗ � c1‖v‖W 2

T� c2‖v‖V2

T

for all u ∈ H00 and v ∈V 2

T , where c1,c2 > 0.

Let the operators A and B satisfy the ellipticity condition:

(B1) There exists a number α > 0 such that 〈Au,u〉Q � α ∑ni=1∫

Q u2xi

dQ for any

function u ∈W 0,12 (Q).

(B2) There exists a number αB > 0 such that ∑ni, j=1 bi jξiξ j � αB ∑n

i=1 ξ 2i for arbi-

trary ξi ∈R, x ∈Ω .

Lemma 4.19. (Main Lemma) Let conditions (A), (B1) and (B2) be satisfied. Thenthere exist positive constants λ , M and c such that for arbitrary functions u ∈ H0

0and v ∈V 2

T related by the formula u(t,x) = eMt(λvtt − vt + v), one has the equality

c〈L0u,v〉V 2T� ‖v‖2

V 2T+‖u‖2

V00.

Proof. Since the functions belonging to the space V 2T have well-defined traces

v(T,x) and vt(T,x) at the point t = T , we have v|t=T = vt |t=T = 0 for v ∈ V 2T .

We take into account the relationship between functions u and v and analyze〈L0u,v〉V 2

T= I1 + . . .+ I10. Let us consider each of the items Ii separately.

By inequality (B1),

I1 =⟨

A(eMt λvtt ),vtt⟩

Q � αλn

∑i=1

∫

QeMtv2

ttxidQ.

From the formula avvt = (av2)t/2 − atv2/2, the symmetry of the matrix{ai j}n

i, j=1, the condition vt(T,x) = 0 and inequality (B1), we have

I2 =−12

⟨

A(eMt vt),vtt⟩

Q =14〈Avt |t=0,vt |t=0〉Ω +

M4

⟨

A(eMt vt),vt⟩

Q

� α4

n

∑i=1

∫

Ωv2

txi|t=0dΩ +

αM4

n

∑i=1

∫

QeMtv2

txidQ.

Consider the item I3. By using the relation vtt =(

e−Mtu+ vt− v)

/λ , the formulafor integration by parts, and the notation

∫ t0 udτ = u, we obtain the estimate

I3 =−12

⟨

A(eMt vt),vtt⟩

Q =1

2λ〈Avtt , u〉Q−

12λ⟨

A(eMt vt),vt − v⟩

Q

=1

2λ 2

⟨

A(e−Mt ut), u⟩

Q +1

2λ 2 〈A(vt − v), u〉Q−1

2λ⟨

A(eMtvt),vt − v⟩

Q .


The item⟨

A(e−Mt ut), u⟩

Q can be estimated by analogy with I2. To estimate tworemaining items, we use the inequality

∣

∣〈Au,v〉Q∣

∣� c0

ε

n

∑i=1

∫

Qu2

xidQ+ ε

n

∑i=1

∫

Qv2

xidQ, ∀u,v ∈ ◦W 0,1

2 (Q),ε > 0, (4.69)

where c0 =(

c∗(n+ c f ))2

. To prove inequality (4.69), one should apply the Cauchy-Buniakovsky inequality and the Friedrichs inequality (4.67) to the definition of〈Au,v〉Q. By using inequality (4.69) for ε = 1, we obtain the estimate

I3 �α

4λ 2

n

∑i=1

∫

Ωe−Mt u2

xi|t=T dΩ +

αM4λ 2

n

∑i=1

∫

Qe−Mt u2

xidQ

− c1(λ )n

∑i=1

∫

QeMt v2

txi+ eMtv2

xidQ− c1(λ )

n

∑i=1

∫

Qe−Mt u2

xidQ,

where c1(λ ) is a sufficiently large constant depending on λ .Let us proceed to the item I4. By using the integration by parts formula, we obtain

I4 =⟨

A(eMtv),vtt⟩

Q =−〈Av|t=0,vt |t=0〉Ω −M⟨

A(eMtv),vt⟩

Q−⟨

A(eMt vt),vt⟩

Q .

To estimate the first term, we use inequality (4.69) with ε = α/8, the second term isestimated by analogy with I2, and for the third term, we again use (4.69) with ε = 1.

I4 �−n

∑i=1

∫

Ω

(

8c0

αv2

xi+

α8

v2txi

)

|t=0dΩ +αM

2

n

∑i=1

∫

Ωv2

xi|t=0dΩ

+αM2

2

n

∑i=1

∫

QeMt v2

xidQ− (c0 +1)

n

∑i=1

∫

QeMt v2

txidQ.

We split the item −(Bu,B∗vt)L2(Q) into three parts:

−(Bu,B∗vt)L2(Q) =−12(Bu,Bvt)L2(Q)−

12(Bu,Bvt)L2(Q) + (Bu,ΔBvt)L2(Q),

where ΔB = B−B∗.By analogy with I2, we have

I5 =−12

(

B(eMt λvtt ),Bvt)

L2(Q)=

λ4‖Bvt |t=0‖2

L2(Ω) +λM

4

∥

∥

∥B(

eMt/2vt)

∥

∥

∥

2

L2(Q).

It is known that the coercivity inequality [45] is valid for an elliptic operator B(αB > 0). Therefore, there exists a C > 0 such that

C‖Bu‖2L2(Q) +C‖u‖2

L2(Q) � ‖u‖2W0,2

2 (Q), (4.70)


for all u ∈ W 0,22 (Q) ∩ ◦W 0,1

2 (Q), where W 0,22 (Q) is the completion of C∞(Q) in

the norm

‖u‖2 =n

∑i, j=1‖uxix j‖2

L2(Q).

By applying the Friedrichs inequality (4.67) to (4.70), we obtain the estimate

‖Bu‖2L2(Q) �

1C‖u‖2

W0,22 (Q)

− c f

n

∑i=1

‖uxi‖2L2(Q). (4.71)

Therefore,

I5 �λ4C

n

∑i, j=1

∫

Ωv2

txix j|t=0dΩ +

λM4C

n

∑i, j=1

∫

QeMtv2

txix jdQ

−λc f

4

n

∑i=1

∫

Ωv2

txi|t=0dΩ − λMc f

4

n

∑i=1

∫

QeMt v2

txidQ.

Further, we have

I6 =12

(

BeMt vt ,Bvt)

L2(Q)� 0,

I7 =−12

(

BeMt v,Bvt)

L2(Q)=

14‖Bv|t=0‖2

L2(Ω) +M4

∥

∥

∥B(

eMt/2v)∥

∥

∥

2

L2(Q).

We again use inequality (4.71) and, arguing by analogy with the estimate I5, obtain

I7 �1

4C

n

∑i, j=1

∫

Ωv2

xix j|t=0dΩ +

M4C

n

∑i, j=1

∫

QeMt v2

xix jdQ

−c f

4

n

∑i=1

∫

Ωv2

xi|t=0dΩ − Mc f

4

n

∑i=1

∫

QeMtv2

xidQ.

To estimate the item I8, we use the integration by parts formula. Then we have

I8 =−12(Bu,Bvt)L2(Q) =−

12(But ,Bvt)L2(Q) =

12(Bu,Bvtt)L2(Q)

=1

2λ(

Bu,B(

e−Mt ut + vt − v))

L2(Q)

=1

4λ

∫

Ωe−Mt(Bu)2|t=T dΩ +

M

4λ

∫

Qe−Mt(Bu)2dQ+

12λ

(Bu,B(vt − v))L2(Q) .


This, together with (4.71) and (4.68), implies the estimate

I8 �1

4λC

n

∑i, j=1

∫

Ωe−Mt u2

xix j|t=T dΩ +

M4λC

n

∑i, j=1

∫

Qe−Mt u2

xix jdQ

− c f

4λ

n

∑i=1

∫

Ωe−Mt u2

xi|t=T dΩ − Mc f

4λ

n

∑i=1

∫

Qe−Mt u2

xidQ

− c1(λ )n

∑i, j=1

∫

Qe−Mt u2

xix jdQ− c1(λ )

n

∑i, j=1

∫

QeMt v2

txix j+ eMtv2

xix jdQ.

Consider the item

I9 = (BeMt λvtt ,ΔBvt)L2(Q) =−λ (Bvt |t=0,ΔBvt |t=0)L2(Ω)

−λ M(

BeMtvt ,ΔBvt)

L2(Q)−λ

(

BeMtvt ,ΔBvtt)

L2(Q).

Since ΔB is a first-order differential operator, we can use the inequality ab �−εa2− 1

4ε b2 and the Friedrichs inequality and readily show, by analogy with thepreceding, that

−λ (Bvt |t=0,ΔBvt |t=0)L2(Ω) �−λ8C

n

∑i, j=1

∫

Ωv2

txix j|t=0dΩ −λc2

n

∑i=1

∫

Ωv2

txi|t=0dΩ ,

−λ M(

BeMtvt ,ΔBvt)

L2(Q)�−λM

8C

n

∑i, j=1

∫

QeMt v2

txix jdQ−λMc2

n

∑i=1

∫

QeMt v2

txidQ,

−λ(

BeMtvt ,ΔBvtt)

L2(Q)�−αλ

3

n

∑i=1

∫

QeMtv2

ttxidQ−λ c2

∫

QeMt

n

∑i, j=1

v2txix j

+eMtn

∑i=1

v2txi

dQ.

We include all remaining items in I10 and estimate it by analogy with the lastthree inequalities.

I10 =(

BeMt(v− vt),ΔBvt)

L2(Q)+⟨

C(

eMt (λ vtt − vt + v))

,v⟩

Q �

−αλ3

n

∑i=1

∫

QeMt v2

ttxidQ− c1(λ )

n

∑i=1

∫

QeMt (v2

txi+ v2

xi

)

dQ

−c2

n

∑i, j=1

∫

QeMt(v2

txix j+ v2

xix j

)

dQ.

We take the sum I1 + . . .+ I10 and set λ = α/(4c f +16c2). Now we can readilychoose a constant M(λ )> 0 large enough to ensure that


c3〈L0u,v〉V 2T�

n

∑i=1

∫

QeMt (v2

ttxi+ v2

txi+ v2

xi

)

dQ+n

∑i, j=1

∫

Ωv2

txix j|t=0dΩ

+n

∑i, j=1

∫

Ωe−Mt u2

xix j|t=T dΩ +

n

∑i, j=1

∫

QeMt v2

txix j+ e−Mtu2

xix jdQ

for some positive constant c3(M,λ )> 0.In view of the inequality∫

Qe−Mt u2

xidQ =

∫

QeMt (λvttxi − vtxi + vxi)

2 dQ � c4

∫

QeMt (v2

ttxi+ v2

txi+ v2

xi

)

dQ,

we conclude that there exists a constant c > 0 such that c〈L0u,v〉V 2T� ‖v‖2

V2T+

‖u‖2V0

0. The proof of the lemma is complete. �

Theorem 4.17. Let conditions (A), (B1), and (B2) be satisfied. Then there exist con-stants ci > 0 such that

‖u‖V00� c1‖L0u‖(V2

T )∗ � c2‖L0u‖(W 2

T )∗ � c3‖u‖H0

0(4.72)

for all u ∈ H00 .

Proof. It suffices to prove the left inequality. For an arbitrary function u ∈H00 , con-

sider the ordinary differential equation (the Cauchy problem)

u(t,x) = eMt(λvtt − vt + v), v|t=T = 0, vt |t=T = 0.

Obviously, the solution v of this equation exists and belongs to the space v ∈V 2T . By

Lemma 4.19, c〈L0u,v〉V 2T� ‖v‖2

V 2T+ ‖u‖2

V00

. By using the Schwarz inequality and

the inequality a2 +b2 � 2ab, we obtain the estimates

c‖L0u‖(V2T )∗‖v‖V2

T� c〈L0u,v〉V 2

T� ‖v‖2

V 2T+‖u‖2

V00� 2‖v‖V2

T‖u‖V0

0,

which imply the assertion of the theorem. �Corollary 4.10. The operator L0 is injective.

4.7.3 Analysis of the System on the Basis of a Single Chainof a Priori Inequalities

Let us show that a meaningful solvability theory of the operator equation L u = fcan already be constructed on the basis of a single chain of a priori inequali-ties (4.72).


4.7.3.1 Adjoint A Priori Estimate

Our method for proving the a priori inequalities (4.72) can also be used in the proofof a similar chain of the adjoint operator

L ∗v≡ A∗(vtt )− (B∗)2(vt)+C∗(v).

Let the coefficients of the problem satisfy the following condition:

(C) ai j,ci j,a,b,c ∈C(Ω ) and bi j,ai,bi,ci ∈C1(Ω ).

Consider the linear operator L2 : V 20 → (H0

T )∗ given by the relation

〈L2u,v〉H0T= 〈Autt ,v〉Q +(But ,B

∗v)L2(Q) + 〈Cu,v〉Q, (4.73)

and the adjoint operator L ∗2 : H0

T → (V 20 )∗ given by the same relation (4.73). Obvi-

ously, L2 is the restriction of the operator L0 : H00 → (V 2

T )∗ and L ∗

2 is an extensionof the operator L ∗

0 : V 2T → (H0

0 )∗.

Theorem 4.18. Let conditions (B1), (B2) and (C) be satisfied. Then there exist con-stants ci > 0 such that

‖v‖V0T� c1‖L ∗

2 v‖(V20 )∗ � c2‖L ∗

2 v‖(W 20 )∗ � c3‖v‖H0

T(4.74)

for all functions v ∈ H0T .

Proof. Consider the leftmost inequality in (4.74). (The remaining inequalities areeasy to prove). For an arbitrary function v(t,x)∈H0

T , consider the function u(τ,x) =v(T−τ,x). One can readily see that u∈H0

0 . Let v(τ,x) be the solution of the Cauchyproblem

u(τ,x) = eMτ (λ vττ − vτ + v), v|τ=T = vτ |τ=T = 0.

Obviously, v ∈ V 2T . Therefore, the functions u and v satisfy the assumptions of

Lemma 4.19. By applying this lemma to the operators A1 = A∗, B1 = B∗, and C1 =C∗ (note that 〈Au,v〉Q = 〈A∗v,u〉Q, where A∗ is defined by analogy with (4.66)) weobtain the inequality

〈A∗u, vττ 〉Q− (B∗u,Bvτ)L2(Q) + 〈C∗u, v〉Q � c−1‖v‖2V 2

T+ c−1‖u‖2

V 00.

By performing the change of variables t = T − τ, we rewrite the last relation inthe form

〈Autt ,v〉Q +(But ,B∗v)L2(Q) + 〈Cu,v〉Q � c−1‖u‖2

V20+ c−1‖v‖2

V0T,

where u(t,x) = v(T − t,x) ∈V 20 .

By using the Schwarz inequality, we obtain

‖L ∗2 v‖(V2

0 )∗‖u‖V2

0� 〈L ∗

2 v,u〉V 20� 2c−1‖u‖V2

0‖v‖V0

T,

which completes the proof. �


Remark 4.26. We could have argued by analogy with Lemma 4.19, directly writingout a relation between functions v and u.

Corollary 4.11. The operator L ∗2 is injective.

By analogy with [59–62, 64] one can prove a theorem on existence and unique-ness on the basis of the chain (4.74).

Theorem 4.19. Let conditions (B1), (B2) and (C) be satisfied. Then for an arbitraryright-hand side f ∈ (V 0

T )∗, there exists a unique solution u∈V 2

0 ⊂H00 of the equation

L0u = f .

Corollary 4.12. The range R(L0) is dense in (V 2T )∗.

Remark 4.27. By analogy with Theorem 4.19, on the basis of estimate (4.72), onecan show that the range R(L ∗

0 ) of the adjoint operator contains (V 00 )∗, which, in

particular, implies that R(L ∗0 ) is dense in the space (H0

0 )∗.

4.7.3.2 Generalizes Solvability

It was shown in remark 4.25 that R(L0) ⊂ (W 2T )∗; in particular, R(L0) �= (V 2

T )∗.

Therefore, for functions f ∈ (V 2T )∗, we face the problem of finding some generalized

solution of the equation L0u = f .

Definition 4.13. A generalized solution of the equation L0u = f with right-handside f ∈ (V 2

T )∗ is an element u ∈ V 0

0 such that there exists a sequence ui ∈ H00 such

that ‖ui−u‖V00→ 0 and ‖L0ui− f‖(V 2

T )∗ → 0 as i→ ∞.

Remark 4.28. If L0u = f for u ∈ H00 , then, obviously, u is a generalized solution in

the sense of Definition 4.13.

Definition 4.14. A generalized solution of the equation L0u = f with right-handside f ∈ (V 2

T )∗ is defined as an element u ∈ V 0

0 such that 〈L ∗0 v,u〉V 0

0= 〈 f ,v〉V 2

Tfor

all v ∈V 2T : L ∗

0 v ∈ (V 00 )∗.

The following assertion can be proved on the basis of the chain of inequali-ties (4.72) by analogy with [62, 64, 75].

Theorem 4.20. Let conditions (B1), (B2) and (C) be satisfied. Then, for an arbi-trary right-hand side f ∈ (V 2

T )∗ of the equation L0u = f , there exists a unique gen-

eralized solution in the sense of Definitions 4.13 and 4.14. Generalized solutions inthe sense of Definitions 4.13 and 4.14 are equivalent.

Corollary 4.13. If a generalized solution u of the equation L0u = f belongs to thespace u ∈ H0

0 , then u is an ordinary solution of this equation and f ∈ R(L0). If u isa generalized solution of the equation L0u = f and f ∈ R(L0), then u belongs tothe space H0

0 and is an ordinary solution.

Proof. It suffices to note that the set of functions v ∈ V 2T : L ∗

0 v ∈ (V 00 )∗ is dense in

V 2T (since L ∗

0 (LT )⊂ (V 00 )∗). �


4.7.3.3 Another Type of Adjoint Inequalities

The chain of a priori inequalities (4.72) permits one to prove similar a priori esti-mates for the adjoint operator. However, there exist other estimates of the operatorL ∗

0 , which can be derived simply “by passing to the adjoint”. Moreover, these esti-mates permit explicitly indicating an extension of L0 corresponding to the definitionof generalized solutions of the equation L0u = f in the sense of Definitions 4.13and 4.14.

Consider the linear operator L0 : V 00 → (H2

T )∗ (and also the linear operator L ∗

0 :H2

T → (V 00 )∗)) given by the relation

〈L0u,v〉H2T= 〈Au,vtt〉Q +(Bu,B∗vtt)L2(Q) + 〈Cu,v〉Q, (4.75)

where u =∫ t

0 udτ .The right-hand side of relation (4.75) specifies an extension L0 : V 0

0 → (H2T )∗ of

the operator L0 : H00 → (V 2

T )∗ to the entire space V 0

0 . It is also obvious that L ∗0 is a

restriction of the operator L ∗0 .

The following assertion can be proved by analogy with Lemma 4.18.

Lemma 4.20. Let condition (A) be satisfied. Then there exist positive constantsci > 0 such that

‖L0u‖(H2T )∗ � c1‖u‖W0

0� c2‖u‖V0

0, ‖L ∗

0 v‖(V00 )∗ � c1‖L ∗

0 v‖(W 00 )∗ � c2‖u‖H2

T(4.76)

for arbitrary u ∈V 00 , v ∈ H2

T .

Therefore, L0 and L ∗0 are continuous operators. By analogy with Remark 4.25,

one can describe the range R(L ∗0 ) of the operator L ∗

0 in more detail. Indeed, itfollows from (4.76) that the range R(L ∗

0 ) is part of (W 00 )∗.


‖v‖V2T� c1‖L ∗

0 v‖(V00 )∗ � c2‖L ∗

0 v‖(W 00 )∗ � c3‖v‖H2

T(4.77)

for all v ∈ H2T .

Proof. It suffices to prove the left inequality. For an arbitrary function v ∈ H2T con-

sider the function u ∈ H00 ⊂ V 0

0 given by the relation u(t,x) = eMt(λvtt − vt + v).Since v ∈ H2

T , it follows that the function v(t,x) satisfies the conditions v(T,x) =vt(T,x) = 0. Therefore, one can use Lemma 4.19 for such functions u(t,x) andv(t,x). We have

c〈u,L ∗0 v〉V 0

0= c〈u,L ∗

0 v〉H00= c〈L0u,v〉V 2

T� ‖v‖2

V 2T+‖u‖2

V00.


By using the Schwarz inequality and the inequality a2 + b2 � 2ab, we obtain theestimates

c‖L ∗0 v‖(V 0

0 )∗‖u‖V0

0� c〈u,L ∗

0 v〉V 00� ‖v‖2

V 2T+‖u‖2

V00� 2‖v‖V2

T‖u‖V0

0,

which implies the assertion of the theorem. �By analogy with Theorem 4.19, on the basis of he chain of inequalities (4.77),

one can prove the solvability theorem.

Theorem 4.22. Let conditions (A), (B1), and (B2) be satisfied. Then for any f ∈(V 2

T )∗, there exists a unique solution u ∈V 0

0 of the equation L0u = f .

We can readily prove a theorem establishing a relationship between the general-ized solvability of the equation L0u = f in the sense of Definitions 4.13 and 4.14and the solvability of the operator equation L0u = f .

Theorem 4.23. Let conditions (B1), (B2) and (C) be satisfied, and let f ∈ (V 2T )∗.

Then the solvability of the equation coincides with the generalized solvability of theequation L0u = f in the sense of Definitions 4.13 and 4.14.

Proof. If u is a generalized solution of the equation L0u = f in the sense if Defini-tion 4.13 and ui ∈H0

0 is a sequence specifying this solution, then L0ui→ f in (V 2T )∗.

By virtue of the continuity of the embedding (V 2T )∗ ⊂ (H2

T )∗, we obtain L0ui→ f

in (H2T )∗. On the other hand, by virtue of the continuity of the operator L0, we have

L0ui = L0ui→ L0u in (H2T )∗. Hence, it follows that L0u = f .

But if u∈V 00 is a solution of the equation L0u = f , then, by Theorem 4.20, there

exists a generalized solution u∗ ∈V 00 of the equation L0u = f , which, as was shown

above, is also a solution of the equation L0u = f and coincides with u by virtue ofthe uniqueness. The proof of the theorem is complete. �

4.7.4 Construction of a “Scale” of Solvability Theorems

Let us show that the a priori inequalities (4.72) permit one to obtain an entire scaleof estimates and prove the corresponding solvability theorems.

4.7.4.1 Shift of the A Priori Inequalities By ∂ k/∂ tk

The a priori inequalities proved above can be “shifted” by the operator of differen-tiation with respect to the time variable. Let us give related considerations.

We introduce the following notation. Let u ∈ L0; by u(k) we denote its kth deriva-tive with respect to the variable t for k ∈ N∪ 0. If k ∈ Z and k < 0, then u(k) isunderstood as the function obtained by the |k|-fold application of the integrationoperator

∫ t0 dτ to the function u(τ,x). The notation v[k] (v ∈ LT ) is introduced in a

similar way with the use of the integral operator∫ t

T dτ .


Obviously, the mapping u→ u(k) (k ∈ Z) is an isometry between the space L0

with the norm Hl0 (or W l

0, V l0) and the space L0 with the norm Hl−k

0 (respectively,W l−k

0 , V l−k0 ). Consequently, this mapping can be extended by the continuity to the

entire space Hl0 (or W l

0, V l0). We preserve the same notation for the extended mapping

which is an isometry between Hl0 (respectively, W l

0 or V l0) and Hl−k

0 (respectively,W l−k

0 or V l−k0 ). One can readily see that (u(k))(l) = u(k+l). A similar property is valid

for the mapping v→ v[k].Note that the above-mentioned mappings should be used with care. For example,

the relation (u(1))(−1) = u can be represented in the form

∫ t

0u(1)(τ,x)dτ = u(t,x), ∀u ∈ H0

0 . (4.78)

However, the space H00 contains also smooth functions u(t,x) not vanishing for

t = 0. Nevertheless, formula (4.78) is valid for such functions2.Consider the operator Lk : Hk

0 → (V 2−kT )∗ (k ∈ Z) (and the adjoint operator L ∗

k :V 2−k

T → (Hk0)∗) given by the relation

〈Lku,v〉V 2−kT

= (−1)k(

⟨

Au(k),v[2−k]⟩

Q−(

Bu(k),B∗v[1−k])

L2(Q)

+⟨

Cu(k),v[−k]⟩

Q

)

.

Obviously, if k = 0, then the above-presented operator becomes the operator L0

considered earlierBy using integration by parts, one can readily justify the following assertion.

Lemma 4.21. The formulas Lk ⊂ Lk−1, L ∗k ⊂ L ∗

k+1 are valid for the operatorsLk, where A⊂ B means that A is a restriction of the operator B.


‖u‖Vk0� c1‖Lku‖

(V2−kT )∗ � c2‖Lku‖

(W 2−kT )∗ � c3‖u‖Hk

0. (4.79)

for all functions u ∈ Hk0 .

2 Relation (4.78) can be explained from the viewpoint of distribution. One can show that it ismeaningful to consider elements of the space H−1

0 as functionals on the space W 1T (more precisely

H−10 ⊂ (W 1

T )∗), where W 1

T is the completion of LT in the norm ‖v‖2W 1

T= ‖vt‖2

L2(Q). Then u(1) ∈H−10

should be treated as a functional acting as 〈u(1) ,v〉W 1T= −(u,vt)L2(Q) for all functions v ∈ W 1

T ,

u ∈ H00 , including smooth functions u ∈ H0

0 not satisfying the condition u(0,x) = 0. An importantdifference of u(1) from the Sobolev generalized derivatives the following: the functional u(1) isdefined on functions v ∈ W 1

T not necessarily vanishing for t = 0.


Proof. Let us prove the leftmost inequality. (The remaining inequalities can beproved in a similar way.) For an arbitrary u ∈ Hk

0 , we consider the ordinary dif-ferential equation (the Cauchy problem)

(−1)ku(k) = eMt (λvtt − vt + v), v|t=T = vt |t=T = 0.

Since u(k) ∈ H00 , it follows that the solution v of this equation exists and belongs to

the space v ∈V 2T . Then v[k] ∈V 2−k

T , and, by Lemma 4.19,

⟨

Lku,v[k]⟩

V 2−kT

= (−1)k(

⟨

Au(k),vtt

⟩

Q−(

Bu(k),B∗vt

)

L2(Q)+⟨

Cu(k),v⟩

Q

)

=⟨

L0

(

(−1)ku(k))

,v⟩

V 2T

� c−1‖v‖2V2

T+ c−1

∥

∥

∥(−1)ku(k))∥

∥

∥

2

V 00

= c−1‖v[k]‖2V 2−k

T+ c−1‖u‖2

Vk0.

Further, by applying the Schwarz inequality to the left-hand side of the last inequal-ity and the inequality a2 +b2 � 2ab to the right-hand side, we obtain

‖Lku‖(V2−k

T )∗‖v[k]‖V 2−kT

�⟨

Lku,v[k]⟩

V 2−kT

� 2c−1‖v[k]‖V 2−kT‖u‖Vk

0,

which completes the proof. �

4.7.4.2 Shift of the Adjoint Estimates and the General Solvability Theorem

On the basis of the general a priori estimates (4.79) , we prove inequalities for theadjoint operator. Consider the linear operator Lk :V k

0 → (H2−kT )∗ (and also the linear

adjoint operator L ∗k : H2−k

T → (V k0 )∗) given by the relation

〈Lku,v〉H2−kT

= (−1)k(

⟨

Au(k),v[2−k]⟩

Q+(

Bu(k−1),B∗v[2−k])

L2(Q)

+⟨

Cu(k−2),v[2−k]⟩

Q

)

.

Obviously, for k = 0 and k = 2, this definition is in agreement with the definition ofthe operators L ∗

0 and L ∗2 introduced above. The following assertion can be proved

by analogy with Lemma 4.21 with the use of integration by parts.

Lemma 4.22. One has Lk ⊂ Lk ⊂Lk−1, L ∗k ⊂ L ∗

k+1 ⊂L ∗k+1.


‖v‖VkT� c1‖L ∗

0 v‖(V 2−k

0 )∗ � c2‖L ∗0 v‖

(W2−k0 )∗ � c3‖v‖Hk

T(4.80)

for all v ∈ HkT .


Proof. We rewrite the desired inequalities in the following form:

‖v‖V2−kT

� c1‖L ∗0 v‖(Vk

0 )∗ � c2‖L ∗

0 v‖(W k0 )∗ � c3‖v‖H2−k

T

for all v ∈ H2−kT .

We only prove the leftmost inequality. Consider an arbitrary function v ∈ H2−kT .

Then v[−k] ∈H2T . Let u(t,x) ∈H0

0 ⊂V 00 be the function given by the relation

u(t,x) = (−1)keMt(

λ v[−k]tt − v[−k]

t + v[−k])

.

Since v[−k] ∈H2T , it follows that the function v(t,x) satisfies the conditions

v[−k](T,x) = v[−k]t (T,x) = 0.

Therefore, one can apply Lemma 4.19 to the functions (−1)ku and v[−k]. We have

c⟨

L0((−1)ku),v[−k]⟩

V 2T

� ‖v[−k]‖2V 2

T+‖u‖2

V00� 2‖v[−k]‖V2

T‖u‖V0

0=2‖v‖V2−k

T‖u‖V0

0.

(4.81)

By using the integration by parts formula, we obtain

⟨

L0((−1)ku),v[−k]⟩

V 2T

= (−1)k(

⟨

Au,v[−k]tt

⟩

Q−(

Bu,B∗v[−k]t

)

L2(Q)

⟨

Cu,v[−k]⟩

Q

)

= (−1)k(

⟨

Au,v[2−k]⟩

Q+(

Bu(−1),B∗v[2−k])

L2(Q)

⟨

Cu(−2),v[2−k]⟩

Q

)

=⟨

Lk(u(−k)),v

⟩

H2−kT

=⟨

L ∗k v,u(−k)

⟩

V k0

� ‖L ∗k v‖(V k

0 )∗‖u(−k)‖V k

0

= ‖L ∗k v‖(Vk

0 )∗‖u‖V0

0,

which implies the desired assertion. �Remark 4.29. Just as in the case of Theorem 4.18, one can give another proof of thechain of adjoint a priori inequalities.

The theorem on the unique solvability of the operator L can be proved by anal-ogy with Theorem 4.19 on the basis of the estimates (4.79) and (4.80).

Theorem 4.26. Let conditions (A), (B1), and (B2) be satisfied. Then for an arbi-trary right-hand side f ∈ (V k

T )∗, there exists a unique solution u∈V 2−k

0 of the equa-tion L2−ku = f .

Remark 4.30. For f ∈ (V kT )∗ one can introduce the notion of generalized solutions

of the equation L2−ku = f , prove their existence and uniqueness, and show thatthey are natural solutions of the equation L2−ku = f .

4.8 Projection Theorem for Banach and Locally Convex Spaces 97

Note that, for the most important special case B= 0, the preceding considerationscan readily be modified; moreover, the corresponding solvability theorems have theform of criteria, since the spaces V k

0 and Hk0 coincide in this case [76].

4.8 Projection Theorem for Banach and Locally Convex Spaces

As a rule, the existence of generalized solutions of boundary value problems maybe reduced to possibility to represent linear continuous functionals using a givenbilinear forms.

The classical Vishik–Lax–Milgram Theorem for Hilbert space is well known [48,113].

Theorem 4.27 (Vishik–Lax–Milgram). Let H be a Hilbert space, b be a bilinearform bounded on H×H. If there exists a number c > 0 such that c‖x‖2

H ≤ |b(x,x)|∀x ∈ H, then for any element f ∈ H there exists a unique element x ∈ H satisfyingthe following identity

b(x,y) = ( f ,y)H ∀ y ∈ H. (4.82)

This theorem is an effective tool for studying elliptic boundary value problems.However, it should be stressed that the natural generalized statements of evolu-tion problems for partial differential equations are not identities (4.82). Moreover,J.-L. Lions proved the following projection theorem for evolution problems.

Theorem 4.28 (J.-L. Lions [50]). Let F be a Hilbert space and Φ be a linear sub-space F with a new inner product (·, ·)Φ . Assume that

‖x‖F ≤ c‖x‖Φ ∀x ∈Φ,

where c > 0. Let b : F×Φ →R be a bilinear form and the following conditions aresatisfied

∀y ∈Φ x �→ b(x,y) ∈ F∗,

∃c1 > 0 : |b(y,y)| ≥ c1‖y‖2Φ ∀y ∈Φ.

Then, for every f ∈Φ∗, there exists x ∈ F that satisfies the identity

b(x,y) = f (y) ∀ y ∈Φ.

The following result is well-known for Banach spaces [24].

Theorem 4.29. Let E be a Banach space, F be a reflexive Banach space, b be abilinear form bounded on E×F. Then the following statements are equivalent:

(1) for any f ∈ F∗ there exists a unique x ∈ E such that

b(x,y) = 〈 f ,y〉F∗,F ∀y ∈ F ;


(2) (i) there exists c > 0 such that

c‖x‖E ≤ supy∈B1(F)

|b(x,y)| ∀x ∈ E;

(ii) if b(x,y) = 0 ∀x ∈ E, then y = 0.

Proof. Let us introduce an operator ET−→F∗ which acts by the following rule

E � x �→ T x = b(x, ·) ∈ F∗.

The operator T is linear and continuous, and

‖T x‖F∗ = ‖b(x, ·)‖F∗ = supy∈B1(F)

|b(x,y)| ≤M‖x‖E .

Let Statement 1 holds, i.e. the operator T is bijective. Then by the Banach theo-rem on inverse operator it is continuously invertible, i.e. ∃c > 0:

c‖x‖E ≤ ‖Tx‖F∗ .

It is equivalent to Statement (2) (i). Condition (2) (ii) means the totality of R(T ) inF∗. It is true when R(T ) = F∗.

Conversely, let Statement (2) holds. Statement (2) (i) imply that

c‖x‖E ≤ ‖Tx‖F∗ ,

i.e. the operator T is continuously invertible over R(T )⊆F∗ and the linear manifoldR(T ) is closed in F∗. Since R(T ) is total in F∗ (Property 2) (ii), then, taking intoaccount that F is reflexive, we obtain R(T ) = F∗. �Remark 4.31. If E is a reflexive Banach space, and F is a Banach space, then Condi-tions (1) and (2) of the Theorem 4.29 are reflexive also. If it is possible to representuniquely all elements of E∗ and F∗ using the norm b, the the spaces E and F arereflexive [24].

Let us generalized the Theorem 4.29 for locally convex linear topological spacesE and F .

Let E be a Hausdorff locally convex space (l.c.s.) with a conjugate space E∗.Recall that the set A∗ ⊆ E∗ is called almost closed, if the set A∗ ∩Uo is closedin topology σ (E∗,E) for every neighborhoods of zero U from E , where Uo ={

y∗ ∈ E∗ : supx∈U 〈y∗,x〉E∗,E ≤ 1}

is a polar of the set U [97].

Definition 4.15 ([97]). The space E is called perfectly complete, if every almostclosed linear subspace in E∗ is σ (E∗,E)-closed.

Remark 4.32. Every Frechet space is perfectly complete [97]. The space E∗ whichis conjugate to the Banach space E and is equipped with the topology σ (E∗,E) is a


perfectly complete space [97]. Strong conjugate space for reflexive Frechet space isperfectly complete [97].

Recall that a barrel in locally convex space is a convex balanced adsorbing andclosed subset. Every locally convex space has a fundamental system of zero neigh-borhoods that consist of barrels. A locally convex space is called barreled of everybarrel in it is a zero neighborhood [97].

The following statements hold [97].

Theorem 4.30 (on an open mapping). A continuous linear mapping of perfectlycomplete space to a Hausdorff barreled space is open.

Corollary 4.14. A bijective continuous linear mapping of a perfectly complete to aHausdorff barreled space is an isomorphism.

Let V be a fundamental system of closed convex and balanced zero neighbor-hoods of the space E , B be a fundamental system of bounded subset in the space F ,x �→ μO (x) be a Minkowski functional of the set O⊆ E .

Theorem 4.31. Let l.c.s. E be perfectly complete and barreled, l.c.s. F be semi-reflexive, b be a bilinear form which is continuous on E ×F. Then the followingstatements are equivalent:

(1) For any f ∈ F∗ there exists a unique x ∈ E such that

b(x,y) = 〈 f ,y〉F∗,F ∀y ∈ F ;

(2) (i) for any neighborhood O ∈V there exists a bounded set P ∈ B such that

μO (x)≤ supy∈P|b(x,y)| ∀x ∈ E;

(ii) if b(x,y) = 0 ∀x ∈ E, then y = 0.

Proof. Let us argue as in (4.29). Introduce a linear operator T : E→ F∗, which actsby the rule

E � x �→ T x = b(x, ·) ∈ F∗.

Let us equip the conjugate space F∗ with the strongest topology of uniform conver-gence β (F∗,F). The linear operator T : E→F∗ is continuous. Indeed, {Po : P ∈ B}is a fundamental system of convex and balanced zero neighborhoods in a strong con-jugate space F∗. Let us take P ∈ B, then the space T−1 (Po) is a convex balancedand adsorbing set. Let us show that the set T−1 (Po) is closed. Then the fact that thespace E is barreled implies thatT−1 (Po) is a zero neighborhood in E . Assume thatT x = b(x, ·) /∈ Po. Then sup

y∈P|b(x,y)| > 1, i.e, ∃ y′ ∈ P: |b(x,y′)| > 1. The fact that

the form b is continuous implies there exists a neighborhood O of the point x ∈ Esuch that |b(x′′,y′)|> 1 ∀ x′′ ∈ O, i.e., T x′′ /∈ Po ∀ x′′ ∈O.

Assume that Condition (1) is satisfied, i.e., R(T ) = F∗ and N (T ) = {0}. Sincethe space E is perfectly complete and the space F∗ equipped of the strongest topol-ogy of uniform convergence β (F∗,F) is barreled [97], then the operator T is anisomorphism (see corollary in 4.14) between E and the space F∗ with the topologyβ (F∗,F). Condition (2) (ii) follows from the fact that the operator T is surjective.


Let us prove that Condition (2) (i) is satisfied. Let us select a space O ∈V . Sincethe operator T is an isomorphism between E and F∗ with the topology β (F∗,F),then there exists a set P ∈ B such that Po = T (O). The fact that the operator T isinjective implies that T−1 (Po)⊆ O. Obviously,

μO (x)≤ μT−1(Po) (x)

for an arbitrary point x ∈ E . Let us show that the following equity holds:

μT−1(Po) (x) = supy∈P|b(x,y)| ∀x ∈ E, (4.83)

This completes the proof.The set Po ⊆ F∗ is a convex and balanced zero neighborhood in a strong conju-

gate space F∗. The space T−1 (Po) ⊆ E is convex, balanced and adsorbing. Let ustake a point x ∈ E . By definition we have

μT−1(Po) (x) = inf{

λ : λ > 0, x ∈ λT−1 (Po)}

.

If T x = b(x, ·) ∈ λPo for some λ > 0, then |b(x,y)| ≤ λ ∀ y ∈ P. Hence, we have

μT−1(Po) (x)≥ supy∈P|b(x,y)|.

Let ε > 0. Then we have 1supy∈P|b(x,y)|+ε |b(x,y)|< 1 ∀ y ∈ P, i.e.,

T x = b(x, ·) ∈(

supy∈P|b(x,y)|+ ε

)

Po.

Hence,μT−1(Po) (x)≤ sup

y∈P|b(x,y)|+ ε.

Taking into account that ε > 0 is an arbitrary value and the inequality proved above,we obtain equity (4.83).

Conversely, assume that Statements (2) (i) and (2) (ii) holds. Let us show thatN (T ) = {0}. Let T x = 0, i.e.

b(x,y) = 0 ∀ y ∈ F.

Then, ∀ O ∈ V μO (x) = 0. Since, ∀ O ∈ V we have that x ∈ O. Due to the fact thatthe space E is Hausdorff, we obtain that x = 0, i.e., the operator T is injective.


Statement (2) (ii) implies that the set R(T ) is total in F∗. Let us show that R(T )is closed linear subspace in a strong conjugate space F∗. Then the fact that the spaceF is semi-reflexive implies that R(T ) = F∗.

Let us take a neighborhoods O ∈ V and O ∈ V such that 2O ⊆ O. Then, since∃P ∈ B we have:

μO (x)≤ supy∈P|b(x,y)|= μT−1(Po) (x) ∀x ∈ E.

Hence, ∀x ∈ T−1 (Po) we have that

2μO (x)≤ μO (x)≤ μT−1(Po) (x)≤ 1.

If x∈ T−1 (Po), than x∈O, i.e., T−1 (Po)⊆O. Thus, R(T )∩Po⊆ T (O). Therefore,T is a continuous open (relatively) and bijective linear operator, which acts from Eonto R(T ). The subspace R(T ) with the topology induced by β (F∗,F) is perfectlycomplete [97], hence, it is closed in F∗. �Remark 4.33. The theorems of Lax–Milgram type were proved in [4, 32, 46, 84, 94,95, 104] also.

Chapter 5Computation of Near-Solutions of OperatorEquations

In previous chapters, we studied the concept of a near-solution of an operatorequation

A(x) = y, x ∈ E,y ∈ F, (5.1)

where A is a bounded injective linear operator, which acts from a Banach space Einto a Banach space F .

Recall that a strong near-solution described in Chap. 2 arises when we introducea topology TA induced in E by the norm ‖x‖E = ‖A(x)‖F , where E is a completionof E with respect to the norm ‖x‖E . In this case, the Banach space E is densely em-bedded into a Banach space E and the operator A can be extended by continuity ontothe entire space E . Denoting this extension by A, we obtain a bounded linear opera-tor A, which acts from E into F . The strong near-solution of (5.1) is such a sequencexn ∈ E , that A(xn) converges to y in the space F . The sequence xn is a Cauchy se-quence in the space E, and hence xn→ x as n→∞, where x is a generalized solutionof (5.1), i.e. A(x) = y.

5.1 Construction of Near-Solutions

Let E be a Banach space with a Schauder basis e1,e2, . . ., where en ∈E and F =H bea Hilbert space. Let us introduce the denotation A(ek) = ek ∈ F = H. It is easy tosee, that every element y ∈ R(A) can be represented in the following form:

y = A(x) =∞

∑k=1

αkek,

where x =∞∑

k=1αkek.

Hence, for any sequence εn of positive numbers that converges to zero and forany element y ∈ F there exists such an element

yn =n

∑k=1

β (n)k ek,


103

104 5 Computation of Near-Solutions of Operator Equations

for which

‖y− yn‖=∥

∥

∥y−n

∑k=1

β (n)k ek

∥

∥

∥< εn.

Consider a finite-dimensional subspace

Ln ={

u : u =n

∑k=1

γkek, γk ∈R

}

⊂ F

Let y∗n = ∑nk=1 β ∗k ek be the element of the best approximation of y in the sub-

space Ln:

‖y− y∗n‖= minβk

∥

∥

∥y−n

∑k=1

βkek

∥

∥

∥,

Then,

‖y− y∗n‖=∥

∥

∥y−

n

∑k=1

β ∗k ek

∥

∥

∥< εn.

Put

xn =n

∑k=1

β ∗k ek

Let us show that xn is a near-solution of operator equation (5.1). Indeed,

A(xn) = A( n

∑k=1

β ∗k ek

)

=n

∑k=1

β ∗k A(ek) =n

∑k=1

β ∗k ek = y∗n→ y

as n→ ∞.Thus, we would obtain near-solutions xn, if we could define the element of the

best approximation of y from the space Ln. This problem is reduced to solving ofthe system of linear algebraical equations with respect to β ∗k in case of the Hilbertspace F = H. Since the element y− y∗n is orthogonal to the subspace Ln, then

(y− y∗n, el) =(

y−n

∑k=1

β ∗k ek, el

)

= (y, el)−n

∑k=1

β ∗k (ek, el) = 0,

where l = 1,2, . . . ,n.Put alk = (ek, el), bl = (y, el). Then we obtain the following system of linear

algebraical equations

n

∑k=1

alkβ ∗k = bl, l = 1,2, . . . ,n. (5.2)

Thus, computation of a near-solution of operator equation (5.1) can be reducedto solving the system of linear algebraical equations (5.2) and this problem isparamount one for such computation.

5.2 Method of Neumann Series 105

5.2 Method of Neumann Series

Let us consider a system of linear algebraic equations

Ax = b, (5.3)

where A ={

ai j}n

i, j=1 is a non-degenerate matrix of order n (i.e. detA �= 0), x =

(x1,x2, . . . ,xn), b= (b1,b2, . . . ,bn) is an element of the space Rn. Using Gauss trans-formations, we obtain

A∗Ax = A∗b, (5.4)

where A∗ is a conjugate matrix. It is clear that system (5.4) is equivalent to originalsystem (5.3), but the matrix M = A∗A is symmetric and positively defined. So, with-out loss of generality we may consider that the matrix A of original system (5.3)is symmetric and positively defined. This implies that the matrix A has n positiveeigenvalues μ1,μ2, . . . ,μn (μk > 0). Let us show that there exists such a positiveconstant μ > 0 that the norm of the matrix U = I− μA (I is a unit matrix) in thespace Rn is less than one: ‖U‖< 1. Indeed, by the Hirsch–Bendixon Theorem

|μk| ≤ nq = δ , (5.5)

whereq = max |ai j|, (5.6)

ai j is an element of the matrix A.Put μ = 1/δ . Then

U = I−μA = I− 1δ

A. (5.7)

Note that inequality (5.5) and, therefore, the parameter μ can be defined moreexactly. According to [16], we will call the norm of matrix A any norm ‖A‖ definedin a vector space of matrices of order n2, for which the multiplicative inequality‖AB‖ ≤ ‖A‖×‖B‖ holds [16]. As it was stated in [16], modulo of every eigenvalueof a matrix does not exceed any possible norm. Note that the functional δ (A) = δ =nq = nmax |ai j| is a so-called M-norm M(A), so Hirsch–Bendixon inequality holds.Taking a N-norm

N(A) =

√

n

∑i, j=1

|ai j|2,

we have|μk| ≤ N(A) (5.8)

Since N(A)≤M(A) [16], inequality (5.8) is more precise than the Hirsch–Bendixoninequality and we can put δ equal to the number N(A) and even an arbitrary normof the matrix A. Analyzing the proof of the inequality ‖U‖ < 1, we note that thematrix U has the following eigenvalues γk = 1−μμk. Therefore, |γk|=

∣

∣1− μkδ∣

∣< 1.


Since 0 < μk/δ < 1, then ‖U‖= max |γk|< 1. Thus, the system of linear algebraicequations

Aμx = μAx = μb, μ = 1/δ (5.9)

is equivalent to system (5.3), but the matrix Aμ = μA can be represented asAμ = I−U , where ‖U‖ < 1. Using well-known results of matrix theory (see, forexample, [16]) we obtain, the Neumann series

A−1μ = (I−U)−1 = I +U +U2 + . . .+Un + . . . (5.10)

converges and since ‖U‖ < 1, then A−1μ can be approximated by a partial sum of

series (5.10)A−1

μ ≈ I+U +U2+ . . .+Uk =Uk. (5.11)

Thus, putting xk = Uk (μb) = μ(Ukb), we obtain that xk → x as k→ ∞, and hencexk can be considered as approximate solution of system (5.3). We will call thisapproach to solving system (5.3) the method of Neumann series.

Let us determine the error of kth approximation of this method. Put q= ‖U‖< 1,where ‖U‖ is a norm of U in the space Rn. Then

‖A−1μ ‖ ≤ ‖I‖+‖U‖+‖U2‖+ . . .+‖Uk‖+ . . .

≤ 1+‖U‖+‖U‖2+ . . .+‖U‖k + . . .

= 1+ q+ q2 + . . .+ qk + . . .

=1

1− q, (5.12)

That is why

‖x− xk‖=∥

∥A−1μ (μb)− μUk(b)

∥

∥= |μ |×∥∥(A−1μ −Uk

)

(b)∥

∥

≤ |μ |×(

‖U‖k+1+‖U‖k+2+ . . .)

×‖b‖

≤ |μ |×‖b‖× qk

1− q=Cqk,

where q = ‖U‖< 1, C = |μ|×‖b‖1−q .

Thus, we have proved the following inequality for xk:

‖x− xk‖ ≤Cqk, (5.13)

so xk converges to an exact solution at a rate of geometrical progression with ratioq < 1. This inequality allows to estimate not only the accuracy of the kth approxi-mation, but the norm of the inverse operator A−1 also, not computing in itself:

‖A−1‖= |μ |×‖A−1μ ‖ ≤

μ1− q

, μ =1δ. (5.14)

5.2 Method of Neumann Series 107

This estimation is based on the possibility of computing the Euclidian norm of theoperator U or, at least, the possibility to majorize this norm ‖U‖= q≤ q∗ < 1. Thisestimation we will call a majorant one.

The process of solving a system of linear algebraic equations can be consideredfrom the other point of view. Indeed, system (5.3) is equivalent to the system

(I−U)(x) = μb = ˜b, μ =1δ

or˜U(x) = ˜b+U(x) = x.

Thus, solving system (5.3) is equivalent to the fixed point problem for the operator˜U . Let us show that the operator ˜U is a contracting mapping, which acts in the spaceR

n with the Euclidean metric. Indeed, for all u,v∈Rn the following inequality holds

ρ(˜U(u), ˜U(v)) =∥

∥

∥

˜U(u)− ˜U(v)∥

∥

∥= ‖U(u)−U(v)‖≤ ‖U‖×‖u− v‖= q×ρ(u,v),

where q = ‖U‖< 1.It is known [29] that a fixed point x of the operator can be obtained as a limit of

a process of successive approximations (iterative process)

xn+1 = ˜U(xn), n ∈ N, (5.15)

where x0 is an arbitrary element from Rn, and the rate of convergence of the se-

quence xn to the solution x is determined by the inequality

ρ(xn,x)≤ qn

1− q×ρ(x1,x0). (5.16)

Thus, the solution of an arbitrary system of linear algebraic equations is reduced tothe fixed point problem.

Let us consider the relation between the iterative process (5.15) and an approxi-mate solution

xn = ˜b+U˜b+ . . .+Un˜b,

determined by the Neumann series method. It is easy to see that this approximatesolution xn can be obtained with the help of the process of successive approxima-tions (5.15), if the initial element x0 of the iterative process is ˜b: x0 = ˜b. Indeed,

xn = ˜b+U˜b+ . . .+Un˜b = ˜b+U˜b+ . . .+Un−2

˜b+Un−1(

˜b+U˜b)

= ˜b+U˜b+ . . .+Un−2˜b+Un−1

[

˜U(˜b)]

= ˜b+U˜b+ . . .+Un−2˜b+Un−1x1.


Further,

xn = ˜b+U˜b+ . . .+Un−2(

˜b+Ux1

)

= ˜b+U˜b+ . . .+Un−2x2

= . . .= ˜b+U(xn−1) = ˜Uxn−1 = xn.

Thus, an approximation xn computed with the help of the Neumann series coincideswith the nth iteration of the element x0 = ˜b. This fact allows to formulate a practicalrecommendation for the approximate solving system (5.3). It is known that one ofthe most difficult problem of the method of successive approximations is the selec-tion of the initial approximation x0. If this selection is successful then x1 ≈ x0 andthe second factor ρ(x1,x0) in (5.16) is small. Thus, we may hope that the right-handside of (5.16) will take small values. Using the Neumann series, we may take thevector ˜b as an initial approximation x0 of the iterative process (5.15), which can befar from the exact solution x in the Euclidean metric, so the value ρ(x1,x0) is notsmall. In such cases the method of the Neumann series can be wrong. However,if we will use these “wrong” approximate solutions as an initial approximation x0

instead of an arbitrary element x0 ∈ Rn, then successive refinements of the solu-

tion obtained with the help of the process of successive approximations can givesatisfactory results.

Let us pass to the solving of the majorant estimation q∗ of the norm of the ma-trix U . It is known that the Euclidean norm of the matrix U , i.e. the norm of thelinear self-conjugate of the operator y =U(x)is computed by the formula

q = M = sup‖x‖=1

(Ux,x) = supx�=θ

(Ux,x)(x,x)

= (Ux∗,x∗),

where x∗ is a normed eigenvector corresponding to the largest eigenvalue λ1 = M.Let us show that an arbitrary sequence xn convergent to an eigenvector x∗ as n→ ∞after normalization yn =

xn‖xn‖ will converge to x∗ and (Uyn,yn)→M as n→ ∞ also.

Indeed, since ‖xn‖→ ‖x∗‖= 1 as n→ ∞, then

‖xn− yn‖=∥

∥

∥

∥

xn− xn

‖xn‖∥

∥

∥

∥

=

∥

∥

∥

∥

xn

(

1− 1‖xn‖

)∥

∥

∥

∥

→ 0,

hence yn = xn +(yn− xn)→ x∗ as n→ ∞.From the other hand,

‖U(x∗)−U(yn)‖= ‖U(x∗)−U(xn)+U(xn)−U(yn)‖≤ ‖U‖×‖x∗− xn‖+‖U‖×‖xn− yn‖→ 0

as n→ ∞.Thus, we have proved that U(yn)→U(x∗) and yn→ x∗ as n→∞ and (Uyn,yn)→

(Ux∗,x∗) = M.

5.3 The Condition Number of Matrix 109

Taking into account that ‖yn‖= 1, we have the inequality

‖U(yn)‖ ≤ ‖U‖< 1,

so U(yn) < 1 and for large n the value q∗ = ‖U(yn)‖ can be used as an estimation‖U‖, but this estimation is not a majorant (the inequality ‖U‖≤ q∗ may not be hold).Yet, the inequality q∗ < 1 which is necessary for computing the condition numberof the matrix A holds.

We can take a sequence xn computed by the method of steepest descent as asequence convergent to x∗ and find the maximum of the functional

l(x) =(Ux,x)(x,x)

,

In this case, the normalizing is not necessary as ‖xn‖= 1 and the rate of convergence(Uxn,xn) = l(xn)→M is geometrical (see [16]).

5.3 The Condition Number of Matrix

The properties of approximate methods for solving a system of linear algebraicequations using Neumann series and successive approximations depend on the con-dition number of the matrix A. Let us give a non-formal definition of this concept.The inverse matrix A−1 is called stable if small changes of elements of the matrixA imply the small changes of elements of the inverse matrix [16]. The matrix A iscalled ill-posed if the inverse matrix is unstable. There are numerical parametersthat describe stability properties of a matrix [16]. These are Turing numbers

N =1n×N(A)×N(A−1), (5.17)

M =1n×M(A)×M(A−1) (5.18)

and Todd numbers

P =max |μi|min |μi| , (5.19)

H = ‖A‖×‖A−1‖, (5.20)

where ‖A‖ is the Euclidean norm of the matrix A. As it was pointed out in [16], thesecondition numbers do not describe the properties of a matrix completely. Thus, wewill start investigation of other condition numbers and consider their relations withequations and (5.17)–(5.20). Since all condition numbers depend on the norm of


the inverse matrix ‖A−1‖ it is desirable to propose conditional numbers that do notdepend on it. Let us introduce the following condition number:

τ∗(A) =1

1−‖U‖, (5.21)

where the matrix U is defined by the formula (5.7).Let τ1(A) and τ2(A) be an arbitrary pair of classical condition numbers

(5.17)–(5.20). It was pointed out in [90] that τ1(A) and τ2(A) are equivalent inthe following sense

τ1(Am)→+∞ ⇔ τ2(Am)→+∞, as m→ ∞,

where Am is a sequence of non-degenerated matrices of order n×n. The problem ofequivalence between τ∗(A) and an arbitrary classical condition number, for exam-ple, τ(A) = ‖A‖×‖A−1‖ arises in a natural way.

Theorem 5.1. Let Am be an arbitrary sequence of symmetric positively defined ma-trix of order n× n. Then τ(Am)→ ∞ as m→ ∞ iff τ∗(Am)→ ∞ as m→ ∞.

Proof. Necessity. Put τ(A) = ‖A‖×‖A−1‖, where ‖A‖ is the Euclidean norm. Thenwe have

τ(Aμ) = τ(μA) = τ(A)

and‖Aμ‖= ‖I−U‖ ≤ ‖I‖+‖U‖≤ 2.

By virtue of (5.12) the following estimation holds

‖A−1μ ‖ ≤

11−‖U‖ = τ∗(A).

Thus,τ(A) = τ(Aμ) = ‖Aμ‖×‖A−1

μ ‖ ≤ 2τ∗(A);

hence, if τ(Am)→ ∞ as m→ ∞, therefore, τ∗(Am)→ ∞ as m→ ∞.Sufficiency. Let τ∗(Am)→ ∞ as m→ ∞, then ‖Um‖ → 1 as m→ ∞, since Um =

I−μ (m)Am, μ (m) = 1δ (m) and δ (m) = M(Am), m ∈ N.

Let us denote by

μ (m)1 ≤ μ (m)

2 ≤ . . .≤ μ (m)n

the eigenvalues of the matrix Am, then μ (m)1 /δ (m)→ 0 as m→ ∞. Since the norms

δ (m) = M(Am) and ‖Am‖ = μ (m)n in R

n2are equivalent, then there exists such a

constant c > 0, that M(Am)≤ c‖Am‖. Therefore,

μ (m)1

δ (m)=

μ (m)1

M(Am)≥ μ (m)

1

c‖Am‖ =1c× μ (m)

1

μ (m)n

.

5.4 Hotteling Method for Correction Inverse Matrix 111

This inequality and the relation μ (m)1 /δ (m)→ 0 as m→∞ imply that μ (m)

1 /μ (m)n → 0

as m→ ∞; hence, τ(Am) = μ (m)n /μ (m)

1 → ∞ as m→ ∞. � The theorem implies that the Turing and Todd numbers (5.17)–(5.20) are equiv-

alent to the condition number (5.21). However, the condition number τ∗(A) is morepreferable than Turing and Todd numbers, since in this case it is not required tocompute the norm of the inverse matrix and it is sufficient to estimate the norm ofthe matrix U = I− μA, μ = 1

δ , δ = M(A).

5.4 Hotteling Method for Correction Inverse Matrix

A partial sum of the Neumann series Uk (5.11) can be considered as an approxima-tion of the inverse matrix A−1

μ of (5.9), since Uk→ A−1μ as n→∞. In this connection

we have to consider the problem of correction elements of the matrix Uk to ob-tain more precise approximation of A−1

μ . This problem was solved by Hotteling andSchultz [26, 99]. Suppose that we have such an approximation D0 of the inversematrix that

‖R0‖ ≤ q < 1, (5.22)

where R0 = I−AμD0, I is the unit matrix.Then, the elements of the inverse matrix A−1

μ can be determined by the followingiteration process:

Dm = Dm−1 (I+Rm−1) , Rm = I−AμDm, m ∈ N, (5.23)

In addition,

‖Dm−A−1μ ‖ ≤ ‖D0‖× q2m

1−q. (5.24)

Thus, we have a sequence of approximations Dm which quickly converges to A−1μ

(a number of exact decimal place increases with rate of geometrical progression)provided that (5.22) holds. Let us show that we can take any partial sum Uk as aninitial approximation D0 provided that k is equal or greater than 1: D0 =Uk as k≥ 1.Indeed,

‖R0‖= ‖I−AμD0‖= ‖I− (I−U)Uk‖

= ‖I− (I−U)(

I +U +U2 + . . .+Uk)

‖

= ‖Uk+1‖ ≤ ‖U‖k+1 = qk+1 = q < 1.


Now, we can formulate the combined method for solving a system of linearalgebraic equations Aμx = μb where μ = 1/δ :

1. Determine the matrix U = I−Aμ2. Determine a partial sum of the Neumann series Uk = I+U +U2+ . . .+Uk where

(k ≥ 1)3. Determine the matrix R0 = I−AμD0 where D0 =Uk

4. Determine the matrix Dm using (5.23)5. Determine an approximation xm by the formula

xm = Dm(μb). (5.25)

The error of the mth approximation is estimated by the formula

‖x− xm‖ ≤∥

∥A−1μ (μb)−Dm(μb)

∥

∥=∥

∥

(

A−1μ −Dm

)

(μb)∥

∥

≤ ∥

∥A−1μ −Dm

∥

∥×‖μb‖ ≤ ‖D0‖× |μ |×‖b‖× q2m

1−q

= ‖D0‖× |μ |×‖b‖× q(k+1)×2m

1− q(k+1)

=Cq(k+1)2m, (5.26)

where

C =‖D0‖×‖b‖× |μ |

1− qk+1 ≤ qk+1×‖b‖× |μ |1− qk+1 =C1, q = ‖U‖< 1.

Note that we used the Euclidean norm of the matrix A instead of the norm N(A)which was used by Hotteling and Schultz. It was necessary because the Euclideannorm of the smallest matrix norm and it allows to determine the initial approxima-tion R0 such that the inequality (5.22) holds; the norm N(A) does not guarantee thatwe can determine such R0.

It is easy to see that the Hotteling method is a partial case of the Newtonmethod: if to apply the Newton method to the equation f (x) = X−1− A = 0 weobtain the Hotteling method. However, the Hotteling method has high rate of con-vergence. Indeed, if we put in formula (5.26) for simplicity c = 1 and k = 1 then‖x− xn‖ ≤ q−2n+1

.

5.5 Exact Solving a System of Linear Algebraic Equations

In many cases the combined method coupled with the orthogonalization allows toobtain an exact solution of a system of linear algebraic equations. Let us considerthe orthogonalization and relative geometrical issues. Consider a non-dereneratedsystem of linear algebraic equations Ax = b:

5.5 Exact Solving a System of Linear Algebraic Equations 113

⎧

⎪

⎪

⎨

⎪

⎪

⎩

a11x1 +a12x2 + . . .+a1nxn = b1,a21x1 +a22x2 + . . .+a2nxn = b2,

. . .an1x1 +an2x2 + . . .+annxn = bn.

(5.27)

We will call the columns of A

a1 =

⎛

⎜

⎜

⎝

a11

a21

. . .an1

⎞

⎟

⎟

⎠

, a2 =

⎛

⎜

⎜

⎝

a12

a22

. . .an2

⎞

⎟

⎟

⎠

, . . . , an =

⎛

⎜

⎜

⎝

a1n

a2n

. . .ann

⎞

⎟

⎟

⎠

the basis vectors of the system Ax = b. Using the basis vectors a1,a2, . . . ,an and thevector b, we can rewrite the system in the following way:

x1a1 + x2a2 + . . .+ xnan = b, (5.28)

so that the solution of (5.27) is equivalent to the expansion of the vector b by thebasic vectors a1,a2, . . . ,an. Using the Hilbert-Schmidt orthogonalization process,we can orthogonalize the system of basis vectors a1,a2, . . . ,an and obtain the systemof vectors z1,z2, . . . ,zn. It is easy to see that we can obtain the exact solution of thesystem Ax = b by the following way:

xn =(b,zn)

(an,zn), xn−1 =

(b,zn−1)− xn (an,zn)

(an−1,zn−1), . . . ,

xn−k =(b,zk)− xk+1 (ak+1,zk+1)− . . .− xn (an,zn)

(an−k,zn−k), (5.29)

where 0≤ k ≤ (n−1).Unfortunately, the Hilbert-Schmidt orthogonalization process is not stable and

can lead to non-exact solution. However, having provided that the basis vectorsa1,a2, . . . ,an are normalized

a1 =a1

‖a1‖ , a2 =a2

‖a2‖ , . . . , an =an

‖an‖ ,

we have the systemx1a1 + x2a2 + . . .+ xnan = b, (5.30)

It is not equivalent to (5.28), but its solution is connected with the solutions (5.28)by the formula xk = ‖ak‖xk, where k = 1,2, . . . ,n.

Definition 5.16. The system of linear algebraic equations (5.27) is called normed,if all its basic vectors ai (i = 1,2, . . . ,n) have unit norm ‖ai‖ = 1. The operation oftransition from an arbitrary system (5.28) to the system (5.30) by normalizing itsbasic vectors is called normalization of a system.


It was mentioned above (see Sect. 5.3), that existing condition numbers do notcharacterize the matrix completely. The volume of simplex consisting of the basicvectors

conv{0, a1, . . . , an}= S(a1, . . . , an) =

{

u =n

∑k=1

αkak,n

∑k=1

αk ≤ 1, αk ≥ 0

}

,

is more informative.The volume Vs can be computed using the Gram determinant of the system

a1, a2, . . . , an. It is maximal if the basic vectors ak, k = 1,n form an orthogonalsystem. Therefore, if the volume Vs is small, then then matrix A is ill-posed. Thecorresponding mathematical tools were developed in [90].

We can reduce the system Ax = b to the equivalent form Aμx = μAx = μb = b(see Sect. 5.2) and compute the matrix Dm. Putting B=Dm, we obtain the equivalentsystem Ãx = DmAμx = Dmb = ˜b. Let us show that the system Ãx = ˜b has flatteningbasic vectors. Indeed, let ε be an arbitrary small positive number and the matrix Dm

be selected in such a way:∥

∥A−1μ −Dm

∥

∥≤ ε∥

∥Aμ∥

∥

.

Denote by a1, a2, . . . , an the basic vectors of the system Ãx = DmAμx = Dmb and lete1,e2, . . . ,en be orts in the space R

n, i.e.

e1 = (1,0,0, . . . ,0), e2 = (0,1,0, . . . ,0), . . . ,en = (0,0, . . . ,0,1).

Then,

‖ek− ak‖=∥

∥A−1μ Aμek−DmAμek

∥

∥=∥

∥

(

A−1μ −Dm

)

Aμek∥

∥

≤ ∥

∥A−1μ −Dm

∥

∥×∥∥Aμ∥

∥×‖ek‖≤ ε

∥

∥Aμ∥

∥

×∥∥Aμ∥

∥= ε.

Thus, the basic vectors of the system Ãx = DmAμx = Dmb = ˜b can be done arbi-trary close to the unit vectors ekand we can use the formulas (5.29) to compute anexact solution of the system Ãx = ˜b and, therefore, Ax = b.

5.6 Solving a System of Linear Algebraic Equationswith Guarantee Precision

At first, let us consider the concept of the precision of an approximate solution of asystem of linear algebraic equations. Let x be an exact solution of the system Ax = band x be an approximate solution. Let us introduce the following denotation:˜b = Axand fix some positive numbers α,β .

5.6 Solving a System of Linear Algebraic Equations with Guarantee Precision 115

Definition 5.17. The precision of an approximate solution x of a system of linearalgebraic equations Ax = b is a number

e(x) = α‖x− x‖2 +β‖b−˜b‖2,

where ‖x− x‖ = ρ(x, x) is the Euclidean distance (or distance in some finite-dimensional Banach space) between exact and approximate solutions, ‖b−˜b‖ isa discrepancy, α,β are the positive numbers.

In many cases (but not always!) we may suppose that α = 1,β = 0, i.e. to de-termine the precision of a solution as a square of a distance between exact andapproximate solutions, so hereinafter we will consider that e(x) = ‖x− x‖2. As arule, the precision e(x) is unknown as we do not know an exact solution x, but wewould be able to estimate e(x) if we would know the norm of the inverse matrix A−1

or majorize it, as far as

‖x− x‖=∥

∥

∥A−1b−A−1˜b∥

∥

∥≤ ‖A−1‖×‖b−˜b‖ (5.31)

In Sect. 5.2, we estimated (5.14) for the norm ‖A−1‖. To do this we used theEuclidean norm of the matrix U = I−μA or the majorant estimation q∗. Since theseproblems are quite difficult, it is helpful to use some easy computable norm forestimation of ‖U‖. It is easy to see, that the norms N(U) and M(U) are not suitablefor this purpose, as N(I) =

√n and M(I) = n, but we may use the operator norm of

the matrix U in the space l(n)1 (the so-called second norm)

‖U‖II = maxk

n

∑i=1|uik| , (5.32)

for which ‖I‖II = 1. In this connection, the following question arises: let A be apositively defined symmetric matrix; can we select such a number λ > 0 that thenorm ‖U‖II of the matrix U = I−λA is less then 1: ‖U‖II < 1?

Definition 5.18. A matrix A = {ai j}i, j=1,n has a diagonal domination, if for all

k = 1,n the following inequality holds

akk > (|a1k|+ |a2k|+ . . .+ |ank|− |akk|) = Ak

Theorem 5.2. If a symmetric matrix A with positive diagonal elements has a diag-onal domination then there exists such a number λ > 0 that the second norm of thematrix U = I−λ A is less than l: ‖U‖II < 1.

Proof. Let Λ be a diagonal matrix with diagonal elements λ1,λ2, . . . ,λn and U∗ =I−AΛ . Then

‖U∗‖II = ‖I−AΛ‖II = max1≤k≤n

{ n

∑i=1|λkaik|+ |1−λkakk|

}

.


Let us consider a function fk(λk):

fk(λk) = |λka1k|+ |λka2k|+ . . .+ |1−λkakk|+ . . .+ |λkank|

After transformations we have

fk(λk) = |λk|×(|a1k|+ |a2k|+ . . .+

∣

∣a(k−1)k

∣

∣+∣

∣a(k+1)k

∣

∣+ . . .+ |ank|)

+ |1−λkakk|= |λk|×Ak + |1−λkakk| ,

whereAk =

(|a1k|+ |a2k|+ . . .+∣

∣a(k−1)k

∣

∣+∣

∣a(k+1)k

∣

∣+ . . .+ |ank|)

.

If 0 < λk <1

akk, then λkakk < 1 and therefore 1−λkakk = |1−λkakk|. Thus,

fk(λk) = λkAk +1−λkakk = 1−λk(akk−Ak)

for all λk ∈(

0, 1akk

)

.

Since the matrix A has a diagonal domination and its diagonal elements are pos-itive, then akk > Ak. Put

γk = min

(

1akk

,1

akk−Ak

)

,

Then, we obtain that for all λk ∈ (0,γk) the inequality 0 < fk(λk)< 1 holds. Put γ =min

1≤k≤nγk > 0, the for all λ ∈ (0,γ) and for an arbitrary k = 1,2, . . . ,n the following

inequality holds:0 < fk(λ )< 1.

Thus, if the matrix Λ has the equal elements λk = λ , where λ ∈ (0,γ), then for thematrix U∗ = I−AΛ = I−λ A =U we obtain the following estimation:

‖U∗‖II = ‖U‖II = max1≤k≤n

fk(λ )< 1.

� This theorem allows to estimate the norm of an inverse matrix for some matrix of

an equivalent system of linear system of algebraic equations and estimate the preci-sion of an approximate solution. In addition, using Theorem 5.2 we can determinean approximate solution with a given precision. Indeed, using the combined methodfor solving a system of linear algebraic equations Ax= b (see Sect. 5.4), we can con-struct approximations D∗m for the inverse matrix A−1, which converge quickly to A−1

as m→ ∞. Let us consider an equivalent system D∗mAx = D∗mb = b∗. The matricesD∗mA converge to the unit matrix, so for some natural m the matrix B∗ = D∗mA willhave positive diagonal elements and a diagonal domination, so we may apply The-orem 5.2 to the matrix B∗ and construct a matrix U = I−λB∗ such that ‖U‖II < 1.In contrast with the Eulidean norm, the second norm ‖U‖II can be easily computed

5.7 Characterization of a Classic Solution Using Neumann Series 117

by the formula (5.32). The norm ‖U‖II allows to estimate the second norm of theinverse matrix (B∗)−1, and therefore the precision of an approximate solution x by

the formula (5.31) in the metric of the space l(n)1 , and therefore in the metric of theEuclidean space Rn.

Now, let us consider how to determine an approximate solution with a givenprecision. Note that the solution of this problem is required for the construction ofnear-solution of an operator equation Ax= y, x∈E,y∈H (see Sect. 5.1). Indeed, wecan compute approximations xn = ∑n

k=1˜βkek of the exact solution xn =∑n

k=1 β ∗k ek ofthe system (5.2) that forms a near-solution of (5.1) with a given precision δn > 0, i.e.‖xn− xn‖E < δn. If δn→ 0 as n→ ∞, the elements xn are the near-solutions of (5.1)also. Indeed,

0≤ ‖Axn− y‖H = ‖Axn−Axn +Axn− y‖H

≤ ‖A‖×‖xn− xn‖E +‖Axn− y‖H ≤ ‖A‖× δn+‖Axn− y‖H → 0

where n→ ∞.

5.7 Characterization of a Classic Solution Using Neumann Series

Let us consider an operator equation

Ax = y, (5.33)

where A is a compact injective linear operator, which acts in a separable Hilbertspace H, x,y ∈ H, y is the known element in H, and x is an unknown solution ofsystem (5.33). Using the Gauss transformations, i.e. applying an adjoint operatorA∗ to the left-hand and right-hand sides of (5.33), we obtain the operator equationA∗Ax = A∗y = y∗, where B = A∗A is an injective symmetric positively defined andcompact operator (we will say that such an operator satisfies the condition α)). Thus,we may suppose that the operator A of system (5.33) is symmetric and positivelydefined also, so it satisfies the condition α).

In an investigation of the operator equation (5.33) the following question arises:for which right-hand sides y∈H does the classical solution exist, and for which doesthe generalized solution exist? To answer this question, we can use the Neumannseries (see Sect. 5.2). Consider the operator U = I− μA, where μ = 1

‖A‖ and ‖A‖ isa norm of the operator A in the Euclidean space Rn. If A satisfies the conditionsα), then ‖U‖< 1. Unfortunately, this fact is not true, if H is a infinite dimensionalseparable Hilbert space. More precisely, if the operator A in an infinite dimensionalseparable Hilbert space H satisfies the conditions α), then for any non-negative μthe inequality ‖U‖= ‖I−μA‖> 1 holds. Hence, the Neuman series

A−1μ = (I−U)−1 = I +U +U2 + . . .+Un + . . . (5.34)

can diverge.


Therefore the formula (5.34) for the inverse operator A−1μ is not correct. However,

if we use the seriesy+Uy+U2y+ . . .+Uny+ . . . (5.35)

instead of (5.34) in the Hilbert space H for a fixed element y ∈ H, then we candetermine exactly when the series (5.35) converges or diverges. Let us determinethe structure of the operator U and its norm. It is known [40] that a self-adjointcompact operator A which acts in a Hilbert space H can be represented in the form

Ax =∞

∑i=1

λi (x,ei)ei, (5.36)

where ei is an orthonormal basis consisting of the eigenvectors of the operator Acorresponding to the non-zero eigenvalues λi �= 0 (if the operator A is definitelydefined, then λi > 0). Hence, the operator U has the form

Ux = Ix−μAx = x−∞

∑i=1

μλi (x,ei)ei

=∞

∑i=1

μ (x,ei)ei−∞

∑i=1

μλi (x,ei)ei

=∞

∑i=1

(1−μλi) (x,ei)ei,

So, the values ˜λi = 1− μλi are eigenvalues of the operator U , and ei are corre-sponding eigenvectors of U . Indeed, if x is an eigenvector of the operator A, andλ is its eigenvalue, then x is an eigenvalue of the operator U with the eigenvalueλ = 1− μλ ; if, vice versa, x is an eigenvalue of the operator U with an eigenvalueλ , then x is an eigenvalue of the operator A, and the corresponding eigenvalue isequal to λ = 1−λ

μ .Let us estimate the operator U . Since U is a self-adjoint operator, then we

have [40]

‖U‖= sup‖x‖=1

|(Ux,x)|= sup‖x‖=1

∣

∣

∣

∣

∣

( ∞

∑i=1

(1− μλi)(x,ei)ei,∞

∑k=1

(x,ek)ek

)

∣

∣

∣

∣

∣

= sup‖x‖=1

∣

∣

∣

∞

∑i=1

(1− μλi) (x,ei)2∣

∣

∣≥ 1, (5.37)

since λi→ 0 as i→ ∞. From the other hand, if μ = 1‖A‖ , then 0 ≤ 1− μλi ≤ 1, as

λi ≥ 0 and λi ≤ ‖A‖ (recall that A is a positively defined self-adjoint operator) [40].Therefore,

‖U‖= sup‖x‖=1

∣

∣

∣

∞

∑i=1

(1− μλi)(x,ei)2∣

∣

∣≤ sup‖x‖=1

∞

∑i=1

(x,ei)2 = 1.

Therefore, ‖U‖= 1.


Theorem 5.3. Let A be an injective linear positively defined self-adjoint compactoperator, which acts in a Hilbert space H and U = I − 1

‖A‖A, then the operatorequation

Ax = y (5.38)

has a classical solution x ∈ H iff the Neumann series

y+Uy+U2y+ . . .+Uky+ . . . (5.39)

converges in the Hilbert space H.

Proof. Sufficiency. Put y1 = μy, where μ = 1‖A‖ and suppose that for an element y

the series (5.39) converges. Let us show that the sum of the series

x = y1 +Uy1 +U2y1 + . . .+Uky1 + . . .

is a classical solution of the operator equation Ax = y. We have

Aμx = μAx = limn→∞

μA( n

∑k=0

Uky1

)

= limn→∞

n

∑k=0

(μA)Uky1

= limn→∞

n

∑k=0

(I−U)Uky1 = limn→∞

(

I+U + . . .+Un−U−U2− . . .−Un+1)y1

= limn→∞

(

I−Un+1)y1 = limn→∞

y1− limn→∞

Un+1y1 = y1 = μy,

as limn→∞

Un+1y1 = 0. Thus, Ax = y and hence the element x is a classical solution

of (5.38).Necessity. Let x be a classical solution of the operator equation (5.38). Denote by

Sny a partial sum of the series (5.39)

Sny = y+Uy+ . . .+Uny.

Then,Sny = Ax+UAx+ . . .+UnAx,

and therefore

Sny1 = μSny = μAx+U(μA)x+U2(μA)x+ . . .+Un(μA)x

= (I−U)x+U(I−U)x+U2(I−U)x+ . . .+Un(I−U)x

= Ix−Ux+Ux−U2x+ . . .+Unx−Un+1x = x−Un+1x.

Thus, μSny = x−Un+1x.Let us show that Un+1x→ 0 as n→ ∞ (it would imply that μSny→ x as n→ ∞,

i.e. the convergence of the series (5.39) to an element 1μ x). Using the method of


mathematical induction, we can prove that the nth order of the operator U has theform

Un(x) =∞

∑i=1

(

1− λi

‖A‖)n

(x,ei)ei.

Put νi = 1− λi‖A‖ , then 0≤ νi < 1 and

Un(x) =∞

∑i=1

νni (x,ei)ei.

Let x be an arbitrary element of the Hilbert space H and ε > 0. Since

x =∞

∑i=1

(x,ei)ei,

there exists such a natural number N, that

∥

∥

∥

∞

∑i=N+1

(x,ei)ei

∥

∥

∥=( ∞

∑i=N+1

|(x,ei)|2)1/2

<ε2.

Further, since νi ∈ [0,1), we can select such a number n0 ∈N, that for all n≥ n0 thefollowing inequality holds

∥

∥

∥

N

∑i=1

νni (x,ei)ei

∥

∥

∥=( N

∑i=1

νni |(x,ei)|2

)1/2<

ε2

for a fixed N. Thus, we have

‖Unx‖=∥

∥

∥

∞

∑i=1

νni (x,ei)ei

∥

∥

∥≤∥

∥

∥

N

∑i=1

νni (x,ei)ei

∥

∥

∥+∥

∥

∥

∞

∑i=N+1

νni (x,ei)ei

∥

∥

∥

≤ ε2+( ∞

∑i=N+1

ν2ni |(x,ei)|2

)1/2 ≤ ε2+( ∞

∑i=N+1

|(x,ei)|2)1/2

≤ ε2+

ε2= ε.

So, we proved that a sequence of operators Un converges to zero in the topology ofpointwise convergence. � Remark 5.34. Theorem 5.3 can be interpreted as a criterium of the belonging of aright-hand side y of an operator equation Ax = y to a range R(A) of the operatorA: y ∈ R(A) iff the Neumann series (5.39) generated by the element y convergesin a Hilbert space H. Therefore, if for a given y ∈ H the Neumann series (5.39) di-verges then the operator equation Ax = y has a generalized solution only; the inversestatement is true also.

Remark 5.35. It is easy to see that in Theorem 5.3 we used the linearity and conti-nuity of the operator A only.


Note that using the spectral theory of the linear operators we can simplify theproof of Theorem 5.3 and generalize it for a Banach space.

Let E be a Banach space, A : E → E be a linear continuous operator, defined ofthe entire space E . Let us consider an operator equation

x−Ax = y, (5.40)

where y is a given element in E .Let us use the method of Neumann series to solve (5.40)

xn = Axn−1 + y, (5.41)

where x0 ∈ E is a given initial approximation. It is easy to see that a sequence xn

generated by the iterative procedure (5.41) can be represented in the form

xn = y+Ay+A2y+ . . .+An−1y+Anx0. (5.42)

Let us show that there exists a simple relation between classical solvabilityof (5.40) and convergence of the sequence xn generated by the equalities (5.41)for a class of operators described below.

Definition 5.19. An operator A is called “correct”, if for any x ∈ E the sequence{Anx} converges in E .

Let us define the following linear operator B corresponding to the “correct” op-erator A in E:

Bx = limn→∞

Anx, x ∈ E.

The Banach–Steinhaus Theorem implies the fact that the operator B is bounded.In addition, for any elements y∗ ∈ E∗ and x ∈ E the following relation holds:

〈(A∗)ny∗,x〉= 〈y∗,Anx〉 → 〈y∗,Bx〉= 〈B∗y∗,x〉 , n→ ∞.

Thus, the sequence {(A∗)ny∗} converges to B∗y∗ in the topology σ(E∗,E). Usingthis fact and the condition of the strong convergence from the definition of a “cor-rect” operator A,we have

⟨

y∗,A2nx⟩

= 〈(A∗)ny∗,Anx〉 → 〈B∗y∗,Bx〉= ⟨

y∗,B2x⟩

, n→ ∞.

This property implies that the sequence {A2nx} converges to B2x in the topologyσ(E,E∗). Thus, we proved that B2 = B. In addition, since BAx = lim

n→∞An+1x = Bx =

ABx, then B = AB = BA. Hence, R(B) = N(I−A).Let us formulate the main result.

Theorem 5.4. Let A be a “correct” linear continuous operator, which acts in a Ba-nach space E. Then y ∈ R(I−A) iff for any x0 ∈ E the sequence (5.42) converges toa solution of (5.40) in a Banach space E.


Proof. Suppose that for any y ∈ E the sequence (5.42) converges to x ∈ E . Let usshow that x is a solution of (5.40) with the right-hand side y, i.e. y∈R(I−A). Indeed,since

Bxn = B(

y+Ay+A2y+ . . .+An−1y+Anx0)

= nBy+Bx0,

thenBy = 0.

Therefore, the following relations hold

(I−A)(y+Ay+A2y+ . . .+An−1y) = y−Any→ y−By = y, n→ ∞.

Further, we havey+Ay+A2y+ . . .+An−1y→ x−Bx0

as n→ ∞. Thus, we proved the equality (I−A)(x−Bx0) = y, but, from the otherhand, taking into account the condition R(B)⊂ N(I−A), we have that (I−A)(x−Bx0) = (I−A)x. Thus, we have

(I−A)x = (I−A)(x−Bx0) = y.

Now, suppose that y ∈ R(I−A). Then for any x ∈ E we have that y = x−Ax. Letus take an arbitrary element x0 ∈ E and consider a sequence

xn = y+Ay+A2y+ . . .+An−1y+Anx0, n≥ 1.

Hence,y+Ay+A2y+ . . .+An−1y = x−Anx.

Since Anx→ Bx and Anx0→ Bx0 as n→ ∞, then

xn→ x−Bx+Bx0, n→ ∞.

Moreover, it is easy to see that x−Bx+Bx0 is a solution of (5.40). Indeed,

(I−A)(x−Bx+Bx0) = x−Ax+(I−A)B(x0− x) = y,

as R(B)⊂ N(I−A). � Note, that the “correct” self-adjoint operators which act in a Hilbert space, are

described simply.

Statement 5.1 ([66]). Let A be a self-adjoint linear continuous operator, which actsin a Hilbert space H. Then the operator A is “correct” iff ‖A‖ ≤ 1 and−1 is not aneigenvalue of the operator A.

Proof. Let ‖A‖ ≤ 1. Then,

A =∫ 1

0λ dEλ .


Let us show that for any x ∈H the sequence

Anx =∫ 1

0λ n dEλ x

converges in H as n→ ∞.Denote by H1 a proper subspace of the operator A corresponding to an eigenvalue

λ = 1 (it is possible that H1 = {θ}), and denote by Hδ2 and Hδ

3 the subspaces

x =∫ 1−δ

0dEλ x.

x =

−1+δ∫

−1

dEλ x+

1∫

1−δ

dEλ x,

respectively (0 < δ < 1). Then, H = H1⊕Hδ2 ⊕Hδ

3 . Let P1, P2 and P3 be the orthog-onal projectors onto the subspaces H1, Hδ

2 , and Hδ3 , respectively. Then,

Anx = AnP1x+AnP2x+AnP3x = P1x+AnP2x+AnP3x (5.43)

Let us estimate the second and the third term in (5.43)

‖AnP2x‖= ‖1−δ∫

−1+δ

λ ndEλ P2x‖ ≤ (1− δ)2 ‖P2x‖,

‖AnP3x‖ ≤ ∥

∥An−1P3x∥

∥≤ ...≤ ‖P3x‖.

Since

‖P3x‖=−1+δ∫

−1

d(Eλ x,x)+

1∫

1−δ

d(Eλ x,x)

and the number−1 is not a point of discontinuity of the function λ �→ (Eλ x,x), thenfor each of the fixed elements x ∈ H we have

limn→∞‖P3x‖= 0.

Let ε be an arbitrary positive number. Select δ > 0 such that the inequality ‖P3x‖ ≤ε2 holds. Then we have for sufficiently large n

‖An(P2x+P3x)‖ ≤ (1− δ )n‖P2x‖+ ε2< ε,

i.e. the sequence Anx converges to P1x.


Conversely, let the operator A be “correct”. Suppose that ‖A‖ > 1. For any ε ∈(0,‖A‖−1) consider a subspace Hε consisting of an element

x =∫ 1+ε

1+ ε2

dEλ x.

Take an arbitrary element x∈Hε . We have the following estimation for the sequence{Anx}:

‖Anx‖=∥

∥

∥

∫ 1+ε

1+ ε2

λ n dEλ x∥

∥

∥≥(

1+ε2

)n‖x‖→+∞,

This contradicts the “correctness” of the operator A.The convergence of the method of simple iteration for solving the equation x−

Ax = f with a linear self-adjoin operator A, which acts in a Hilbert space H wasstudied in [38]. It was supposed that ‖A‖ = 1. Let us formulate the main result –Krasnoselskii’s Theorem.

Theorem 5.5 (M. A. Krasnoselskii). Let −1 be not an eigenvalue of the operatorA. Let the equation x−Ax = f have a solution for a given f ∈ H (possibly, non-unique). Then for any initial approximation x0 ∈ H the successive approximations

xn+1 = Axn + f , n = 0,1,2, ...

converge to a solution of the operator equation x−Ax = f .

It is clear that Theorem 5.4 and Statements 5.1 imply the Krasnoselskii’sTheorem stated above.

� Theorem 5.6. Let A be a self-adjoint non-negatively defined linear continuous op-erator, which acts in a Hilbert space H and U = I− 1

‖A‖A. Then, y ∈ R(A) iff for anyx0 ∈ H the sequence

xn = y+Uy+U2y+ . . .+Un−1y+Unx0

converges in the space H.

Proof. The operator U is “correct” and R(I−U) = R(A). �

Chapter 6General Scheme of the Construction ofGeneralized Solutions of Operator Equations

In this chapter, we will consider a general approach to the construction of general-ized solution of a linear operator equation.

Let E,F be linear topological spaces and L : E→ F be a linear operator definedon the set D(L ) = E . Suppose that the operator L : E → F is invertible, i.e. theequation L u = f have no more than one solution. In theory, we always can achievethis by considering a narrowing of the operator L onto the factor space E/KerL . Ifthe range R(L ) of the operator L does not coincide with F then for the right-handside f ∈ F \R(L ) there arises a problem of the construction of some generalizedsolution of the equation L u = f (if f ∈ R(L ), we will call the element u ∈ E :L u = f a classical solution as before). The natural approach to this problem isthe following one: let us introduce in E and F the topologies TE , TF , which areconsistent with the structures of the linear spaces E and F , so that in the lineartopological spaces (E,TE), (F,TF ) the operator L acts continuously and the right-hand side f ∈ F \R(L ) of the equation L u = f belongs to the closure R(L ) of theset R(L ) in the linear topological space (F,TF) (it’s an ideal case when R(L )=F).Further, let us extend the operator L by continuity onto the completion of the spaceE in “topology TE”, where we will look for a generalized solution of the equationL u = f with the right-hand side f ∈ F \R(L ).1

6.1 Generalized Solution of Linear Operator Equationsin Locally Convex Topological Spaces

Let E,F be linear spaces; E ′,F ′ be the corresponding algebraic conjugate spaces.Let E∗,F∗ be such linear spaces that (E,E∗), (F,F∗) form dual pairs. It is clear, thatE∗ ⊂ E ′, F∗ ⊂ F ′.

Let us consider a linear injective operator L : E→ F , defined on the entire spaceE , L ′ : F ′ → E ′ is an algebraically adjoint operator.

1 Namely, such schemes were implemented in the previous chapters.


125

126 6 General Scheme ...

Let us suppose that the operator L is weakly continuous (i.e. it is continuousin the weak topologies σ(E,E∗) and σ(F,F∗)). Under the assumptions made thenarrowing of the operator L ′ onto the space F∗ specifies the operator L ∗ : F∗ →E∗ ⊂ E ′, where ϕ ∈ D(L ∗) = F∗, i.e. for all ϕ ∈ F∗ the element L ′ϕ = L ∗ϕbelongs to the space E∗ [97].

In addition, suppose that the range R(L )⊂ F of the operator L is a total subsetof the space F in duality (F,F∗). Consider the linear equation

L u = f (6.1)

and pass to the determination of a generalized solution for this equation.Let us consider U = {α} which is the system of non-empty centrally symmetric

subsets of the space F∗ satisfying the following conditions:

(1) The union of two arbitrary sets from U is contained in some set from U .(2) The product of an arbitrary set α ∈ U by an arbitrary real number λ > 0 is a

set from U .(3) Every set α and U is bounded in F∗ with respect to the topology σ(F∗,R(L )).(4) The set N =

⋃

α∈Uα is total in F∗ with respect to the duality (F∗,R(L )).

It is easy to prove that every of the sets L ∗(α), where α ∈ U , is bounded inthe space E∗ with respect to the topology σ(E∗,E). Indeed, if we suppose thatsome of the sets L ∗(α0) are unbounded in E∗, then there exists such a sequenceln ∈L ∗(α0) and an element u ∈ E that ln(u) ≥ n. This implies the existence of asequence ϕn ∈ α0 ⊂ F∗ (L ∗ϕn = ln) such that

ln(u) = (L ∗ϕn)(u) = ϕn(L u)≥ n.

The latter inequality contradicts the condition (3).It is also easy to prove, that M=L ∗(N) is a total subset E∗ in duality (E∗,E). In-

deed, if there exists u∈E , such that l(u) = 0 for all l ∈M=L ∗(N) then ϕ(L u)= 0for all ϕ ∈N. By virtue of the totality of the set N in duality (F∗,R(L )) we get thefact that L u = 0, and the injectivity of the operator L implies that u = 0.

Let us consider on a linear set E the topology TE of the uniform convergencedefined by a system of neighborhoods of zero

oα = {u ∈ E | |(L ∗ϕ)(u)| ≤ 1,ϕ ∈ α}, α ∈U

in other words, by the system of semi-norms

pα(u) = infλ>0, 1

λ u∈oαλ = sup

ϕ∈α|(L ∗ϕ)(u)| u ∈ E, α ∈U .

The set E with this topology we will denote by ET .It is easy to see that ET is a Hausdorff locally convex linear topological space.

Let us denote by ET a completion ET with respect to the topology TE (or, tobe more precisely, with respect to the corresponding Hausdorff uniform structure

6.1 Generalized Solution of Linear Operator Equations in Locally Convex... 127

[8, 97]). The semi-norms pα(u) allow the extension by continuity to ET (we willdenote these extensions by pα ) and the system of semi-norms pα specifies the topol-ogy of the space ET .

Similarly, let us consider a topology TF defined by a system of neighborhoodsof zero in the set R(L )

Oα = { f ∈ R(L ) | |ϕ( f )| ≤ 1,ϕ ∈ α}, α ∈U

or by a system of semi-norms

Pα( f ) = infλ>0, 1

λ f∈Oαλ = sup

ϕ∈α|ϕ( f )|, f ∈ R(L ), α ∈U .

The set R(L ) with respect to this topology (let us denote it by RT ) turns intoa Hausdorff locally convex linear topological space. Denote by RT the comple-tion RT .

It is easy to see that the operator L realizes an isomorphism between ET andRT (i.e. an isomorphism between linear locally convex topological spaces). Indeed,for an arbitrary α ∈U

Pα(L u) = supϕ∈α|ϕ(L u)|= sup

ϕ∈α|(L ∗ϕ)(u)|= pα(u), u ∈ E.

Thus, the operator L : ET → RT is continuous and the topology TE is the weakestof all topologies on E conserving continuity of the operator L : E→ RT .

Let us extend the operator L onto the whole space ET . Let E be the minimalCauchy filter in the space ET [8]. Taking into account the isomorphism mentionedabove, F = L (E ) is the minimal Cauchy filter in the space RT . Such extensionof the operator L , which we shall denote hereinafter by L , is a linear continuousinjective operator, which realizes an isomorphism between the spaces ET and RT .

If to require the following additional condition:

(5) Every of the sets α ∈U is bounded in the space F∗ with respect to the topologyσ(F∗,F) and N is a total subset F∗ in duality (F∗,F),

then the topology TF is naturally extended onto the whole space F , and it makessense to compare elements from the spaces F and RT with each other. Let us denotethis topology by TF as before, but now it is defined on the whole space F . The spaceF with the topology TF we will denote by FT , and corresponding completion byFT . Obviously, RT is a closed linear subset of FT , which can coincide with FT .

Definition 6.20. A generalized solution of the equation L u = f is such an elementu ∈ ET that L u = f .

It is easy to see that a classic solution of the equation L u = f is a generalizedsolution also. If u is a generalized solution and f ∈ R(L ) (or u ∈ E) then u is aclassic solution.

Thus, the following theorem holds.


Theorem 6.1. Let conditions (1)–(5) be true, then for an arbitrary element f ∈ F ∩RT there exists a unique generalized solution of the equation L u = f .

Proof. The theorem follows from the fact that the operator L specifies an isomor-phism between ET and RT . �

Note that an arbitrary functional l ∈ M allows extension by continuity to thewhole space ET . Indeed, there exists such α0 ∈U that ϕ ∈ α0, L ∗ϕ = l, i.e. in theneighborhood

oα0 = {u ∈ E | |(L ∗ϕ)(u)|< 1,ϕ ∈ α0} ∈ TE

the functional l is bounded. The extension of l to ET we will denote by l, and the setof all extended functionals l, where l ∈M we will denote by [M]. Thus, M⊂ (ET )∗,[M] ⊂ (ET )∗, where (ET )∗,(ET )∗ are conjugate spaces to ET , ET respectively.Similarly, the functionals ϕ ∈ N allow extension onto FT , i.e. N ⊂ (FT )∗. We willdenote extended functionals by ϕ and the set of all extended functionals by [N].

Let us consider one more definition of generalized solution, the analogue ofwhich was studied in [5, 33, 62, 63, 89, 91].

Definition 6.21. A generalized solution of the equation L u = f is such an elementu ∈ ET that for all l ∈M the following equality holds

l(u) = ϕ( f ), L ∗ϕ = l.

It is easy to prove that if u is a classic solution, then u ∈ ET is a generalizedsolution in the sense if Definition 6.21. Indeed, if L u = f then for all ϕ ∈ F∗ wehave ϕ(L u) = ϕ( f ) or l(u) = (L ∗ϕ)(u) = ϕ( f ) for all l ∈ R(L ∗) in particular foran arbitrary l ∈M also. This implies that l(u) = ϕ( f ) for all l ∈M.

Theorem 6.2. Let the system of sets U satisfy conditions (1)–(5). Then for an arbi-trary right-hand side f ∈ F ∩ RT there exists a generalized solution u ∈ ET of theequation L u = f in the sense of Definition 6.21.

Proof. Let O( f ) = {Oα( f )}α∈U be a set of neighborhoods of the point f ∈ RT intopology TF of the space FT . Since the set R(L ) is dense in RT in topology TF

then the set {R(L )∩ Oα( f )}α∈U forms a basis of some filter F , which majorizesthe filter O( f ). Thus, F converges to f in FT . In addition, since R(L )∩Oα( f )⊂Fthen {R(L )∩ Oα( f )}α∈U is a basis of a Cauchy filter in the space FT . It is clearthat the set of pre-images {R(L )∩ Oα( f )}α∈U under the mapping L : E → Fforms a basis of some filter E in the space ET .

It is clear that E is a Cauchy filter. Indeed,it is sufficient to show that for everyα0 ∈U there exists such a neighborhood O∗( f ) ∈ O( f ) that the set {L −1(R(L )∩O∗( f )) } is small of order oα0 . Take as O∗( f ) the following neighborhood of thepoint f :

O∗( f ) = {g ∈ FT | |ϕ( f −g)| ≤ 1, ϕ ∈ 2α0}.

6.1 Generalized Solution of Linear Operator Equations in Locally Convex... 129

Then for arbitrary elements u1,u2 from the set L −1(R(L )∩ O∗( f )) we have thatL u1,L u2 ∈ O∗( f ), i.e |ϕ( f −L u1)| ≤ 1, |ϕ( f −L u2)| ≤ 1 for all ϕ ∈ 2α0.Hence, |ϕ(L u1 −L u2)| ≤ 2 for all ϕ ∈ 2α0 or |(L ∗ϕ)(u1−u2)| ≤ 1 for allϕ ∈ α0. Thus, the difference (u1−u2) belongs to oα0 , whence it follows that E is aCauchy filter in ET .

Since ET is a complete space, then a basis E of some Cauchy filter in ET con-verges to an element u0 ∈ ET . In addition, limE L = f in FT .

That is why limE l = limE l = l(u0) for all l ∈M. From the other hand, for everyfunctional l ∈M⊂ R(L ∗) there exists ϕ ∈ F∗ such that l = L ∗ϕ = ϕ ◦L . That iswhy

limE

l = limE(ϕ ◦L ) = ϕ

(

limE

L)

= ϕ(

limE

L)

= ϕ( f ).

Thus, u0 is a generalized solution of the equation L u = f in the sense ofDefinition 6.21. �

Let us study the uniqueness of a generalized solution in the sense ofDefinition 6.21.

Lemma 6.1. A generalized solution of the equation L u = f in the sense ofDefinition 6.21 is unique iff the set [M] is total in the duality ((ET )∗, ET ).

Proof. If u1,u2 ∈ ET are different generalized solutions of the equation L u = fthen l(u1−u2) = 0 for all l ∈M. That is why the condition u1 = u2 is equivalent tothe totality of the set [M] in the duality ((ET )∗, ET ). �Remark 6.36. It is easy to specify a simple sufficient condition of the totality of theset [M] (Lema 6.1). Namely, if L ∗(α) are compact sets in M =L ∗(N) with respectto the topology σ(M,E) for every α ∈U , then by the Mackey–Arens Theorem [8]the topology TE is matched with the duality (E,M). This means that (ET )∗ =M, hence (ET )∗ = [M]. Thus, [M] is a total set in the duality ((ET )∗, ET ), i.e. ageneralized solution in the sense of Definition 6.21 is unique.

Let us consider the extension of this condition. This will allow us to guaranteethe totality exactly of the set M. At first, let us remind the statement on embeddingsof completions of uniform spaces.

Statement 6.2. Let us suppose that over the set L two Hausdorff uniform structuresU1 andU2 are set and the uniform structure U1 majorizes U2

2, L1, L2 is a completionof the set L with respect to these uniform structures. The set L1 is embedded into thespace L2 densely and continuously iff the following condition holds.

π) Let E be the minimal Cauchy filter in L with respect to the uniform structure U1,then E is the minimal Cauchy filter in L with respect to the uniform structure U2.

This statement is an analogue of the condition π) from [41]. Let us illustrate thiscondition by the following commutative diagram.

2 Note that in common case topologies T1, T2 of different uniform structures may be the same.


L1j1−−−−→ L1

∥

∥

∥

⏐

⏐

�j

L2j2−−−−→ L2

(6.2)

Let L1 be the set L with the topology of the uniform structure U1. The operator j1realizes embedding of the whole space L1 into the space L1. In a similar manner wedefine the set L2 and the operator j2. The operators j1, j2 are injective. Diagram (6.2)specifies the operator j : L1 → L2 defined on j1(L1). The operator j is injectiveand since the uniform structure U1 majorizes U2 then j is a uniformly continuousoperator.

The uniformly continuous operator j can be continued from the set j1(L1) whichis dense in L1 to uniformly continuous operator j, which is defined on the wholespace L1 [8]. However, this continuation can be non-injective. Indeed, the minimalCauchy filter E in L1 is a Cauchy filter in L2, but it can by not minimal (the minimalCauchy filter which corresponds to E we denote by E ). In the space L1, there canexist several minimal Cauchy filters, and every of them majorizes E . Therefore, allthese minimal filters are mapped by the operator j to one element E . When thecondition π) holds true the operator j is injective, and hence, we can say aboutembedding L1 ⊂ L2. Otherwise, the embedding L1 ⊂ L2 does not exist. At first,we have to factorize the space L1 (with respect to the above-mentioned equivalencerelation), and then we may map the equivalence classes of factorized space to theelements L2.

In the case of linear locally convex topological spaces, a uniform structure is setby a topology (i.e. a topology in such space is induced by a uniform structure), sothe condition (π) can be rewritten in the following form.

(π1) Let E be the minimal Cauchy filter in a locally convex linear topological spaceL1, which majorizes the filter O2(0) of neighbors of the point 0 in L2. ThenE = O1(0) is a filter of neighborhoods of the point 0 in L1.

Now, we can formulate the condition of the uniqueness of a generalized solution.

Theorem 6.3. Let the condition π1) hold for the set E and the topologies TE andσ(E,M), then the generalized solution is unique.

Proof. By Lemma 6.1, it is sufficient to prove that the set [M] is total in the duality((ET )∗, ET ). From the proved above it follows that the linear spaces (E,M) form adual pair. Let EM be a set E with the topology σ(E,M). It is clear that (EM)∗ = M.Let us denote by EM the completion of EM . Then (EM)∗ = [M] 3. It is easy to seethat the topology (and the separable uniform structure) TE majorizes the topologyσ(E,M); hence, by the conditions of the theorem, we have that ET ⊂ EM . This em-bedding is continuous and dense. Thus, there exists the embedding (EM)∗ ⊂ (ET )∗,

3 More precisely, (EM)∗ coincides with the set of linear continuous functionals, every of which isa continuity of some functional from M to the whole space EM .

6.2 Examples of Generalized Solutions 131

i.e. [M] ⊂ (ET )∗. Since (EM , [M]) are in duality then for an arbitrary u ∈ EM , inparticular, for an arbitrary u∈ ET , the condition l(u) = 0 for all l ∈ [M] implies thatu = 0? which was to be proved. �Remark 6.37. Instead of the topology σ(E,M), we can consider any other topol-ogy which is conformed with the duality (E,M) and is connected with TE via thecondition π1). An example of this situation is described in Remark 6.36.

In [33, 62, 63, 89, 91], the concept of a near-solution was introduced. Let us con-sider a general analogue of this definition.

Definition 6.22. An element u ∈ ET for which there exists a filter E of the spaceET is called a generalized solution of the equation L u = f if the filter of the spaceET with the basis E converges to u ∈ ET and lim

EL = f in FT .

In addition, in many cases a sequential analogue of this definition play an impor-tant role.

Definition 6.23. An element u ∈ ET for which there exists a sequence un ∈ ET iscalled a generalized solution of the equation L u = f , if un converges to u ∈ ET inthe space ET and lim

n→∞L un = f in FT .

It is clear that if u is a generalized solution in the sense of Definition 6.23, thenu is a generalized solution in the sense of Definition 6.22. If the space ET (or FT )satisfies the first axiom of countability (the neighborhood system of every point hasa countable base), then the inverse statement is also true.

The definition of the operator L directly implies that generalized solutions in thesense of Definition 6.20 and 6.22 are equivalent, i.e. under the conditions (1)–(5) forany right-hand side f ∈ F ∩ RT there exists a unique generalized solution u ∈ ET

of the equation L u = f in the sense of Definition 6.22.By Theorem 6.2 the following statements are true.

Corollary 6.15. If u is a generalized solution in the sense of Definitions 6.20 or 6.22,then u is a solution in the sense of Definitions 6.21 also.

Corollary 6.16. If a solution by Definitions 6.21 is unique, then Definitions 6.20–6.22 are equivalent.

Corollary 6.17. Let u ∈ ET be a unique generalized solution in the sense of Defini-tion 6.21 and f ∈ R(L ). Then, u ∈ E and L u = f .

Remark 6.38. Let the linear spaces N and RT ∩ F be in duality and u ∈ E be ageneralized solution in the sense of Definition 6.21, then L u = f . Note that N andRT are in duality, e.g., when α ∈ U are compact spaces in N with respect to thetopology σ(N,F).

6.2 Examples of Generalized Solutions

Since the topology TE is specified by the structure U , then comparing differentstructures U1, U2 we can study relations between the spaces ET1 , ET2 with differ-ent topologies T1,T2, and therefore between generalized solutions in these spaces.


Thus, there is an opportunity to construct a detailed classification of spaces ofgeneralized solutions in terms of structures U . We will not do this since we con-sider that the applications of the method proposed are far more important then theconstruction of a complete theory.

In applications, it is important to know the relation between topologies of thespaces ET and FT , from the one hand, and natural topologies of the spaces E and F ,from the other hand. Let us consider the most common cases and construct examplesof specific structures U which lead to the topological spaces ET and FT .

Hereinafter, we will consider that E and F are Banach spaces and L : E → F isan injective linear continuous operator (it is well-known that such an operator isweakly continuous [97]), D(L ) = E , R(L ) is a dense subset of F , E∗, and F∗ areconjugate spaces.

It is clear that (E,E∗) and (F,F∗) are dual pairs, and the set R(L ) ⊂ F is totalin the duality (F,F∗). Since R(L ) is densely embedded into F , the adjoint opera-tor L ∗ : F∗ → E∗ is injective and continuous. It is easy to prove that under theseconditions the set R(L ∗) is total in E∗ with respect to the duality (E∗,E) [33, 35].

Since E,F are Banach spaces, then the following definition of a generalized so-lution has the great importance.

Definition 6.24. A generalized solution of the equation L u = f is such an elementu ∈ ET for which there exists a sequence un ∈ E which is convergent to u ∈ ET inthe space ET and lim

n→∞L un = f in the space F .

It must be stressed that this definition is different from the previous: the conver-gence of L un is considered in the space F , rather than FT . If the topologies of thespaces F and TF are comparable, then it is easy to establish the relation betweenDefinitions 6.23 and 6.24.

Now, let us pass to consideration of examples of specific structures U .

6.2.1 Classical Solvability

Let the set R(L ∗) have zero characteristics, for example, when E is a quasireflexivespace [86]. Put

U = {αλ |αλ = (L ∗)−1(Sλ (E∗)∩R(L ∗)),λ ∈R+},

where Sλ (E∗) is a closed ball of radius λ in the space E∗ (E∗ is the space con-

jugate to E) with the center in the point of origin. Conditions (1) and (2) aresatisfied obviously. In addition, N = ∪αλ = F∗, i.e. Condition (4) is satisfied.Let us test Condition (3). Let some set αλ ∈ U not be a bounded set in F∗ intopology σ(F∗,R(L )), i.e. there exists such a sequence ϕn ∈ αλ and f ∈ R(L )that ϕn( f ) ≥ n. Since f ∈ R(L ), then there exists such an element u ∈ E , thatϕn(L u) ≥ n or (L ∗ϕn)(u) ≥ n, but L ∗ϕn ∈L ∗(αλ ) ⊂ Sλ (E

∗). This contradictsto boundedness of the ball Sλ (E

∗).

6.2 Examples of Generalized Solutions 133

Thus, the topology TE is determined by the system of semi-norms

pλ (u) = supl∈Sλ (E∗)∩R(L ∗)

|l(u)|= supl∈Sλ (E∗)∩R(L ∗)

|l(u)|, λ ∈ R+,

where Sλ (E∗)∩R(L ∗) is a closure of the set Sλ (E∗) ∩ R(L ∗) in topology

σ(E∗,E).Since the set R(L ∗) is total in the duality (E∗,E), then it is dense in E∗ in

weak-* topology σ(E∗,E), and since R(L ∗) has non-zero characteristics, thenSλ (E∗)∩R(L ∗) contains some ball Stλ (E

∗) of smaller radius (0 < t < 1). Thenwe have that

supl∈Stλ (E∗)

|l(u)| ≤ pλ (u)≤ supl∈Sλ (E∗)

|l(u)|.

By the Hahn–Banach Theorem, the topology TE is induced by the norm ‖u‖E andthe space ET coincides the the Banach space E . Since the operator L realizes anisomorphism between ET and RT and the space ET = E is complete, then RT is acomplete space also, i.e. RT = R(L ).

Thus, under such selection of the structure U the generalized solvability co-incides with the classical solvability. This means that the concept of the classicalsolvability of linear operator equations is described in terms of the structure U .

6.2.2 Generalized Strong Solvability

Let us suppose that U = {α |α = Sλ (F∗),λ ∈R}. The fact that the operator L ∗ is

bounded implies that the set L ∗(Sλ (F∗)) is bounded. It is easy to see that Condi-

tions (1)–(5) are satisfied.By the Hahn–Banach Theorem, the topology TE is induced by the norm

‖u‖1 = supl∈L ∗(S1(F∗))

|l(u)|= supϕ∈S1(F∗)

|ϕ(L u)|= supϕ∈F∗

|ϕ(L u)|‖ϕ‖F∗

= ‖L u‖F ,

and the topology TF is induced by the norm ‖ f‖F . This means that ET is a com-pletion of E with respect to the norm ‖L u‖F . The space F = FT is complete andRT = F .

Note that under these conditions N=F∗ and M=R(L ∗), since the vector spacesN and RT are in duality (see Remark 6.38).

Let us prove that ([M], ET ) is a dual pair. Suppose that there exists such anelement u ∈ ET , that l(u) = 0 for any l ∈M. Let un be a sequence of elements fromE , which converges to u in the norm of the space ET . By definition an extendedfunctional l ∈ [M] on the element u takes the value

l(u) = limn→∞

l(un) = limn→∞

(L ∗ϕ)(un) = limn→∞

ϕ(L un) = ϕ( limn→∞

L un) = ϕ( f ),


where l = L ∗ϕ , ϕ ∈ F∗, f = limL un. The latter limit exists by virtue of the factthat the sequence un converges in the topology TE . Thus, ϕ( f ) = 0 for all ϕ ∈ N =F∗. Whence it follows that, f = 0 in F or u = 0 in ET .

Thus, when we select the structure U = {α |α = Sλ (F∗),λ ∈ R} a generalized

solution in the sense of Definition 6.21 is unique.4

Note that in this case the topologies TE and TF are normed and naturally con-nected with the original spaces E and F , since a generalized solution specified by thestructure U = {α |α = Sλ (F

∗),λ ∈R} is called a strong generalized solution. Suchapproach was exposed in Chap. 2. In this important particular case we will denotethe space ET by E1, and the extended operator L we will denote by L1 : E1→ F .Taking into account the common results, we can formulate the following statementfor the structure U = {α |α = Sλ (F

∗),λ ∈ R}.Theorem 6.4. For an arbitrary right-hand side f ∈ F there exists a unique gener-alized solution u ∈ E1 of the equation L u = f in one of the following equivalentsenses:

1. L1u = f ,2. l(u) = ϕ( f ) for all l = L ∗ϕ , ϕ ∈ F∗,3. ∃un ∈ E, so that un→ u in E1 and L un→ f in F as n→ ∞.

6.2.3 Generalized Weak Solvability

Let U = {α} be a collection of sets, consisting of all finite centrally symmetricsubsets F∗. It is easy to see that Conditions (1)–(5) are satisfied and the topologyTE is specified by the system of neighborhoods of zero

oα = {u ∈ E | |(L ∗ϕi)(u)| ≤ 1}, α = {ϕ1,ϕ2, . . . ,ϕn} ⊂ F∗.

Thus, in this case the topology TE coincides with weak topology σ(E,R(L ∗)),therefore the space ET coincides with the space studied in Chap. 2. As it was notedin [89], this approach generalizes the known concepts of a generalized solutionwhich were considered, for example, in [5]. The similar cases were considered alsoin [33, 91]. Let us denote the space ET by E2, and the space RT by F2, and theextended operator L by L2 : E2→ F2.

Note that N = F∗, M = R(L ∗). In addition, the sets α are compact in N withrespect to the topology σ(N,F), and the set L ∗(α) is compact in M with respect tothe topology σ(M,E), i.e. the conditions from Remarks 6.36, 6.38 are satisfied.

If U = {(L ∗)−1(β )}, where β are finite centrally symmetric subsets of somelinear set A⊂ R(L ∗), which is total in the duality (E∗,E), then ET coincides with˜M from Chap. 3.

4 The proof of the uniqueness may be conducted using Theorem 6.3. The corresponding reasoningswere formulated in Theorem 6.5 under the proving the existence of the embedding E1 ⊂ E2.

6.3 Properties of the Generalized Solvability in the Spaces E1,E2 135

6.2.4 A Priori Inequalities

Let M be some equable convex set bounded by norm F∗, which is total in the duality(F∗,F). If

U = {λM |λ ∈ R+},then Conditions (1)–(5) are obviously satisfied and the topology TE is specified bythe norm

‖u‖M = supl∈L ∗(M)

|l(u)|,

for which the following estimation is true

‖u‖M = supϕ∈M|(L ∗ϕ)(u)| ≤ sup

ϕ∈Sc(F∗)|ϕ(L u)|= c‖L u‖F .

Estimations of such types are called the a priori ones. There are a number of pub-lications, where a priori estimations are used to study properties of operator (see,e.g., [62, 63] and the bibliographies in these monographs) in order to construct thetheory of generalized solvability of linear partial differential equations (in the senseof analogous of Definitions 6.21 and 6.22).

6.3 Properties of the Generalized Solvability in the Spaces E1,E2

Let us consider the relations between the concepts of generalized solutions in themost important cases – in the spaces E1,E2. We remind that if E,F are Banachspaces, L : E→ F is an injective linear continuous operator, D(L ) = E , R(L ) is adense subset of F , then E1 is a completion of the set E in the norm ‖u‖E1 = ‖L u‖F ,and E2 is a completion E in the weak topology σ(E,R(L ∗)). It should noted thatsome proofs cited above duplicate the reasonings of Chap. 2. However, now weconsider them from the new point of view – in terms of the structures U .

Theorem 6.5. The space E1 is densely and continuously embedded into the E2.

Proof. Since in any finite subset of the space F∗ there exists a ball Sλ (F∗), which

contains this set, then the structure U specifying the space E1, majorizes the corre-sponding structure for E2. So, the topology of the space E1 majorizes the topologyE2. Thus, to prove the existence of the embedding E1 ⊂ E2 it is necessary to test thecondition π). Since E1 is a normed space, then we can consider a Cauchy sequenceun ∈ E in E1, which converges to zero in E2. Then L un is a Caushy sequence in F ,which converges to some f (F is a complete space). For every ϕ ∈ F∗ we have

(L ∗ϕ)(un) = ϕ(L un)→ ϕ( f ) n→ ∞.


But since un converges to zero in σ(E,R(L ∗)),

(L ∗ϕ)(un)→ 0.

That is why for any ϕ ∈ F∗ we have ϕ( f ) = 0. Whence it follows that f = 0, i.e.un ∈ E converges to zero in E1, which was to be proved. �Corollary 6.18. The space F is densely and continuously embedded into the spaceF2.

Proof. It is sufficient to equate the element u ∈ E with L (u) ∈ R(L ) and to repeatthe reasonings of the main theorem. �

The following statement is true.

Theorem 6.6. For any right-hand side f ∈ F there exists a unique generalizedsolution u ∈ E2 of the equation L u = f in one of the following equivalent senses:

1. L2u = f ,2. l(u) = ϕ( f ) for all l = L ∗ϕ , ϕ ∈ F∗,3. ∃un ∈ E, such that un→ u in E2 and L un→ f in F.

Proof. It is necessary to test only latter statement of the theorem. Indeed, sinceR(L ) is a dense subset of the space F , then there exists a sequence L un = fn ∈R(L ), which converges to f in F . However, the topology of the space F ma-jorizes the topology of the space F2, so fn converges to f in the space F2 also.The operator L2 realizes an isometry between F2 and E2, so un converges to someu ∈ E2 in E2. �Corollary 6.19. The definitions of generalized solutions in the spaces E1 and E2

(see Theorems 6.4 and 6.6) are equivalent.

Chapter 7Concept of Generalized Solution of NonlinearOperator Equation

In this chapter, we consider the concept of a generalized solution of a nonlinearoperator equation A(x) = y in metric spaces x ∈ E,y ∈ F , according to which ageneralized solution is an element of a completion of an original metric space Ein a metric specified by this operator and the metric of the space F [34]. We studythe existence and uniqueness of the generalized solution, and its correctness in thecase when the operator A is continuous or uniformly continuous. Also, we considerthe problems related with embedding of the completion of the metric space E withrespect to two comparable metrics.

Suppose E and F are metric spaces with metric ρE and ρF , respectively, F is acomplete space, and A : E → F is an injective operator whose domain D(A) coin-cides with the entire space E and the range R(A) ⊂ F in dense in F . Consider theoperator equation

A(x) = y, x ∈ E,y ∈ F, (7.1)

in the metric spaces E and F . If y ∈ R(A), then there exists a unique solution x ∈ Eof (7.1). We will call it a classic solution. But if y /∈ R(A) then classic equation x∈ Edoes not exist. That is why it is necessary to introduce the concept of a generalizedsolution of a nonlinear operator equation.

To determine a generalized solution of (7.1) let us consider a new metric on E

ρ∗(x,y) = ρF(A(x),A(y)). (7.2)

The fact that the functional ρ∗ defined on the Cartesian product E×E is a metricon E follows immediately from the injectivity of the operator A.

7.1 Generalized Solution of Nonlinear Operator Equation

Let us denote by E∗ the completion of E with respect to the metric ρ∗. Let y bean element of F that does not belong to R(A). Since R(A) is everywhere densein F , there exists a sequence yn ∈ R(A) converging to y in F as n → ∞. Let


137

138 7 Concept of Generalized Solutions...

xn = A−1(yn) ∈ E. It is easy to see that this sequence is Cauchy in the metric ρ∗.Indeed,

ρ∗(xn,xm) = ρF(A(xn),A(xm)) = ρF(yn,ym)→ 0 as n,m→ ∞,

since yn is Cauchy. Hence, in the complete metric space E∗ the sequence xn con-verges to some element

x ∈ E∗ : x = limn→∞

xn. (7.3)

Definition 7.25. The element x which was described above is called a generalizedsolution of (7.1).

Let us show that the generalized solution x is defined correctly. Indeed, let yn besome other sequence in R(A) converging to y in F and xn = A−1(yn). It is easy tosee that xn is equivalent to xn in E∗. Indeed,

ρ∗(xn, xn) = ρF(A(xn),A(xn)) = ρF (yn, yn)→ 0 as n→ ∞,

sinceρF(yn, yn)≤ ρF(yn,y)+ρF(y, yn)→ 0 as n→ ∞.

Thus, x = limn→∞

xn = limn→∞

xn in E∗.

7.2 Near-Solution of Nonlinear Operator Equation

Definition 7.26. A sequence of elements xn = A−1(yn) ∈ E , where yn ∈ F is an ar-bitrary sequence converging to y ∈ F is called a near-solution of the operator equa-tion (7.1), and x = lim

n→∞xn in E∗ is called a limit element of a near solution.

It is easy to see that x is a generalized solution of the operator equation (7.1) iffit is a limit element of a near-solution. In some cases, the elements of the sequencexn can be referred to as a near-solutions themselves.

The term “near-solution”, is justified by the following reasons. As it was shownbefore, the sequence xn converges to the generalized solution x in the complete spaceE∗, so that ∀ε > 0 if n>N we have ρ∗(x,xn)< ε. Since for small ε > 0 two elementsx and xn with distance ρ∗(x,xn) < ε can be considered almost identical (equal) inE∗. Hence, we may suppose that xn almost coincides with x, and it is natural to referto the element xn as to the near-solution of the operator equation (7.1).

It is interesting to note that generally in the original space E the sequence xn is notconvergent, but there are arguments in support of rationality of the term “near solu-tion” in E not referring to the completion E∗ and generalized solution x. Indeed, inmany practically important problems it is difficult or impossible to define the right-hand side y of (7.1) exactly. Therefore, we have to consider its ε-approximation, i.e.an element yε ∈ R(A) such that ρ(y,yε) < ε . In this case, there exists an elementxε = A−1(yε ) in D(A), that can be considered as an ε-approximation of a solution

7.4 Correctness of Generalized Solution 139

of (7.1) in the sense that its image yε = A(xε) deviates slightly from the right-handside (7.1). Thus, it is naturally to consider xε as an ε-solution or near-solution. Itmust be stressed that in many cases the quality of a solution xε depends on the prox-imity between its image yε = A(xε) and the element y. As it will be shown below, theeffect of stabilization of the family of elements xε is a derivative one, i.e. it followsfrom the fact that yε converges to y.

7.3 Existence and Uniqueness of a Generalized Solution

From the construction of the generalized solution x it follows that this solution existsfor every y∈ F . Let us prove that for an injective operator A : E→ F the generalizedsolution is unique. Assume the contrary: let there be two generalized solutions x andx corresponding to y. Then there exist sequences xn = A−1(yn) and xn = A−1(yn),converging yo x and x, respectively (it is natural to suppose that yn→ y and yn→ yin F as n→ ∞). Now, we have

ρ∗(x, x) = limn→∞

ρ∗(xn, xn) = limn→∞

ρF (A(xn),A(xn)) = limn→∞

ρF(yn, yn) = 0.

Therefore, x = x.

7.4 Correctness of Generalized Solution

Usually, correctness of a solution means that it is continuously dependent on theright-hand side:

∀ε > 0 ∃δ > 0 ρ(y, y)< δ ⇒ ρ(x, x)< ε,

where x and x are generalized solutions corresponding to right-hand sides y and y.Let us denote by E∗0 a set E with the metric ρ∗, then E∗ is a completion of E∗0 .

The operator A(x) maps E∗0 into R(A)⊂ F and is an isometry E∗0 and R(A), since forany x1,x2 ∈ E∗0 the following equality holds.

ρ∗(x1,x2) = ρE∗0 (x1,x2) = ρF (A(x1),A(x2)) = ρF(y1,y2).

Here, A(x1) = y1 and A(x2) = y2. We can extend the operator A onto the entire spaceE∗ so that it will map E∗ into F in the following way. Let x be an arbitrary elementin E∗. Then there exists a sequence xn from E∗0 converging to x. The sequence xn isCauchy in E∗0 ; hence, the sequence yn = A(xn) is Cauchy also

ρF(yn,ym) = ρF(A(xn),A(xm)) = ρ∗(xn,xm)→ 0

as n,m→ ∞.


Since F is complete, there exists an element y ∈ F such that limn→∞

yn = y (with

respect to the metric ρF ). Let us define the extension A of the operator A on thecompletion E∗ by the formula

A(x) = y, x ∈ E∗, y ∈ F.

We can see that this extension is correct and the element y is determined uniquely.If y /∈ R(A), the element x is a generalized solution. Both A and A are isometries ofthe spaces E∗ and F . Indeed, for all x, x ∈ E∗

ρE∗(x, x) = limn→∞

ρE∗(xn, xn) = limn→∞

ρF (A(xn),A(xn)) = limn→∞

ρF(yn, yn),

where y = A(x), y = A(x), xn, xn ∈ E , xn → x, xn → x as n→ ∞ in E∗. It followsthat A−1 maps F onto E∗ and is an isometry between F and E∗. Therefore, thegeneralized and classic solutions fill up E∗, and the generalized solutions form a setE∗ \E (a complement of E up to E∗). Moreover, the generalized solution x= A−1(y)is correct in E∗, since the isometry A−1 is a continuous mapping. Note that in E theexistence of continuous inverse operator is not guaranteed and classic solution maynot be correct.

7.5 Pseudo-Generalized and Essentially Generalized Solutions

The construction of generalized solution described above does not use the propertiesof the metric space E , so there is no connection between E and E∗. This follows fromthe fact that we do not impose any restrictions on the original operator A, except itsinjectivity. But if we assume that it has some additional topological properties (forexample, continuity) then this connection arises. Let us study these properties.

Lemma 7.1. If A is a continuous injective operator mapping a metric space E intoa metric space F, then E is densely embedded into E∗, where E∗ is the completionof E in the metric (7.2).

Proof. Indeed, since A : E → F is a continuous injective operator, D(A) = E , andR(A) is a dense subset of F , then the operator A defines a dense and continuous em-bedding of E into F . From the other hand, as it was established above, the operatorA is an isometry between the metric spaces E∗ and F . Thus, we have the followingcommutative diagram

EA−−−−→ F

∥

∥

∥ A

�

⏐

⏐

Ej−−−−→ E∗,

where the operator j : E→ E∗, defined as j = A−1 ◦A, specifies a dense and contin-uous embedding of the metric space E into the metric space E∗. ��

7.5 Pseudo-Generalized and Essentially Generalized Solutions 141

Lemma 7.1 implies that the topology of E∗0 is weaker that the topology of E .Under the investigation of generalized solutions of the operator equation (7.1)

the completion E of the original space E with respect to the metric ρE plays animportant role (in contrast to the completion E∗ of the space E with respect to themetric ρ∗).

Definition 7.27. A generalized solution x of the operator equation (7.1) is called apseudo-generalized solution if x ∈ E .

Definition 7.28. A generalized solution x of the operator equation (7.1) is called anessentially generalized solution, if x /∈ E (and x ∈ E∗).

At a first glance, it seems that pseudo-generalized solution differs slightly fromthe classical one and the completeness of metric space E is not essential. Indeed,each metric space E has a completion E and it is obvious that we can extend theoperator A by continuity onto the whole space E . But the latter statement it not true,i.e. by far not every continuous operator A : E→ F may be extended onto the wholeE (in contrast to linear continuous operators which act in linear topological spaces).In addition, if such an extension yet exists, it can be not an injective operator evenif A is an injective itself. Let us consider these issues in details.

Let us recall that the operator A : E→ F is called uniformly continuous on E , if

∀ε > 0∃δ (ε)> 0 : ρE(x,y)< δ (ε)⇒ ρF (A(x),A(y))< ε. (7.4)

Let us prove that a uniformly continuous operator A can be extended onto thewhole space E preserving its properties (i.e. uniform continuity).

Theorem 7.1. Let A : E → F be a uniformly continuous operator on E. Then it canbe extended to uniformly continuous operator A, which acts from E into F.

Proof. Let x be an arbitrary element in E . Let us take an arbitrary sequence xn ∈ Econverging to x in E as n→ ∞. The sequence xn ∈ E is Cauchy in E , i.e.

∀δ > 0∃N(δ ) : ∀n,m > N(δ ) ⇒ ρE(xn,xm)< δ .

By virtue of (7.4)

∀ε > 0∃N(δ (ε)) : ∀n,m > N(δ (ε))⇒ ρ∗(xn,xm) = ρF(A(xn),A(xm))< ε.

Therefore, the sequence xn is Cauchy in the space E∗0 also. Since the operator A :E → F specifies an isometry between E∗0 and R(A) with the metric ρF , then thesequence yn = A(xn) is Cauchy in R(A), hence in F also. Since F is a completemetric space, there exists y = lim

n→∞yn.

Put A(x) = y. Let us justify the correctness of this definition. Let xn ∈ E be otherarbitrary sequence converging to x in E as n→ ∞. Then by the triangle inequalityρE(xn, xn)→ 0 as n→ ∞. Therefore,

∀δ > 0∃N : ∀n > N ⇒ ρE(xn, xn)< δ .


Whence it follows that, taking into account the uniform continuity of A, we have

∀ε > 0∃N : ∀n > N⇒ ρ∗(xn, xn) = ρF(A(xn),A(xn))< ε.

This means that the sequences yn = A(xn) and yn = A(xn) have a common limit –the element y ∈ F .

Let us prove that A is a uniformly continuous operator. For an arbitrary ε > 0 weselect arbitrary elements x, x ∈ E satisfying the inequality ρE(x, x) < δ (ε), wherethe value δ (ε) is determined from the equality (7.4). Let xn and xn be arbitrarysequences in E converging to x and x respectively. Then,

∃N : ∀n > N⇒ ρE(xn, xn)< δ (ε),

and, taking into account the uniform continuity of A, we have that for all n > Nthe equality ρF(A(xn),A(xn))< ε holds true. Since the metric ρF is continuous, wehave

ρF(A(x), A(x)) = ρF( limn→∞

A(xn), limn→∞

A(xn)) = limn→∞

ρF(A(xn),A(xn))≤ ε.��

Note that the continuity of the operator A : E → F is not sufficient for the exis-tence of a continuous extension A of the operator A onto the whole space E .

Indeed, consider the continuous (but not uniformly continuous) operator A(x) =sin 1

x , which acts between the metric spaces E = (0,1], F = [−1,1] of the real num-bers R with natural metric. The completion of the space E = (0,1] is E = [0,1]. Butsince the limit lim

x→0A(x) does not exist, then a continuous extension of the operator

A onto E = [0,1] does not exist also.It is obvious that since we have considered the construction of extension of an

injective operator A : E→ F onto the whole space E∗, then the cause of the absenceof the continuous extension of A onto E in the previous example was the fact thatthe operator A was not injective. However, this is also not the case. Let us considerthe example of a continuous injective operator A with a range R(A) that is a densesubset of the complete metric space F , which nevertheless allows the extension ontothe the whole space E . Consider a continuous (but not uniformly continuous again)injective operator

A(x) =

(

x,sin1x

)

,

which acts from the metric space E = (0,1]⊂ R into the metric space F = A(E)⊂R

2 with the usual Euclidean metric. Here, under F = A(E) we mean a completionof the metric space A(E) with the usual metric, i.e.

F ={

(x,y) ∈ R2 |x ∈ (0,1], y = sin

1x

}

∪{

(0,y) ∈R2 |y ∈ [−1,1]

}

.

7.5 Pseudo-Generalized and Essentially Generalized Solutions 143

Since the limit limx→0

A(x) in F does not exist (the set of partial limits consists of the

point of the segment{

(0,y) ∈ R2 |y ∈ [−1,1]

}

), then a continuous extension doesnot exist.

Let us study the problem described from the other point of view. Suppose thatsome injective continuous operator A : E→ F allows the continuation by continuityon the completion E . Let us establish the criteria of the injectivity of the extendedoperator A : E→ F . Suppose that the following condition holds.

˜π) if xn and x′n are Cauchy sequences from E in metric ρE and ρ∗(xn,x′n) =ρF(A(xn),A(x′n))→ 0 as n→ ∞, then ρE(xn,x′n)→ 0 as n→ ∞ (cf. analogouscondition π) in [41]).

Recall that two sequences xn and x′n are equivalent in the metric ρ(x,y) ifρ(xn,x′n)→ 0 as n→∞. Using this concept, we can reformulate ˜π) in the followingway: two Cauchy sequences xn and x′n from E which are equivalent in the metric ρ∗Eare equivalent in the metric ρE(x,y) also.

Theorem 7.2. Let A be a uniformly continuous injective operator mapping the met-ric space E into a complete metric space F and A be an extension by continuity of Aon the completion E of the metric space E. The extension A is an injective operatormapping E into F iff the condition ˜π) holds.

Proof. Necessity. Let A be an injective mapping, xn and x′n be two Cauchy sequencesin E (xn→ x and x′n→ x′ in E as n→ ∞), which are equivalent with respect to themetric ρ∗(x,y), then

0 = limn→∞

ρ∗(xn,x′n) = lim

n→∞ρF(A(xn),A(x

′n)) = ρF(y,y

′),

where y = limn→∞

A(xn), y′ = limn→∞

A(x′n). Since y = A(x), y′ = A(x′), then the equal-

ity A(x) = A(x′) and injectivity A imply that x = x′. This means that ρE(xn,x′n)→ρE(x, x

′) = 0 as n→ ∞.Sufficiency. Suppose that A(x) = A(x′) and xn, x′n are a Cauchy sequences E

converging to x and x′ in E as n→ ∞ respectively. These sequences are equivalentin the metric ρ∗(x,y), since

limn→∞

ρ∗(xn,x′n) = lim

n→∞ρF(A(xn),A(x

′n)) = ρF(A(x), A(x

′)) = 0.

By virtue of condition ˜π) the sequences xn and x′n are equivalent in the metricρE(x,y) also. Therefore, x = x′, and the injectivity of the extension A of the operatorA onto E is proved. ��

From the point of view of the previous theorem let us consider the followingissue. Suppose that the extension A is weakly injective: if x �= x′ and at least oneelement of x and x′ belongs to the space E , then A(x) �= A(x′). Using the reasons


as in the proof Theorem 7.2, we can show that this property is equivalent to thefollowing condition:

π∗)if the sequence xn is Cauchy in the metric ρE(x,y), x ∈ E and xn→ x as n→ ∞in the metric ρ∗(x,y), then xn→ x in the metric ρE .

It is easy to see that condition (π∗) follows from condition ˜π), and in addition,if E and F are linear normed spaces and A is a continuous linear operator, then theequivalence of conditions ˜π) and π∗) is clear. Let us prove that in a general casethese conditions are not equivalent [78].

Indeed, let us consider the sets E = F = (0,1). Let us take as the operator A theidentity mapping A(x) = x. Let us consider on E,F the following metrics

ρE(x,y) = |x− y|, ρF(x,y) = min{|x− y|,1−|x− y|}.

It is easy to see that the functionals ρE and ρF satisfy the metric axioms.Let us show that condition π∗) holds. Indeed, let xn ∈ (0,1) be a Cauchy sequence

in E . Since E is the interval (0,1)⊂R with usual metric, then xn converges to somex ∈ [0,1] in R. Since ρF(x,y) ≤ ρE(x,y), then xn converges to x with respect to themetric ρ∗(x,y) = ρF(x,y) also. That is why x ∈ (0,1) = E , which was to be proved.

From the other hand, condition ˜π) does not hold. Indeed, let xn = 1/n and x′n =(n−1)/n. Then it is easy to see that ρ∗(xn,x′n)→ 0 as n→ ∞, but ρE(xn,x′n) doesnot converge to zero.

Basing on this example, it is easy to construct analogous counterexamples foroperators which act in other metric spaces.

7.6 Relation Between Pseudo-Generalizedand Generalized Solutions

The investigation of the concept of a pseudo-generalized solution leads to a quiteinteresting situation, which at first glance contradicts to the uniqueness of a gen-eralized solution. Indeed, suppose that condition ˜π) does not hold true. Then theextension A of a uniformly continuous operator A onto the whole space E is not aninjective operator, so there exists a pair x and x′ consisting of different elements ofE , for which A(x) = A(x′) = y. Now, it is obvious that there are at least two pseudo-generalized (and hence generalized) solutions x and x′ corresponding to the elementy (the right-hand side of the operator equation (7.1)). This would have been pos-sible if the completion E was embedded into the completion E∗ of the space E∗0 .However, actually there is no any contradiction, since by far not always E ⊂ E∗;therefore, at least one of the elements x, x′ does not belong to E∗. So, this elementis not a generalized (and pseudo-generalized) solution, and the supposed contra-diction is solved. Thus, there is an important theoretical problem of embedding ofcompletions E and E∗.

7.6 Relation Between Pseudo-Generalized and Generalized Solutions 145

Let us recall the exact definition.

Definition 7.29. Let E and F be metric spaces and j be an injective mapping of E toF . The metric space E is said to be continuously (uniformly continuously) embeddedinto F using the embedding operator j if j is a continuous (uniformly continuous)operator. If in addition the subspace j(E) ⊂ F is dense in the metric space F , thenthere exists a dense embedding of E into F using the embedding operator j.

Note, that the embedding operator j in this definition plays an extremely impor-tant role, since the space E can be embedded into F by different embedding opera-tors j, so the phrase “the space E is embedded into F” without clear understandingof the nature of the operator j is incorrect.

Let E be an arbitrary metric space and E be its completion. Is is well-known thatthe completion E consists of classes x of equivalent Cauchy sequences xn in E . Letus define the canonical embedding operator j1 in E in the following way: we mapevery element x ∈ E to a class x containing the stationary sequence xn ≡ x, n ∈ N.Then the space E is embedded into its completion E by canonical embedding op-erator j1. In this case, the metric space E is equipped with two metrics: originalmetric ρE and a metric ρ∗ (the operator A is supposed to be continuous). The com-pletion E∗0 = E in the metric ρ∗ is denoted by E∗, therefore E = E∗0 is embeddedinto E∗ by an analogous canonical embedding operator j3. If the original operatorA is uniformly continuous, we can define a canonical embedding mapping j2 of Einto E∗ in the following way: every element x ∈ E is mapped to an element x∗ ∈ E∗containing the element x: j2(x) = x∗, x⊂ x∗.

Let us work out all the details of j2 and show that it is correct. Let x be anarbitrary element of E and x0

n be a Cauchy sequence in the metric ρE that belongs tothe class x. Since the sequence xn is an element of x, then it is equivalent to x0

n in themetric ρE , but xn and x0

n are equivalent in the metric ρ∗ also. Indeed, let ε > 0 bean arbitrary positive number. Since A is a uniformly continuous operator, then thereexists such δ > 0, that the condition ρE(x,y) < δ implies that ρF (A(x),A(y)) < ε.Let us select a natural number N in such a way that ρE(xn,x0

n)< δ for every n > N,then ρF(A(xn),A(x0

n)) < ε for every n > N. Therefore, xn and x0n are equivalent

sequences in metric ρF(x,y). Hence, every sequence xn in x belongs to the classx∗; hence, x ⊂ x∗ and the mapping j2 is defined correctly. Canonical embeddingoperators j1, j3 and the operator j2 are connected by the following commutativediagram.

Ej1−−−−→ E

∥

∥

∥j2

⏐

⏐

�

Ej3−−−−→ E∗

Recall that due to commutative property this formula can also be expressed in theform j3(x) = j2 ( j1(x)).

In order for the canonical mapping j2 to be embedding it should be injective. Thecriterion for injectivity of j2 can be formulated in terms of condition ˜π).


Theorem 7.3. Let E and F be metric spaces, F be complete and A : E → F be auniformly continuous injective operator, which acts from E into F. The canonicalmapping j2 : E→ E∗ is injective iff condition ˜π) holds.

Proof. Necessity. Let j2 be an injective mapping, xn and x′n be Cauchy sequencesin metrics ρE , xn ∈ x, x′n ∈ x′ and ρ∗(xn,x′n)→ 0 as n→ ∞. By definition of j2 theelement j2(x) is such an element x∗1 ∈ E∗ that x⊂ x∗1. Similarly, j2(x′) = x∗2: x′ ⊂ x∗2.This implies that xn ∈ x∗1 and x′n ∈ x∗2, but from the other hand ρ∗(xn,x′n)→ 0 asn→ ∞. This means that x∗1 = x∗2, i.e. j2(x) = j2(x′), hence by the injectivity of theoperator j2 we have x = x′. Thus, the sequences xn and x′n are equivalent in themetric ρE also.

Sufficiency. Let x and x′ be two different elements of the completion E and letthe condition ˜π) hold. Suppose that j2(x) = j2(x′) = x∗. Consider two Cauchy se-quences xn ∈ x and x′n ∈ x′, then xn ∈ j2(x) = x∗ and x′n ∈ j2(x′) = x∗, so thesesequences are equivalent in metric ρ∗. By virtue of the condition ˜π) the sequencesxn and x′n are equivalent in metric ρE also, so x = x′, but this contradicts to ourassumption. The theorem in proved. ��Definition 7.30. An injective canonical mapping j2 of completion E into E∗ iscalled a canonical embedding of completion E into E∗.

Theorem 7.3 implies that the canonical mapping j2 is a canonical embedding ifthe condition ˜π) holds. The existence of canonical embedding j2 of E into E∗ allowsus to extend the metric ρ∗ from the space E onto its completion E by the formula

ρ(x, x′) = ρ∗(x∗,x∗∗) = ρF(

A(x), A(x′))

,

where x∗ = j2(x), x∗∗ = j2(x′). If the condition ˜π) does not hold, then the functionalρ∗ is only a quasi-metric. This fact is very useful in the study of generalized solu-tions of operator equations. Thus, when the condition ˜π) holds true, the extensionA of the operator A onto the completion E is an injective operator (as well as theoperator A), and the space E (as well as the space E) is embedded into E∗ with thehelp of canonical embedding. Therefore, this case differs sharply from the originalproblem.

Now, pass to the classification of solutions of an operator equation according tovarious types of convergence in the metric space E:

1. If the right-hand side y of (7.1) belongs to the range R(A) of the operator A, thenthere exists a classical solution of (7.1).

2. If the right-hand side y of (7.1) does not belong to the range R(A) of the operatorA, there exist following fundamentally different possibilities:

(a)Let xn = A−1(yn), yn→ y as n→∞ be a near-solution of (7.1); if the sequencexn is Cauchy in metrics ρE , then for this element a pseudo-generalized solutionexists, and the limit element x of this solution belongs to the completion E .

(b)If a near-solution xn = A−1(yn) is not Cauchy in metric ρE(x,y), then x /∈ E ,and x ∈ E∗ is an essentially generalized solution of (7.1).

7.6 Relation Between Pseudo-Generalized and Generalized Solutions 147

Let us examine the conditions under which generalized solutions arise and shouldbe investigated. If a metric space E is compact (relatively compact) and the oper-ator A : E → F is continuous (uniformly continuous), then generalized solutionsoccur very seldom. Indeed, in this case by virtue of well-known results [8] the in-verse mapping A−1 : R(A)→ E is continuous and, and since R(A) is everywheredense in F , the uniformly continuous operator A−1 can be extended by continuityto the whole space F . If E is a complete metric space, then for any y ∈ F the el-ement x = A−1y is a classical solution. If E is an incomplete metric space, thenthere arise pseudo-generalized solutions which differ only slightly from classicalsolutions. In both cases, the essentially generalized solutions do not arise. However,generalized solutions can arise when E and F are non-compact infinite-dimensionalspaces (topological spaces or differentiable manifolds, spaces of distributions, clas-sical Banach spaces, Hilbert spaces an so on). We need such spaces to investigatelinear and nonlinear integral equations and infinite systems of linear and nonlinearalgebraic equations. In this case function spaces and spaces of sequences play therole of space E .

The problem of necessity of the investigation of generalized solutions appearsfor the following reasons. As it was shown earlier, the generalized solution x of theequation A(x) = y is a limit element of a near-solution xn, where xn = A−1(yn), andyn→ y as n→ ∞. If y /∈ R(A), then the near-solution xn does not converge in E toany element (for essentially generalized solution this sequence is not even a Cauchysequence in E). Therefore, when solving approximately the operator equation (7.1)it is not obvious which metric is expected to stabilize the near-solution xn and whenthe element xn can be considered as an approximate solution of (7.1). Moreover, itis not clear what the generalized solution x is and to which classical function spaceof sequences it belongs. From the reason we have mentioned it can be deduced thatstabilization of the near-solutions takes place in the metric ρ∗ in E , and in this casethe generalized solution belongs to the completion E∗ of the space E in metrics ρ∗.However, this result represents only a principal solution of this problem, since itdoes not answer the question how to define the sense of convergence of a sequenceof near solutions xn in the metric ρ∗ and what elements constitute the space E∗.Unfortunately, the space E∗ depends on the operator A. This makes the problemmore complicated, because investigating the family of operators A depending onparameter λ involves a family of spaces E∗(λ ) (in some cases we may have a wholescale of spaces). It is very difficult to obtain interesting results for such family ofspaces. Fortunately, we can use the theory of embedded spaces [41]. Indeed, if wesuppose that for some family of operators Aλ , (λ ∈ I) all metric spaces E∗(λ ) arecontinuously embedded, with the help of an operator of natural embedding, intosome metric or topological space Σ well-defined structure, then each generalizedsolution x of the operator equation Aλ (x) = y belongs to the space Σ and we canidentify the element x (for example, if Σ is a space of measurable functions, then xis some measurable function). We will refer to the metric space Σ as to the basic one.Moreover, under certain restrictions imposed on a near-solution xn ∈ E∗(λ )⊂ Σ theconvergence of xn in the topology (or metric) of the space Σ (this convergence holdstrue always) implies convergence of xn in the space E∗(λ ). For example, by the


classic Lebesgue theorem on majorants, the convergence of a sequence of integrablefunctions xn(t) by measure with conditions |xn| ≤ g(t), where g(t) is an integrablefunction implies convergence of xn in the space of integrable functions L1. Thisfact allows us to investigate the conditions of stabilization of near-solution xn inthe space Σ (and in the space E∗(λ )) and to obtain an approximate solution of theoperator equation Aλ (x) = y. Metric or topological vector spaces with the weakesttopology (convergency) may play the role of the basic space Σ . For many Banachfunction spaces (e.g., for Banach spaces of measurable functions and, in particular,for ideal spaces [41]) the basic space Σ is a space S(a,b), consisting of measurablefunctions x(t) defined on a segment [a,b] with the metric

ρ(x,y) =∫ b

a

|x(t)− y(t)|1+ |x(t)− y(t)| dt, x(t),y(t) ∈ S(a,b).

In this space, convergence is equivalent to convergence by measure. Also, the spaceof Schwarz distribution may be considered as the basic space [33]). In the space ofsequences, we may select as the basic space Σ the space s of all numerical sequenceswith the metric

ρ(x,y) =∞

∑n=1

12n ×

|xn− yn|1+ |xn− yn| , x = (x1,x2, . . .), y = (y1,y2, . . .),

where the convergence is coordinate-wise.

7.7 Example of Operators

Let us consider examples of nonlinear integral and differential operators which mayinduce a concept of generalized solutions (see, e.g., [37]).

(1) Nemytskii operator. It is said that a function f (s,u) of two arguments−∞ <u < ∞, s ∈ G satisfies the Caratheodory conditions if it is continuous with respectto u almost for all s ∈ G and is measurable by s for all u. Here, G is a subset of an-dimensional Euclidean space with finite measure. Denote by f an operator on theset of real functions defined on G by the equality

f [u(s)] = f (s,u(s)),

where f (s,u) satisfies the Caratheodory conditions. The operator f is called theNemytskii operator. The operator f [u(s)] is continuous and maps the space S(G) ofall measurable functions onto G. In addition, the continuity and boundedness of theoperator f follows from the fact that f acts from Lp1 into Lp2 .

(2) Uryson operator. Let K(x, t,u), s, t ∈ G,−∞ < u < ∞ be a function of threearguments. The nonlinear integral operator

A[ϕ(s)] =∫

GK(s, t,ϕ(t))dt

7.7 Example of Operators 149

is called the Uryson operator. If K(s, t,u) is continuous with respect to every variablein the aggregate, where s, t ∈ G and u ≤ a, then A[ϕ] is defined in the ball of radiusa in the space C(G) and is completely continuous. Under quite common conditions,the operator is defined in Lp and is completely continuous.

(3) Hammerstein operator. One class of the Uryson operators is studied morecarefully: that is the Hammerstein operators.

A[ϕ(s)] =∫

GK(s, t) f [t,ϕ(t)]dt.

Let us denote by B the linear integral operator induced by the kernel K(s, t)

B[ϕ(s)] =∫

GK(s, t)ϕ(t)dt,

If the operator f mapping Banach space E2 into E2 is continuous and bounded, andthe operator B mapping from E1 into E2 is completely continuous, then the Ham-merstein operator acts from E1 into E2 and is completely continuous. Note, thatthe concept of generalized solution is especially important for completely continu-ous operators mapping infinite-dimensional spaces, since the inverse operator A−1

is not continuous and there exists only essentially generalized solution for everyy ∈ R(A)⊂ F .

(4) Nonlinear parabolic equation. Consider the nonlinear initial-boundary valueproblem which occurs in the theory of mass transport in porous media:

A(u)≡ ∂u∂ t−

2

∑α=1

∂∂ xα

(

kα(u)∂u

∂xα

)

= f (x, t),

u|x∈∂Ω = u|t=0 = 0

where (x, t)∈Q = Ω×(0,T ], Ω ⊂R2, u∈W+l

2,bd(Q), f ∈V ⊂W−l2,bd(Q), W+l

2,bd(Q)

is the Sobolev space consisting of the functions of W+l2,bd(Q), that satisfy the

boundary conditions (bd), W−l2,bd(Q) is a negative space constructed on L2(Q) and

W+l2,bd(Q), V is a dense subset of W−l

2,bd(Q). Similar problem, e.g., describes the wa-

ter transport in unsaturated soil during the drip irrigation. In this case the right-handside has the following form:

f (x, t) =m

∑i=1

Qiδ (x− xi).

In conclusion, let us consider the problem of construction of the basic space Σfor one-parametric family of operators Aλ , λ ∈ I. Each operator Aλ induces on E themetric ρ∗λ (x,y) = ρF (Aλ (x),Aλ (y)) and the completion E∗λ in this metric. Denoteby Σ the union of all spaces E∗λ :

Σ =⋃

λ∈I

E∗λ .


Operator of natural embedding of E∗λ into Σ induces mappings fλ : E∗λ → Σ . We candefine in Σ the strongest topology T , for which all the mappings fλ are continu-ous [8]. Then, the space Σ with the topology T plays the role of the basic space forthe family Aλ , λ ∈ I.

7.8 Computation of Generalized Solution

Both classical and generalized solutions can be found exactly only in exceptionalcases. One of the complications in this process is the difficulty of construction ofthe basic space Σ for a given class of nonlinear operators. In this connection, theproblem consists of finding approximate solutions which are close to generalizedsolution x in the metric ρ∗ or in the metric of basic space Σ . It is interesting that theseapproximate solutions belong to the original space E , since we may consider themas elements of near solution xn with limit element x. Thus, to compute approximatesolutions xn we may not need to define the basic space Σ . It is sufficiently onlyto prove that their images yn = A(xn) converge to the right-hand side y of (7.1) inthe metric of space F as n→ ∞. At a first glance, the problem of constructing theapproximate solutions xn appears to be quite simple: we have to select a sequenceyn from the range R(A), which converges to y, and then the elements xn = A−1(yn)would be approximate solutions of the operator equation A(x) = y. For that purpose,let us introduce the following definition.

Definition 7.31. Let ε be an arbitrary positive number. We will refer to an elementxε in the space E as ε-approximation of the generalized solution x of the operatorequation A(x) = y, if ρF(A(xε ),y) = ρF(A(xε), A(x))< ε .

However, careful analysis of computation of ε-approximations shows great defi-ciencies of such a “direct approach”. First, in many cases it is difficult or impossibleto describe the range R(A) of an operator A in a space F . Therefore, it is not clearhow to select a sequence yn converging to the element y in F . Second, even if el-ements xn = A−1(yn) are known exactly, computation of the elements yn ∈ R(A)frequently is a very hard problem. That is why it is necessary to find a more effec-tive way of constructing approximate solutions xn of the operator equation A(x) = ywhen y /∈ R(A). Let us describe one method which is realistic (although not nec-essarily optimal) for constructing of a sequence converging to the solution in themetric ρ∗(x,y). Note, that this sequence of ε-approximations converges only in themetric ρ∗, whereas in original metric ρE it will be divergent, since the near-solutionxn does not converge to any element in the metrics ρE . Thus, the proposed approxi-mate solution differs from all other approximations of exact solution of the operatorequation (7.1).

Denote by E a separable metric space. Let S = (a1,a2, . . .) be a countable every-where dense set in E . Let us introduce the following notation: bn = A(an), n ∈ N,B = (b1,b2, . . . ,bn, . . .). It is easy to see that B is everywhere dense in F if A is acontinuous operator and R(A) is everywhere dense in F . Indeed, let ε > 0 and y be

7.8 Computation of Generalized Solution 151

a fixed element (the right-hand side of (7.1)), then there exists an element y ∈ R(A)such that ρF(y, y) < ε

2 . Let x = A−1(y). Since A is a continuous operator, there is anumber δ > 0 such that ρE(x, x)< δ implies ρF(A(x, x))< ε

2 . Since S is everywheredense in E , there exists a subsequence ank of S converging to x. Hence, there existssuch an element ank for that ρE(x,ank)< δ , and therefore

ρF(y,bnk)≤ ρF(y, y)+ρF(y,bnk)≤ε2+

ε2= ε.

Denote by an1 the first element of S, for which

ρF(y,bn1) = ρF(y,A(an1))< 1.

Existence of the element an1 follows from the density of B in F . Let an2 be the nextelement in S after an1 such that

ρF(y,bn2) = ρF(y,A(an2))<12.

(existence of an2 follows from the density of the set {bn1,bn1+1, . . .} in F), and soon. Let ank be the next element in S after ank−1such that

ρF(y,bnk) = ρF(y,A(ank))<1k. (7.5)

Thus, we obtain a sequence of the elements ank , which is a near-solution whose limitelement is the generalized solution x corresponding to the right-hand side y of (7.1).Note, that in constructing this near-solution we did not resort to the operator A anddid not check whether yk = A(xk) ∈ R(A).

One of the ways to find the near-solution xn using this procedure is the Monte-Carlo method [109]. Indeed, we can number the elements of the everywhere count-able dense subset S in E not only by natural numbers, but also by rational numbersfrom the segment [0,1], i.e. map a rational number r ∈ [0,1]∩Q = Q[0,1] to an ele-ment ar ∈ S. (In some cases such enumeration is more suitable than the enumerationby natural numbers.) In the set Q[0,1] (more precisely, in the class SQ[0,1]

of all subsetsof Q[0,1]) we can introduce the uniform distribution of probabilities p(M), M ∈ SQ[0,1]

so that

p(r ∈ (α,β )) = β −α, 0≤ α,β ≤ 1

(note that this distribution of probabilities p is not a measure!).Randomly selecting rational numbers from the segment [0,1] according to the

distribution p we can select elements ar from S and construct a near solution asabove: the first element x1 of the near-solution is the first element ar, which underrandom sampling r from Q[0,1] satisfies the inequality ρF(y,A(ar)) < 1, the secondelement x2 is an element ar satisfying the inequality ρF (y,A(ar))<

12 and so on, the

kth element xk is an element ar satisfying the inequality ρF(t,A(ar))<1k . It is easy to


see that the sequence xk obtained as a result of repetition of this random experimentwith probability 1 converges to the generalized solution x in the metric ρ∗.

If in the spaces E and F there are additional structures besides the structure of ametric space (for example, structure of Hilbert space) a near-solution may be foundwith the help of modified Galerkin method.

Remark 7.39. The theory of generalized solutions of nonlinear operator equationscan be extended from the metric spaces E and F onto uniform spaces introducedby A.Weil. In this case, the analogue of the metric ρ∗(x,y) = ρE(A(x),A(y)) is apre-image of the uniform structure of the space F in the space E with respect toinjective uniformly continuous mapping A : E→ F .

7.9 Uniform Structures and Generalized Solutionsof Operator Equations

We developed the abstract theory of generalized solutions of linear operator equa-tions in previous chapters and obtained the series of results for equations with op-erators defined in metric spaces. In this section we give a brief description of thetheory of generalized solutions in uniform spaces basing on [69, 106].

7.9.1 Definition of a Generalized Solution of Operator Equation

Let (E,UE) and (F,UF ) be Hausdorff uniform spaces, A : E→ F be injective oper-ator, whose domain D(A) coincides with the whole set E and range R(A) is densein F in topology TF induced by uniform structure UF

1

Let us consider an operator equation

A(u) = h, (7.6)

where h ∈ F ; if h ∈ R(A) ⊆ F , then there exists a unique solution u ∈ E of (7.6),which we will call a classical solution. If h ∈ F\R(A), then equation (7.6) has notclassical solution. There is a need to extend the notion of a solution of an operatorequation (7.6) and to introduce generalized solutions.

Let us pass to definition of generalized solution of (7.6).Consider the following sets on a Cartesian square E×E:

{(u,v) ∈ E×E : (A(u) ,A(v)) ∈ O}, O ∈UF . (7.7)

1 Hereinafter, all topological notions are meant respectively to a topology induced by uniformstructures, and all uniform structures introduced are meant Hausdorff. By completion of a uniformspace we mean a Hausdorff completion.

7.9 Uniform Structures and Generalized Solutions of Operator Equations 153

The set of all subsets having the form (7.7) is a base of Hausdorff uniform structure.We denote it by UA (original with respect to A and uniform structure UF ). Theuniform structure UA is the weakest uniform structure in E . The operator A : E→ Fis uniformly continuous with respect to this uniform structure (F is equipped withthe uniform structure UF ).

It is easy to understand that the operator A realizes an isomorphism (E,UA) in(F,UF). Denote by

(

EA,UA)

and (F ,UF) completions (F,UF ) and (F,UF ), respec-tively. Since the set R(A) is dense in F in topology TF , then the completion of the

space(

R(A) ,˜UF

)

, where ˜UF = UF ∩ (R(A)×R(A)) is a uniformity induced on

R(A) by the uniformity UF , can be equate with the space (F,UF ).Let us consider the operator A : EA→ F which is a uniformly continuous exten-

sion of the operator A : E→ F .Recall that the operator A is defined in the following way. Let u be an arbitrary el-

ement from EA, i.e., it is a class of UA-equivalent UA-Cauchy nets of nets consistingof elements of the space E 2. The net (A(uα)) is a UF -Cauchy net with respect to(uα) ∈ u. Consider the class h ∈ F consisting the net (A(uα)). Let us put A(u) = h.

The operator A is a uniform isomorphism of the uniform spaces(

EA,UA)

and(F ,UF). Therefore, for any h ∈ F the operator equation

A(u) = h (7.8)

has a unique solution u ∈ EA. Moreover, (7.8) is correctly solvable. In other words,for any surrounding O ∈ UF there is a surrounding V ∈ UA such that the inclusion(h′,h′′) ∈ O implies the inclusion (u′,u′′) ∈V , where A(u′) = h′ and A(u′′) = h′′.

Definition 7.32. A generalized solution of an operator equation A(u) = h is suchan element u ∈ EA that A(u) = h.

Obviously, a classical solution of (7.6) is generalized. If u is a generalized solu-tion of (7.6) and h ∈ R(A) (or u ∈ E), then u is a classical solution.

Theorem 7.4. For any element h ∈ F there exists a unique generalized solution ofan equation A(u) = h.

Proof. The operator A realize a uniform isomorphism between uniform spaces(

EA,UA)

and (F ,UF). ��Remark 7.40. The following equivalent definition of a generalized solution of equa-tion (7.6) has significant importance in applications: an element u ∈ EA is called ageneralized solution of equation A(u) = h, if there exists a net (uα) of elementsof space E such that lim

αA(uα) = h in F . Note that the net (uα) is UA-convergent

to u ∈ EA.

2 Two U0-Cauchy nets (uα ),(

vβ)

of elements of a uniform space (E0,U0) are called U0-equivalent, if for any symmetric surrounding O ∈ U0 there exist α0, β0 such that

(

uα ,vβ) ∈ O

as soon as α ≥ α0, β ≥ β0.


Remark 7.41. Let A = {Aλ }λ∈Λ be a one-parameter family of injective operatorsAλ : E → F . Consider a uniform structure, which is a UA -initial uniform structurewith respect to the family A in E [8]. The completion E by the uniform structureUA is a natural object for construction of the theory of generalized solvability ofthe family od equations Aλ (u) = h, λ ∈Λ .

7.9.2 Generalized Solutions and Embeddings of Uniform Spaces

Assume that the operator A is continuous. Then the spaces (E,UE ) and(

EA,UA)

are related in the following way.

Theorem 7.5. Let A : E → F be a continuous injective operator. Then a uni-form space (E,UE) is continuously and densely embedded into a uniformspace

(

EA,UA)

.

Proof. The operator A : E→F specifies a dense and continuous embedding (E,UE)in (F,UF). The operator A : EA→ F is a uniform isomorphism between the spaces(

EA,UA)

and (F,UF ). Let j : F→ F be a canonical embedding of the space (F,UF)into (F ,UF).

Fj−−−−→ F

A

�

⏐

⏐ A−1

⏐

⏐

�

Ei−−−−→ EA.

Then the operator i = A−1 ◦ j◦A specifies a dense and continuous embedding of thespace (E,UE ) into the space

(

EA,UA)

(see the commutative diagram). ��The completion (E,UE) of the space E with respect to the uniformity UE plays a

significant role in studying of solvability of (7.6). It is quite natural to try to consideran element u ∈ E as a generalized solution of the equation A(u) = h, h ∈ F , if thereexists a net (uα) of elements in E such that lim

αuα = u in E and lim

αA(uα) = h in F .

Remark 7.42. In studying equations with operators, which act in metric spaces, wecalled such solutions pseudo-generalized.

However, to obtain a theorem on correct solvability of equations in the space Ewhich is similar to Theorem 7.5 it is needed that the operator A : E → F allows anextension by continuity up to uniform isomorphism between the spaces (E,UE ) and(F ,UF).

Statement 7.3 ([8]). Let A : E → F be a uniformly continuous operator. Then theoperator A can be uniquely extended up to uniformly continuous operator A′, whichacts from E to F.


As we mentioned above, the continuity of the operator A : E → F is insufficientfor the existence of a continuous extension A′ : E → F . In addition, if neverthelesssuch extension exists then it cannot be an injective operator, even if the operatorA : E→ F is injective [8].

Let us formulate a criterium of injectivity of an extension by continuity A′ : E→F of an injective uniformly continuous operator A : E→ F .

Theorem 7.6. Let A : E → F be a uniformly continuous injective operator and A′ :E→ F is a uniformly continuous extension of the operator A onto the completion Eof the space E. The operator A′ : E→ F is injective iff the following condition holds

(π) if nets (uα),(

vβ)

of elements E are UE -Cauchy nets and UA-equivalent, thenthey are UE -equivalent.

Proof. Let A′ : E → F be an injective operator and (uα),(

vβ)

be two UE -Cauchyand UA-equivalent nets of elements in the space E . Let us put

u = limα

uα ∈ E, v = limβ

vβ ∈ E, h′ = limα

A(uα) ∈ F , k′ = limβ

A(

vβ) ∈ F .

Since

∀OF ∈UF ∃ α0, ∃ β0 : ∀ α ≥ α0, ∀ β ≥ β0(

A(uα) ,A(

vβ)) ∈ OF ,

then h′ = k′.The injectivity of the operator A′ : E → F and equalities A′ (u) = h′, A′ (v) = k′

imply that u = v, whence

∀OE ∈UE ∃ α ′0, ∃ β ′0 : ∀ α ≥ α ′0, ∀ β ≥ β ′0(

uα ,vβ) ∈ OE ,

i.e., the nets (uα),(

vβ)

are UE -equivalent.Let the conditions π) and A′ (u) = A′ (v) hold for u ∈ E and v ∈ E . Consider two

UE -Cauchy nets of elements in the space E (uα) and(

vβ)

such that u = limα

uα ∈ E ,

v = limβ

vβ ∈ E . The nets (uα) and(

vβ)

are UA-equivalent, since A′ (u) = limα

A(uα),

A′ (v) = limβ

A(

vβ)

in F . By virtue of the condition π) the nets (uα) and(

vβ)

are

UE -equivalent, therefore, u = v in E and the operator A′ is injective. ��Let us consider the following property of the extension A′ : E→ F.

Definition 7.33. The extension A′ : E → F of the operator A : E → F is calledweakly injective, if A′ (u′) �= A′ (u′′) for all u′ and u′′ such that u′ �= u′′ and at leastone of the elements u′ or u′′ belong to the space E .

Note that in this definition we equate an element u ∈ E with a class of UE -equivalent UE -Cauchy nets, which contains a stationary (u,u, ...).

Theorem 7.7. Let A : E → F be a uniformly continuous injective operator and A′ :E→ F be a uniformly continuous extension of the operator A onto the completions


E of the space E. The extension A′ : E → F is weakly injective iff the followingcondition holds:

(π∗) if the set (uα) of elements E is UE -Cauchy and UA-convergent to u∈ E, thenit is UE -convergent to u ∈ E.

Proof. Let the extension A′ : E→ F be weakly injective and u ∈ E , and the net (uα)of elements in E be UE -Cauchy and UA-convergent to u. Let us put

u = limα

uα ∈ E, h′ = A′ (u) = limα

A(uα) ∈ F , k′ = A′ (u) = A(u) ∈ F.

Since∀OF ∈UF ∃ α0 ∀ α ≥ α0 (A(uα) ,A(u)) ∈ OF ,

then h′ = k′. The fact that A′ : E→ F is weakly injective implies that u = u, i.e., thenet (uα) is UE -convergent to u ∈ E .

Let the condition π∗ holds and A′ (u) = A′ (u) = A(u) for u∈ E and u∈ E . Let usconsider an arbitrary net (uα) ∈ u. The net (uα) is UE -Cauchy and UA-convergentto u∈ E . The condition π∗ implies that u = limα uα in E . Therefore, u = u in E , andthe operator A′ is weakly injective. ��Remark 7.43. The condition π) implies π∗). In general case these conditions are notequivalent.

We saw repeatedly that in the studying of generalized solutions an importantproblem of embedding of completions E with respect to two uniformities UE andUA arises. Let us show that the embedding E into EA exists iff the operator A′ (i.e.the extension of uniformly continuous operator A : E→ F) is injective.

Recall the accurate definition of an embedding of a uniform space into a uniformspace.

Definition 7.34. A uniform space (E0,U0) is to be said uniformly (continuously)embedded into a uniform space (E1,U1) by an embedding operator j : E0 → E1,if j is a uniformly continuous (continuous) injective operator. Moreover, if the setj (E0) is dense in E1, then there exists a dense embedding of E0 into E1 using theoperator j3.

Let (E,UE) be a uniform space and (E,UE) be its completion by uniformity UE .Recall that the completion E consists of classes of u UE -equivalent and UE -Cauchynets (uα) of elements of the set E , and the base of uniformity UE is given by the sets

O ={

(u, v) ∈ E× E : ∃ (uα) ∈ u,(

vβ) ∈ v such that

(

uα ,vβ) ∈ O

}

,

where O run over UE [12].

3 It should be stressed that the operator j in this and similar definitions plays extremely importantrole: if the space E0 can be embedded into the space E1 using different operators j, then the phrase“the space E0 is embedded into the space E1” is incorrect if there is no clear understanding forwhat exactly operator j is used.


Let us define the operator j1 of canonical embedding of E into E in the followingway: an element u ∈ E is mapped to a class j1 (u) ∈ E , which contains a stationarysequence (u,u, ...,u, ...). Then the space E is densely and uniformly continuous em-bedded into its completion E using the operator of canonical embedding j1.

Besides UE , we consider the uniformity UA on E . The space (E,UA) is embed-ded into

(

EA,UA)

using the similar operator of canonical embedding j2.If the operator A : E → F is uniformly continuous, then we can define a canon-

ical mapping E into EA in the following way: every class u ∈ E is mapped to anelement ¯u = j (u) ∈ EA, which is a class containing u: ¯u ⊇ u. The map j : E → EA

is defined correctly. Indeed, let u ∈ E, (uα) and(

vβ)

belong to u. Then (uα) and(

vβ)

are UA-Cauchy nets (an image of a UE -Cauchy net under a uniformly contin-uous mapping A : E → F is a UF -Cauchy net). In addition, the nets (uα) and

(

vβ)

are UA-equivalent. Indeed, let OF be an arbitrary symmetrical surrounding fromUF . The uniform continuity of A : E → F implies that there exists a surroundingOE ∈UE such that (A(u) ,A(v)) ∈ OF , if (u,v) ∈ OE . Let us select α0 and β0 suchthat

(

uα ,vβ) ∈ OE for α ≥ α0, β ≥ β0. Then

(

A(uα) ,A(

vβ)) ∈ OF . Therefore,

every class u belongs to some class ¯u ∈ EA, and the mapping j is defined correctly.The definition of canonical embedding j implies the density of the set in j (E) is thespace EA.

The relations between the canonical embeddings j1, j2 and the canonical embed-ding j are illustrated by the following commutative diagram.

Ej1−−−−→ E

∥

∥

∥j

⏐

⏐

�

Ej2−−−−→ EA.

Definition 7.35. Let us say that the completion (E,UE) is canonically embeddedinto

(

EA,UA)

, if the canonical embedding j : E→ EA is injective.

Theorem 7.8. Let A : E → F be a uniformly continuous injective operator. Thespace (E,UE) is uniformly continuously and densely embedded into

(

EA,UA)

iffthe following condition is satisfied:

(π) if the nets (uα) and(

vβ)

of elements E are UE -Cauchy and UA-equivalent,then they are UE -equivalent.

Proof. Let the condition π) holds and u and v are two different elements E. Assumethat j (u) = j (v) = ¯u ∈ EA. Let us consider two nets (uα) ∈ u and

(

vβ) ∈ v. Then,

(uα)∈ j (u) = ¯u and(

vβ)∈ j (v) = ¯u. In other words, (uα) and

(

vβ)

are UA-Cauchy.The condition π) implies that the nets (uα) and

(

vβ)

are UE -equivalent. Therefore,u = v, that contradicts to the assumption.

Let us prove the necessity. Let the canonical mapping j : E → EA be injective,and (uα) and

(

vβ)

be two UE -Cauchy and UA-equivalent nets of elements of the setE . Let u and v be elements E containing (uα) and

(

vβ)

, respectively. By definition,


j (u) ∈ EA: j (u)⊇ u and j (v) ∈ EA: j (v)⊇ v. Therefore, (uα) ∈ j (u),(

vβ) ∈ j (v).

From the other hand, the nets (uα) and(

vβ)

are UA-equivalent. Then, j (u) = j (v);hence, by virtue of the injectivity of the mapping j the equality u = v holds. Thus,(uα) and

(

vβ)

are UE -equivalent nets. ��In a similar way we can prove the following general theorem on embedding of

completions of uniform spaces.

Theorem 7.9. Let U0, U1 be two uniformity on a set E;(

E0,U0)

and(

E1,U1)

be completions of E by uniformities U0 and U1, respectively. The space(

E0,U0)

is uniformly continuously and densely canonically embedded into(

E1,U1)

iff thefollowing conditions are satisfied:

(1) ∀O1 ∈U1 ∃O0 ∈U0 : O0 ⊆ O1.(2) If the nets (uα) and

(

vβ)

of elements E are U0-Cauchy and U1-equivalent,then they are U0-equivalent.

Remark 7.44. Let the operator A−1 : R(A)→E be uniformly continuous and the fol-lowing condition holds: the facts that (uα) and

(

vβ)

are UE -equivalent and (A(uα))

and(

A(

vβ))

UF -Cauchy imply that (A(uα)) and(

A(

vβ))

UF -equivalent. Thenthe space EA is uniformly continuously canonically embedded into E. This condi-tion in a nonlinear analogous of the known property of linear operators which allowclosuring.

7.9.3 Examples of Generalized Solutions

Let us consider two known approaches to generalized solvability of linear operatorsequations from the stated point of view (see Chap. 2). Assume that E and F areBanach spaces with norms ‖·‖E and ‖·‖F , respectively. An operator A : E → F isinjective, linear and continuous, and the set R(A) is dense in F . Then the adjointoperator A∗ : F∗ → E∗ is injective and continuous. Moreover, the set R(A∗) is densein E∗ in topology σ (E∗,E).

Let us consider a Hausdorff uniform structure UA in E with a base of surround-ings induced by the sets

{(u,v) ∈ E×E : ‖Au−Av‖F < ε}.

The uniformity of UA is an pre-image of the uniformity induced by the strong topol-ogy of the space F with respect to the operator A.

Let us complete the the space E by the uniformity UA. Denote the correspondingcompletion by EA. The continuous extension of the operator A onto EA is an isomor-phism between the spaces EA and F , more precisely, taking into account the fact that


the uniformity above are normed, it is an isometry of EA onto F . The completion ofthis kind and corresponding notion of a generalized solution of an operator equation

Au = h , h ∈ F , (7.9)

are considered in Chaps. 2 and 3.Let us consider a uniform structure U σ

F induced by the weak topology σ (F,F∗)on the subspace R(A) ⊆ F . Consider R(A) ⊆ F on the subspace. Denote by U σ

A apre-image of U σ

F with respect to the operator A. The base of uniformity U σA consist

of the sets

n⋂

k=1

{

(u,v) ∈ E×E :∣

∣

∣〈A∗ϕk,u− v〉E∗,E

∣

∣

∣< εk

}

, n ∈N, ϕk ∈ F∗, ε > 0.

Note that the topology induced by U σA on E coincides with the weak topology

σ (E,R(A∗)). Let ˜F be a completion R(A) with respect to U σF , and ˜EA is a com-

pletion of E with respect to U σA , and Ã : ˜EA → ˜F is an isomorphism of ˜EA onto ˜F

induced by the extension of A onto ˜EA.Let us show that the space F is continuously and densely embedded into ˜F .

The spaces F and ˜F are induced by the completion R(A) using the correspondinguniform structures. To finish the proof it is necessary to show only that the factsthat hn ∈ R(A) is strongly Cauchy sequence and converges to zero in the topologyσ (F,F∗) imply that ‖hn‖F → 0. Since the space F is Banach, then there exists h∈ Fsuch that ‖hn−h‖F → 0 as n→ ∞. For every h∗ ∈ F∗, we have

〈h∗,hn〉F∗,F → 〈h∗,h〉F∗,F = 0.

Therefore, h = 0, i.e., ‖hn‖F → 0.Note that ˜F = F if F is a reflexive space.We proved that the space˜EA is densely and continuously embedded into the space

EA in Sect. 6.3.The problem

u ∈˜EA : Ãu = h , h ∈ F. (7.10)

is a general statement of (7.9).For any h ∈ F there exists a unique solution u ∈ ˜EA of the operator equation

(7.10). Note that u ∈˜EA may be considered as an element, for which there exists asequence un ∈ E such that ‖Aun−h‖F → 0, un→ u in ˜EA. Indeed, if the set R(A) isdense in the space F , then there exists a sequence un ∈ E such that ‖hn−h‖F → 0as n→∞ (hn = Aun). The strong uniformity of the space F majorizes the uniformityof the space ˜F . Then, hn→ h in ˜F . The operator Ã is an isomorphism of ˜EA onto ˜F ,then there exists u ∈˜EA such that un→ u in ˜EA.

We shown that the definition of generalized solutions from the spaces ˜EA and EA

are equivalent in Chap. 7.Thus, we have done the studying of generalized solvability of abstract operator

equations.


7.9.4 Generalized Solution of Operator Equationin Proximity Spaces

In the 1930s, V. Efremovich [13, 14] tried to describe spaces in which it is possi-ble to define geometrical structure that allows to introduce the concept of uniformcontinuity of a map besides the uniform structures of A. Weil. These attempts wereassociated with the concept of proximity between two sets in connection with thefollowing considerations. It is well known that one of the main concept of generaltopology is an adherent point of set; recall that a point x in a topological space Eis called an adherent point of a set A ⊂ E if every neighborhood Vx of the point xhas a non-empty intersection with A: A∩Vx �= /0. Thus, this concept characterizesthe proximity between the point x and the set A. The map f between topologicalspaces E and F is continuous if it preserves this proximity, i.e. the image y = f (x)of any adherent point x of the set A is an adherent point of the set f (A): y ∈ f (A),where f (A) is a closure of f (A). V.Efrempovich introduced a space where the con-cept of proximity between sets is defined and he called it a proximity space; thisspace like topological one consists of elements of arbitrary nature (points) and inthis space it is possible to say whether two any subsets are proximal or not. In thiscase, there are maps like continuous maps between topological spaces. After themanner of Yu.Smirnov we will call them δ -maps (and the proximity spaces we willcall δ -spaces). The map f which acts from δ -space P into δ -space Q is called aδ -map if it preserves the proximity between sets, i.e. any two proximal sets A and Bfrom P are mapping by f into the sets f (A) and f (B) which are proximal in Q. If fis a bijective map of δ -space P onto the δ -space Q wherein the inverse map f−1 is aδ -map also then f is called a homeomorphism and the δ -spaces P and Q are calledδ -homeomorphous.

Let us pass to the concise definition of the concept of a proximity space. Afterthe manner of V.Efremovich let us call a set P a proximity space (δ -space) if for anytwo its subset it is defined whether they are proximal or not (in latter case they aresaid to be remote sets) such that the following conditions are satisfied:

1. If a set A is proximal to a set B then B is proximal to A.2. The sum of sets A and B is proximal to a set C iff at least one one of the sets A or

B is proximal to C.3. Two points of a set P are proximal iff they are equal.4. The entire set P is remote from the empty set.5. For every remote sets A and B there exist sets C and D, such that C∪D = P and

A remote from C and B is remote from D.

Substituting Condition 3 by weaker condition3’. Every point x ∈ P is proximal to itself

we obtain general δ -spaces.The examples of natural δ -spaces are metric and topological groups: sets A and

B of a metric space P are proximal if the distance between them equals to zero:

ρ (A,B) = in f {ρ (x,y) : x ∈ A,y ∈ B}= 0


Sets A and B in a topological group G are proximal if for any neighborhood U ofthe unit of the group G the intersection UA∩B ) AU ∩B, respectively) is not empty.

In any δ -space P, it is possible to introduce a topological structure consider-ing that a set A ⊆ P is closed if it contains all its proximal points. In this case allproperties of topological space are easily checkable. For any δ -space P, it is possi-ble to introduce the concept of a Cauchy sequence: it is such a sequence that anytwo its subsequences are proximal sets. As V. Efremovich stated, the it is naturallyto apply the completion process with the help of Cauchy sequences to metrizableδ -spaces. In the most general case it is possible to point out the completion processconsidering generalized Cauchy sequences (nets) {xα} where the index α runs overdirected set. Unfortunately, these vain wishes and hopes associated with the pro-cess of completion of δ -spaces were not justified. As it was shown by Yu.Smirnov,the results were very unexpected and they do not agree with the classical conceptof completion. In Chap. 2 of the paper [108], Yu.Smirnov defined a δ -extension ofthis space as a δ -space containing this space as a everywhere dense subset. It isnaturally to consider a δ -space which have not any δ -extension different from it ascomplete spaces, as it was made in the report of Yu.Smirnov to Moscow mathemat-ical society (March 11, 1952). However, as we will see hereinafter, this concept ofa complete δ -space does not equivalent to the concept of metric completeness since(see Theorem 8 in Chap. 2 [108]) if to define the concept of completeness in a nat-ural way a δ -space P will be complete iff it is bicompact (in own topology). If P isa Hausdorff space, it is just a compact space. This result contradicts to the classicaltheorems on completeness of almost all metric spaces which occur in topology andfunctional analysis. In this connection, Yu.Smirnov rejected from the term “com-pleteness” and called completions “absolutely closed” δ -spaces. However, this doesnot change the matter.

Let us pass to investigation of generalized solution of equations with continu-ous maps (operators) which act in δ -spaces. Let A is a continuous operator whichacts from a complete Hausdorff δ -space P into a Hausdorff δ -space Q. Since P iscomplete, hence it is a compact topological space, then by Tikhonov’s Theorem thismap is a homeomorphism, i.e. the inverse map A−1 : Q← P is continuous, there-fore equations with such operators have not generalized solutions. From our pointof view, the reason of this quite strange phenomenon consists in the fact that theproperties 1–5 in the definitions of proximity spaces are not correct despite that theyseem natural and obvious. This incorrectness is absent in the definition of uniformstructures of A.Weil where the concept of completeness well conforms with theclassical concept of metric completeness.

Chapter 8Generalized Extreme Elements

In the previous chapters, we introduced and investigated the concept of a generalizedsolution of a linear operator equation. In this chapter, we will give a definition ofgeneralized extreme elements of functionals, investigate the existence of generalizedextreme elements of a convex continuous functional defined in a Banach space,and illustrate this concept by examples. The chapter contains also auxiliary resultshaving independent significance. The presentation is based on the papers [74, 102].

8.1 Examples of Generalized Extreme Elements

Let E be a Banach space, M be a bounded and closed set in E , and ϕ be a boundedcontinuous function on E . We will say that the function ϕ attains a supremum (orinfimum) on M if there exists such an element x∗ ∈M (x∗ ∈M) that

supx∈M

ϕ(x) = ϕ(x∗) ( infx∈M

ϕ(x) = ϕ(x∗)).

If E is an infinite-dimensional space, then M can be non-compact, so not eachof the bounded continuous functions x �→ ϕ(x) (x ∈M) attains a supremum (or in-fimum) on M. For example, if E is a non-reflexive Banach space and M = S1(E) ={x : ‖x‖E ≤ 1} is a unit ball in E , then there exists a linear continuous functionalf ∈ E∗, which does not attain a supremum on the unit ball S1(E) (this fact is awell-known reflexivity criterion for a Banach space [40]).

In this connection, the following problem arises: to construct the extension bycontinuity for the function ϕ on M (M⊂ M, M ⊂ E) such that the extended functionϕ attains a supremum (infimum) on generalized extreme elements x∗ (x∗), i.e.

supx∈M


ϕ(x) = ϕ(x∗)),

where x∗, x∗ ∈ M, but x∗, x∗ /∈M.


163

164 8 Generalized Extreme Elements

Let us formulate a rigorous definition of a generalized extreme element. At first,note that a topology T , matching the structure of the vector space E , induces auniform structure on E , so hereinafter we will say about a completion M of the setM in the topology T , having in mind the completion of M in corresponding uniformstructure.

Definition 8.36. A generalized extreme element x∗ (x∗) of a bounded continuousfunctional ϕ on a bounded closed set M in a Banach space E is an element x∗ (x∗) ofa completion M of the set M by some Hausdorff topology T of the space E whichhas the following properties:

(1) The topology T agrees with the structure of the vector space E .(2) The topology T is weaker than the original topology of the Banach space E .(3) The functional ϕ is continuous on M in the topology T , and

supx∈M


ϕ(x) = ϕ(x∗)),

where ϕ is an extension of ϕ by continuity on the set M in the topology T .

Note that the idea of extension of a solution of an extreme problem posedby D. Hilbert1, was realized completely in 1930th years by L. Young [117] andE. McShane [71] in the case of one-dimensional problems of variational calculus inthe form of “generalized curves” (see also multidimensional extensions in [118]).Similar constructions of extensions were proposed and investigated in the optimalcontrol theory by R.V. Gamkrelidze [22] (“sliding regimes”), J. Warga [115, 116](“generalized curves” and “generalized control functions”), E. McShane [71] (“re-laxed controls”), A. Chouila-Houri [10] (“limit controls”) and other mathematicians.

Let us study the issue of the existence of generalized extreme elements for convexcontinuous functionals on a Banach space, formulate some auxiliary results havingindependent significance also, and give examples of extreme elements.

Let E be a Banach space which is densely embedded into a Banach space F (i.e.there exists a linear continuous injective operator k : E→F with a dense range in F).Recall that in such case E is called compactly embedded into F , if the operator k iscompact, i.e. a closure of an image of the unit ball S1(E) in E with respect to themetric F is a compact subset of F . In this case, a conjugate space F∗ is embeddedinto E∗ (with the help of the operator k∗ : F∗ → E∗); therefore, for every continuouslinear functional f ∈ F∗ the relation f ∈ E∗ makes sense. In addition, there existssuch an element x∗ ∈ F , that the following equality holds:

‖ f‖E∗ = supx∈S1(E)

| f (x)| = supx∈S1(E)

| f (x)| = | f (x∗)|, x∗ ∈ S1(E),

since the closure S1(E) of the unit ball S1(E) in F is a compact set, and a narrowingof the functional f ∈ F∗ on S1(E) is a continuous function. Thus, x∗ is a generalized

1 D.Hilbert posed his twentieth problem: “Do all variational problems with certain boundary con-ditions have solutions... if to use an extended interpretation of the solution?” in 1900.

8.1 Examples of Generalized Extreme Elements 165

maximal element, on which the functional f attains its norm. Note that in the caseof a non-reflexive Banach space E there exist functionals f in an conjugate spaceE∗, which do not attain a supremum of the unit ball S1(E) by virtue of the JamesTheorem. However, if such a functional f belongs to the space F∗ then there existsthe corresponding generalized maximal element in the closure S1(E) of the unit ballS1(E) (and even sphere) in topology of the space F . Later we will prove that allfunctionals f ∈ E∗ have a generalized maximal element in some Banach space.

If there exists a compact embedding E ⊂ F , then for f ∈ F∗ it is possible to pointout some intermediate Banach space H (E ⊂H ⊂ F), containing a generalized max-imal element. Namely, let us consider a completion H of the space E with respectto the space F [41]. Recall that H is a set of elements x ∈ F , for which there areexist such a real number R > 0 and a sequence xn ∈ E , ‖xn‖E ≤ R, that xn→ x in thenorm of F . Let us define a functional on each element x ∈ H

‖x‖H = inf{R ∈ R : ∃xn ∈ E,‖xn‖E ≤ R,‖xn− x‖F → 0} .

Such functional x→ ‖x‖H is a norm on the space H, the space H is complete, andthe unit ball S1(H) in the space H is a closure of the unit ball S1(E) in the space F(in the norm F) [41]. Thus, the generalized maximal element x∗ belongs to the unitball S1(H) in the space H.

Let us generalize the previous example and find a generalized maximal elementinvestigating a norm of a linear operator A, which acts from E into a Banach spaceG. Let E be densely embedded into F . Denote by EF a vector space E with norm ofthe space F . Suppose that A∈L (EF ,G), where L (EF ,G) is a space of all boundedlinear operators mapping EF into G. Such operator A can be extended by continuityon the entire space F (this extension we will denote by A : F → G). By definition,

‖A‖E→G = supx∈S1(E)

‖Ax‖G = supx∈S1(E)

‖Ax‖G,

where S1(E) is a closure of the unit ball S1(E) in the norm of the space F .Let us show that the functional ϕ(x) = ‖Ax‖G, where x ∈ E , is continuous in EF .

Indeed,

|ϕ(x)−ϕ(y)|= |‖Ax‖G−‖Ay‖G| ≤ ‖Ax−Ay‖G

= ‖A(x− y)‖G ≤ ‖A‖EF→G‖x− y‖EF =C‖x− y‖EF ,

where x,y ∈ E .If E is compactly embedded into F , the the set S1(E) would be compact in F ;

therefore, there would exists such an element x∗ ∈ S1(E), that

‖A‖E→G = supx∈S1(E)

‖Ax‖G = supx∈S1(E)

ϕ(x) = ϕ(x∗), x∗ ∈ S1(E),

where ϕ is a continuous extension ϕ from E onto F .


Thus, in this case, the functional ϕ(x) has a generalized extreme element x∗ ∈S1(E) ⊂ F , on which it attains the norm of the operator A ∈ L (E,G). Later wewill prove that in a reflexive Banach space E every compact operator A has suchproperty.

8.2 Generalized Extreme Elements for Linear and PositivelyHomogeneous Convex Functional

In the previous section, we gave the examples of generalized extreme elements whenthe space E was compactly embedded into a Banach space F . In this connection theproblem of the constructive construction of the space F arises. Note that we wouldtake the space E∗∗ (the second conjugate space) with a weak-* topology σ(E∗∗,E∗).Then the space E would be densely and compactly embedded in the space F andthe problem of the existence of generalized extremal elements would be solved.However, in this case the space F is not normed, and that would not be preferrable.

Theorem 8.1. For every separable Banach space E there exists such a separableBanach space F, that E is densely and compactly embedded into F.

Proof. Let us consider the space E∗ that is conjugate to a separable space E andhas weak-* topology σ(E∗,E). The unit ball S1(E∗) in the space E∗ is a separa-ble in weak-* topology and metrizable space [7]. Let { fn}∞

n=1 be a countable andeverywhere dense subset in S1(E∗) with respect to the topology σ(E∗,E). Let usconstruct an equable convex set W ∗ ⊂ E∗, which is compact with respect to thenorm of the space E∗ and absorbs every of the one-point sets fn. For that, considerthe set V ∗ consisting of the functionals fn/n and − fn/n, n ∈ N. Since fn ∈ S1(E∗),then the set U∗ =V ∗ ∪{ΘE∗}, where ΘE∗ is a zero element of the space E∗, is com-pact in the strong metric E∗. Since U∗ is a compact and symmetric set with respectto ΘE∗ , then a closure W ∗ = conv U∗ of a convex hull of the set U∗ in the norm E∗is equable and compact set in the metric E∗ [40]. The topology σ(E∗,E) is weakerthan the original topology of the space E∗, so W ∗ is compact in weak-* topologyσ(E∗,E) also. Since fn/n∈U∗ ⊂W ∗, then the set W ∗, clearly, absorbs all elementsfn. Since U∗ ⊂ S1(E∗), then

W ∗ = convU∗ ⊂ convS1(E∗) = S1(E∗)

and, therefore,

S1(E) = (S1(E∗))◦E ⊂ (W ∗)◦E , (8.1)

where (A)◦E is a polar of the set A⊂ E∗ in the duality (E∗,E).The inclusion (8.1) implies that the polar (W ∗)◦E of the set W ∗ in the duality

(E∗,E) is a neighborhood of zero ΘE in the linear normed space E .

8.2 Generalized Extreme Elements for Linear and Positively Homogeneous... 167

Let us show that the polar (W ∗)◦E does not contain any linear manifold. Indeed, ifto suppose that a linear set {λ x∗ : λ ∈ (−∞,∞)} belongs to (W ∗)◦E for some x∗ ∈ E(x∗ =ΘE), then

supf∈W∗| f (λx∗)| ≤ 1, ∀λ ∈ R;

hence, f (x∗) = 0 for all f ∈W ∗. However, there exists such a functional f ∗ ∈ S1(E∗)in E∗ that f ∗(x∗) = ‖x∗‖E > 0 and by virtue of the density of the subset { fn}∞

n=1in the unit ball S1(E∗) in the weak-* topology σ(E∗,E) there exists a sequence fnk

converging to f ∗ in the topology σ(E∗,E), hence fnk (x∗)→ f ∗(x∗) > 0. However,

fnk/nk ∈W ∗, so ( fnk/nk)(x∗) = 0 and fnk(x

∗) = 0. We have reached a contradiction.Now, we can see easily that the set (W ∗)◦E can be taken as the unit ball in some

normed topology, defined on E . Let ‖x‖F be a norm on E , induced by the polar(W ∗)◦E (denote the corresponding linear normed space by EF ). The embedding (8.1)implies that the norm ‖x‖F is weaker than the original norm of the space E and thefollowing equality holds:

‖x‖F ≤ ‖x‖E ∀x ∈ E.

Denote by F a Banach space obtained as a result of completion of the vector

space EF (with the norm ‖x‖F ). Also, denote by (W ∗)◦E the unit ball S1(F) in F .It is easy to see that the unit ball S1(EF) = (W ∗)◦E in EF is dense in the unit ball

S1(F) = (W ∗)◦E in F , since the ball S1(F) is obtained by closuring of the set S1(EF)in F .

Let us prove that W ∗ ⊂ F∗, i.e. every functional f0 ∈W ∗ ⊂ E∗ can be extendedby continuity (in the norm of the space F) onto the entire space F . Indeed, since

(W ∗)◦E =

{

x ∈ E : supf∈W ∗| f (x)| ≤ 1

}

,

then for an arbitrary element x ∈ (W ∗)◦E ⊂ E we have | f0(x)| ≤ 1. Since the polar(W ∗)◦E is a unit ball S1(EF), then f0 ∈ (EF )

∗ and ‖ f0‖(EF)∗ ≤ 1. Since EF is a densesubset of the space F (i.e. f0 is defined and continuous on a dense subset of thespace F), then f0 allows the extension by continuity onto the entire space F (withconservation of the norm). This proves the embedding W ∗ ⊂ F∗ (moreover, W ∗ ⊂S1(F∗)).

Let us prove that

S1(F) = (W ∗)◦E = (W ∗)◦F . (8.2)

Let x ∈ S1(F) = (W ∗)◦E ⊂ F . Since S1(EF) is dense in S1(F), then there exists asequence xn ∈ S1(EF ) = (W ∗)◦E , such that xn→ x in F . Hence,

supf∈W ∗| f (xn)| ≤ 1.


Ifsupf∈W∗| f (x)| > 1,

then there exists f0 ∈W ∗, such that | f0(x)|> 1. The latter inequality gives a contra-diction, since 1≥ | f0(xn)| and f0 ∈ F∗, i.e. f0(xn)→ f0(x).

Hence,supf∈W∗| f (x)| ≤ 1,

i.e. x ∈ (W ∗)◦F .

Now, let us prove that (W ∗)◦F ⊂ S1(F) = (W ∗)◦E . Suppose the contrary. Let thereexists x ∈ (W ∗)◦F ⊂ F , such that ‖x‖F > 1. Then ‖x‖F > 1+2ε for some ε > 0 and| f (x)| ≤ 1 for any f ∈W ∗. Let us consider the element x′ = x/(1+ ε). Then

‖x′‖F =‖x‖F

1+ ε>

1+2ε1+ ε

> 1+ ε1

and

supf∈W∗| f (x′)|= sup

f∈W ∗

| f (x)|1+ ε

≤ 11+ ε

< 1− ε2,

where ε1 > 0,ε2 > 0.Since EF is densely embedded into F , then there exists such a sequence xn ∈ EF ,

that xn→ x′ in F and ‖xn− x′‖F < ε2 for any n ∈N. Then, since W∗ ⊂ S1(F∗), then

supf∈W ∗| f (xn)|= sup

f∈W ∗| f (x′)+ f (xn− x′)| ≤ sup

f∈W∗| f (x′)|+ sup

f∈W∗| f (xn− x′)| ≤

≤ 1− ε2 + supf∈W∗‖ f‖F∗‖xn− x′‖F ≤ 1− ε2 + ε2 = 1.

Thus, xn ∈ (W ∗)◦E = S1(EF) and since xn→ x′ in F , then x′ belongs to the closureof the set (W ∗)◦E in F , i.e. x′ ∈ S1(F). So, we have the contradiction: ‖x′‖F > 1.

This way, we prove the inclusion (W∗)◦F ⊂ S1(F) = (W ∗)◦E . This finishes the proofof (8.2).

The unit ball S1(F∗) in the space F∗ is a polar of the set S1(F) = (W ∗)◦E = (W ∗)◦Fin the duality (F∗,F), since S1(F∗) is a bipolar ((W ∗)◦F )◦F∗ of the set W ∗ in theduality (F∗,F), where (A)◦F∗ – is a polar of the set A⊂ F in the duality (F∗,F).

Since W ∗ is a convex equated set, then the bipolar ((W ∗)◦F)◦F∗ is a closure of W ∗in the weak-* topology σ(F∗,F). However, by construction W ∗ ⊂ E∗ is a compactset in the topology σ(E∗,E), and since W ∗ ⊂ F∗, then W ∗ is a compact set in thetopology σ(F∗,E) also, hence W ∗ is closed in this topology also. Since the topologyσ(F∗,F) is stronger than σ(F∗,E), then W ∗ is closed in the topology σ(F∗,F),hence ((W ∗)◦F)◦F∗ =W ∗ = S1(F∗).

By construction, the set W ∗ = S1(F∗) is compact with respect to the norm of theconjugate space E∗. Therefore, the set F∗ is compactly embedded into the space E∗.By the Schauder theorem [119] the embedding E ⊂ F is compact also. ��


Remark 8.45. Theorem 8.1 implies that there exists such a space F , that for any setof functionals {g1,g2, . . .} ⊂ E∗, the space E is compactly embedded into F and thefunctionals {g1,g2, . . . ,} ⊂ E∗ are continuous in the norm of the space F .

Indeed, to construct the set W ∗ we can use the following subset which is count-able and dense in S1(E∗) with respect to the weak-* topology σ(E∗,E):

{

f1,g1

‖g1‖E∗, f2,

g2

‖g2‖E∗, . . .

}

⊂ S1(E∗).

In this case, the set W ∗ absorbs every of the functional {g1,g2, . . .} ⊂ E∗; therefore,they are continuous in the norm of the space F .

Theorem 8.2. The narrowing of any linear continuous functional defined of a unitball of a separable Banach space has a generalized extreme (maximal) element, onwhich this functional attains its norm.

Proof. Let E be a separable Banach space and f ∈ E∗. By Theorem 8.1 the set E iscompactly embedded into some Banach space F using an operator k : E → F . Letus introduce new norm in E:

‖x‖ f = | f (x)|+‖k(x)‖F , x ∈ E.

Let us denote by Ef and EF the linear set E with norm ‖x‖ f and with norm ‖k(x)‖F ,respectively. Then, for an arbitrary element x ∈ E the following inequality holds:

‖x‖ f = | f (x)|+‖k(x)‖F ≤ ‖ f‖E∗‖x‖E + c‖x‖E = M‖x‖E ,

where M does not depend on x∈E . Thus, the space E can be considered as naturallyembedded into a completion E f of the space Ef .

It is easy to see, that the functional f is linear and continuous in E f , since | f (x)| ≤‖x‖ f , and hence f allows the extension by continuity onto the entire space E f .

There exist two alternatives: (1) the functional f ∈ E∗ is continuous in the norm‖k(x)‖F (i.e. f ∈ (EF)

∗), (2) the functional f ∈ E∗ is not continuous in the norm‖k(x)‖F ( f /∈ (EF)

∗). Note, that Remark 8.45 allows to construct such a space F , thatE is compactly embedded into F and f ∈ F∗, i.e. to guarantee the first alternative.However, in some cases it is necessary to find a generalized extreme element insome a priori specified space F (only if E is compactly embedded into F). This factmake us study the second alternative also.

Under the first alternative we have that | f (x)| ≤ C‖k(x)‖F and it is easy to seethat the norm ‖k(x)‖F and ‖x‖ f are equivalent and hence the spaces E f and F areisomorphous. Indeed,

‖k(x)‖F ≤ ‖x‖ f ≤C‖k(x)‖F +‖k(x)‖F = (C+1)‖k(x)‖F , ∀x ∈ E.

Denote by f (x) the extension by continuity of a functional f (x) onto a Banachspace F , and denote by S1(E) the closure of the unit ball S1(E) in E with respectto the norm F . Then, by the Weierstrass Theorem there exists such an element x∗ in


a compact set S1(E), on which the continuous functional f (x) attains its maximalvalue. This element is a generalized extreme element of the functional f .

Let us consider the second alternative: the functional f ∈ E∗ is unbounded withrespect to the norm ‖k(x)‖F . In this case, in spite of the inequality ‖k(x)‖F ≤ ‖x‖ f

for all x ∈ E , the space E f is not embedded (using the extension of the operatorl : Ef → EF of the natural embedding l(x) = x, x ∈ E = E f = EF ) into a completionof the space E with respect to the norm ‖k(x)‖F , i.e. into the space F . Indeed, thecondition π) does not hold for the norms ‖x‖ f and ‖k(x)‖F . Recall that the conditionπ) means the following implication: if a sequence xn ∈E is Cauchy in the norm ‖·‖ f

and k(xn)→ΘF as n→ ∞ in the norm F , then xn→Θ f , n→ ∞ in the norm ‖ · ‖ f .Since the functional f (x) is unbounded in the norm ‖k(x)‖F , then there exists sucha sequence xn ∈ E , that ‖k(xn)‖F ≤ 1 and f (xn)≥ n. Put

yn =xn

f (xn)∈ E.

Then,

‖k(yn)‖F =‖k(xn)‖F

f (xn)≤ 1

n→ 0, n→ ∞;

i.e. k(yn)→ΘF , n→ ∞.Further,

‖yn− ym‖ f =

∣

∣

∣

∣

f

(

xn

f (xn)− xm

f (xm)

)∣

∣

∣

∣

+

∥

∥

∥

∥

k

(

xn

f (xn)− xm

f (xm)

)∥

∥

∥

∥

F

= ‖k (yn− ym)‖F → 0,

as n,m → ∞. Thus, the sequence yn is a Cauchy sequence in the norm ‖ · ‖ f .However,

‖yn‖ f = | f (yn)|+‖k(yn)‖F =

∣

∣

∣

∣

f

(

xn

f (xn)

)∣

∣

∣

∣

+‖k(yn)‖F = 1+‖k(yn)‖F → 1,

as n→ ∞.Thus, yn does not converge to Θ f , i.e. the condition π) does not hold.Nevertheless, the completion E f of the set E in the norm ‖ ·‖ f can be considered

as a space F ⊕R. More precisely, E f is isometrically isomorphous to the spaceF⊕R. Indeed, let us consider a mapping j, which acts from E into F⊕R by formula

j(x) = (k(x), f (x)), x ∈ E.

Since‖x‖ f = ‖ j(x)‖F⊕R,

the operator j, mapping the linear space E with the norm ‖x‖ f onto R( j) ⊂ F⊕R

is an isometry. Let us show that the range R( j) of the operator j : E f → F ⊕R iseverywhere dense in the space F ⊕R. Let (y,c) be an arbitrary element in F ⊕R.


Since there exists a dense embedding E ⊂ F , then there exists a sequence yn ∈ E ,such that k(yn)→ y, n→ ∞ in F . Since the functional f is unbounded on EF byhypothesis, then there exists a sequence xn ∈ E , such that

k(xn)→ΘF , f (xn)≥ | f (yn)|+n.

Let us consider elements

zn = yn +αnxn ∈ E,

where the numerical sequence αn satisfies the condition

f (zn) = f (yn)+αn f (xn) = c,

i.e.

αn =c− f (yn)

f (xn).

It is easy to see, that

‖k(zn)− y‖F =

∥

∥

∥

∥

k(yn)+c− f (yn)

f (xn)k(xn)− y

∥

∥

∥

∥

F

≤ ‖k(yn)− y‖F +

∣

∣

∣

∣

c− f (yn)

f (xn)

∣

∣

∣

∣

‖k(xn)‖F

≤ ‖k(yn)− y‖F +|c− f (yn)|f (yn)+ n

‖k(xn)‖F → 0, n→ ∞.

Thus, the sequence zn satisfies the condition j(zn) = (k(zn),c)→ (y,c) in F ⊕R.This implies that the range R( j) of the operator j in F⊕R is dense.

Extending the operator j : Ef ↔ R( j) onto the entire space E f by continuity, weconclude that the extended operator is an isometry between the Banach spaces E f

and F⊕R.Since the operator k : E→ F is compact, then the image of the unit ball S1(E) in

the space E under the mapping

j (S1(E))⊂ k (S1(E))× [−‖ f‖E∗ ,‖ f‖E∗ ]

is a relatively compact set. Hence, the mapping j is compact.Thus, the space E is compactly embedded into E f and f ∈ (E f )

∗. So, the func-tional f has a generalized maximal element. ��

Let E and F be Banach spaces and A be a bounded linear operator, which actsfrom E into F . We will say that the operator A attains its norm, if there exists suchan element x∗ ∈ E , that ‖x∗‖E = 1 and ‖Ax∗‖F = ‖A‖.Theorem 8.3. Let E be a reflexive Banach space, and F be an arbitrary Banachspace, and A be a compact linear operator, which acts from E into F. Then, Aattains its norm.


Proof. Consider the functional f (x) = ‖Ax‖F defined on the set x ∈ E . Let us showthat

supx∈S1(E)

f (x) = supx∈S1(E)

‖Ax‖F = ‖A‖=C.

Let {xn} be a sequence of the elements from the unit ball S1(E), such that f (xn) =‖yn‖F → C as n→ ∞, where yn = A(xn). Since the unit ball S1(E) in the reflex-ive Banach space E is a compact set in the weak topology σ(E,E∗) [40], then bythe Eberlein–Schmulian theorem [40] we can derive from the sequence {xn} a sub-sequence (denote it by {xn} again), which is weakly convergent to some elementx∗ ∈ S1(E). Since A is a compact operator, then yn = Axn converges to some ele-ment y ∈ F in the norm of the space F , therefore, for any functional f ∈ F∗ wehave

f (y) = limn→∞

f (Axn) = limn→∞

(A∗ f )xn = (A∗ f )x∗ = f (Ax∗),

i.e. y = Ax∗. Thus,

‖Ax∗‖F = limn→∞‖Axn‖F = lim

n→∞‖yn‖F =C.

��Remark 8.46. The compactness of the operator A is a very significant condition,since for any continuous operator A and reflexive Banach space E (and even F)Theorem 8.3 does not hold. Indeed, let E = F = �2, and an operator A : �2→ �2 isdefined by the formula

y = Ax =

(

12

x1,23

x2, . . . ,n

n+1xn, . . .

)

∈ �2,

where x = (x1,x2, . . .) ∈ �2. It is clear that ‖A‖ ≤ 1. From the other hand,

‖Aen‖�2 =

∥

∥

∥

∥

(

0, . . . ,n

n+1,0, . . .

)∥

∥

∥

∥

�2

=n

n+1→ 1, n→ ∞,

where e1 = (1,0,0, . . .), e2 = (0,1,0,0, . . .) and so on. Thus, ‖A‖ = 1, but for anyx ∈ S1(�2) we have ‖Ax‖< 1, i.e. the operator A does not attain its norm on the unitsphere.

Yet, if the Banach space E is reflexive, then, as it was proven by J. Linden-strauss [49], the set of linear continuous operators, which act from E into a Banachspace F and attain their norm, is strongly dense in the space L (E,F) of all linearcontinuous operators, which act from E into F . In [1, 98, 101, 105, 110, 121], somerefinements of this statement were get.

Remark 8.47. If a bounded linear operator A maps a reflexive Banach space E intoa reflexive Banach space F and it does not attain its norm on the unit sphere inthe Banach space E , then it does not have a general maximal element in any lo-cally convex topology T . Indeed, suppose the contrary: let T be a locally convextopology (see Definition 8.36) and x∗ be an element from M = S1(E), such that

8.3 On Compact Embedding into a Banach Space 173

‖Ax∗‖F = ‖A‖, where A is an extension of the operator A onto the completion E ofthe space E with respect to the topology T by continuity. Since S1(E) is a compactset in the weak topology σ(E,E∗), and the topology σ(E,(ET )∗) is weaker thatthe topology σ(E,E∗) (since (ET )∗ ⊂ E∗), then S1(E) is compact in the topologyσ(E,(ET )∗) also. Therefore, the ball S1(E) is compact in the space ET in topologyσ(ET ,(ET )∗), where ET is a completion of E with respect to the topology T .Whence, it follows that S1(E) is a closed set in ET in the topology σ(ET ,(ET )∗),therefore, S1(E) is a closed set in the topology T , so that the closure S1(E) of theunit ball S1(E) in the topology T coincides with S1(E). Thus, the extremal elementx∗ ∈ S1(E) belongs to the set S1(E). This contradicts to the assumption that theoperator A does not attain its norm on the unit ball S1(E).

8.3 On Compact Embedding into a Banach Space

In connection with results obtained in the previous section, let us consider the con-ditions under which a linear normed space can be densely and compactly embeddedin a Banach space.

Recall the concept of embedding of a linear normed space E into a Banachspace F .

Definition 8.37. A linear normed space E is said to be embedded into a Banachspace F , if there exists a bounded injective linear operator j : E → F (embeddingoperator).

Among all embeddings dense and compact embeddings are the most importantones.

Definition 8.38. If the set j(E) is dense in F , then the space E is said to be denselyembedded into the space F .

Definition 8.39. If the operator j : E → F from Definition 8.37 is compact 2, thespace E is said to be compactly embedded into F .

Let us consider spaces E and F . The investigated problem is the following:

When the space E is densely and compactly embedded into the Banach space F?

I.e., we must ascertain which conditions provide the existence of a bounded linearoperator j : E→ F with the following properties:

Ker( j) = {0} , Im( j) = F , j ∈ K(E,F) ,

where K(E,F) is a set of linear compact operators E → F .

2 Recall that a linear continuous operator T : E→ F is called compact if it maps a closed unit ballS1(E) into the set T (S1(E)) ⊆ F with a compact closure.


Since the range of a linear compact operator is a separable linear subspace, thennone of linear normed spaces E can be embedded densely and compactly into anon-separable Banach space F .

The following theorem demonstrates when a linear normed space E can bedensely and compactly embed into a Banach space F .

Theorem 8.4. Let E and F be infinite dimensional linear normed spaces, and aspace F be a Banach space. Then the folowing statements are equivalent:

(1) The space F is separable, the space E∗ is separable in the topology σ (E∗,E).(2) The space E can be densely and compactly embedded into F.

Remark 8.48. The separability of the conjugate space E∗ in a weak-* topologyσ (E∗,E) is equivalent to the existence of a countable total set M ⊆ E∗.

The proof of Theorem 8.4 is based on the following statement.

Lemma 8.1. In a infinite-dimensional linear normed space E there exists a se-quence of elements (xn) such that:

(1)∞∑

n=1‖xn‖E <+∞.

(2) for α = (αn) ∈ �∞ with∞∑

n=1αnxn = 0 implies αn = 0 for all n ∈N.

Remark 8.49. Obviously, the elements xn from Lemma 8.1 form a linearly indepen-dent system.

Proof (Lemma 8.1). Let us consider a linearly independent system of elements yn ∈E such that ‖yn‖E = 1 for all n ∈ N. Put

mn = min

{∥

∥

∥

∥

∥

n

∑k=1

ckyk

∥

∥

∥

∥

∥

E

: 12 ≤

n

∑k=1

|ck| ≤ n

}

.

Obviously, 1 > mn ≥ mn+1 > 0.Let us define a sequence of elements xn ∈ E in the following way:

xn = λnyn,

where

λn+1 =λnmn

23 ∀n ∈ N , λ1 = 1.

Let us show that the sequence (xn) has the desirable properties.We have:

0 < λn+k =λn+k−1mn+k−1

23 =

λn+k−2mn+k−223 mn+k−1

23

<λn+k−2mn+k−2

23+3 <λnmn

23k

<1

23k ∀n,k ∈N.

8.3 On Compact Embedding into a Banach Space 175

Therefore,∞

∑n=1

‖xn‖E =∞

∑n=1

λn ≤∞

∑n=1

123n <+∞.

Statement 1 is proved.Let us prove Statement 2. Let us consider an arbitrary sequence α = (αn) ∈

�∞\{0}. Let us show that∞

∑n=1

αnxn = 0.

Let α = supn∈N |αn|> 0 and n ∈ N such that |αn|> α/2. We have∥

∥

∥

∥

∥

∞

∑n=1

αnxn

∥

∥

∥

∥

∥

E

≥∥

∥

∥

∥

∥

n

∑n=1

αnxn

∥

∥

∥

∥

∥

E

−∥

∥

∥

∥

∥

∞

∑n=n+1

αnxn

∥

∥

∥

∥

∥

E

. (8.3)

Let us estimate the sums in the right-hand side of (8.3). We have

∥

∥

∥

∥

∥

n

∑n=1

αnxn

∥

∥

∥

∥

∥

E

=

∥

∥

∥

∥

∥

n

∑n=1

αnλnyn

∥

∥

∥

∥

∥

E

= max1≤n≤n

|αnλn|∥

∥

∥

∥

∥

n

∑n=1

αnλn

max1≤n≤n |αnλn|yn

∥

∥

∥

∥

∥

E

≥ max1≤n≤n

|αnλn|×mn ≥ |αnλn|×mn

>α×λn×mn

2. (8.4)

Further,∥

∥

∥

∥

∥

∞

∑n=n+1

αnxn

∥

∥

∥

∥

∥

E

≤ α∞

∑n=n+1

λn ≤ α×λn×mn

∞

∑k=1

123k <

α×λn×mn

4. (8.5)

The estimations (8.4), (8.5), and (8.3) imply that∥

∥

∥

∥

∥

∞

∑n=1

αnxn

∥

∥

∥

∥

∥

E

> 0.

Proof (Theorem 8.4). Let a space F be separable, a space E∗ be separable in thetopology σ (E∗,E). Then there exists countable linearly independent sets {φn} ⊆S1 (F), {ψn} ⊆ S1 (E∗) with the following properties:

(1) The linear span of the set {φn} is dense in F .(2) The set {ψn} is total in E∗.


Let us construct sequences of elements fn, e∗n, respectively, that satisfy conditionsof Lemma 8.1, using {φn} and {ψn} . Obviously, the linear span { fn} is dense in Falso, and {e∗n} is total in E∗.

Let us consider the operator

j (x) =∞

∑n=1〈e∗n,x〉 fn ∀x ∈ E.

Let us show that j ∈ K (E,F), Ker ( j) = {0} and Im( j) = F , i.e. j is the oper-ator of dense and compact embedding of E into F . Let us consider the followingsequence of operators

jm (x) =m

∑n=1〈e∗n,x〉 fn.

It is clear that jm are bounded and linear operators and dim Im( jm)<+∞. Therefore,jm ∈ K (E,F). Since

‖ j− jm‖E→F = supx∈S1(E)

‖ j (x)− jm (x)‖F

= supx∈S1(E)

∥

∥

∥

∥

∥

∞

∑n=m+1

〈e∗n,x〉 fn

∥

∥

∥

∥

∥

F

≤∞

∑n=m+1

‖e∗n‖E∗‖ fn‖F → 0 as m→ ∞,

then j ∈ K (E,F).Let x ∈ E be such that

j (x) =∞

∑n=1

〈e∗n,x〉 fn = 0.

Then 〈e∗n,x〉 = 0 for all n ∈ N. The totality of {e∗n} implies that x = 0. Therefore,Ker ( j) = {0}.

Let us show that Im( j) = F . Arguing by contradiction, assume that there existf ∗ ∈ F∗\{0} such that 〈 f ∗, j (x)〉= 0 for all x ∈ E , i.e.,

⟨

f ∗,∞

∑n=1

〈e∗n,x〉 fn

⟩

=∞

∑n=1

〈e∗n,x〉〈 f ∗, fn〉=⟨

∞

∑n=1

〈 f ∗, fn〉e∗n,x⟩

= 0 ∀x ∈ E.

Thus, we have∞∑

n=1〈 f ∗, fn〉e∗n = 0. Hence, 〈 f ∗, fn〉 = 0 for all n ∈ N. Since a linear

span of the set { fn} is dense in F , then f ∗ = 0. This contradicts to the assumptionf ∗ = 0.

Let there exist an operator j ∈ K (E,F) such that Ker ( j) = {0} and Im( j) = F .Since the subspace Im( j) ⊆ F is separable and dense in F , then the space F isseparable. Further, the conjugate space Im( j)∗ has a countable total subset. Since

8.4 Generalized Extreme Elements for General Convex Functionals 177

j ∈ L(E, Im( j)), Ker ( j) = {0}, then E∗ has a countable total subset also. It is equiv-alent to the separability of the space E∗ in the topology σ (E∗,E).

Theorem 8.5. Let E and F be infinite dimensional separable spaces, and the spaceF is a Banach space. Then the spaces E and E∗ can be densely and compactlyembedded into the space F.

Proof. The assumption implies that the space E∗ is separable in the topologyσ (E∗,E), and the space E∗∗ be separable in the topology σ (E∗∗,E∗). Next, it isnecessary to apply Theorem 8.4. ��

8.4 Generalized Extreme Elements for GeneralConvex Functionals

Let F∗ be a space conjugate to a Banach space F . By Banach–Alaoglu Theoremthe ball S1(F∗) is compact in the topology σ(F∗,F). This statement is true for anyconvex, closed, and bounded set X ⊂ F∗. Therefore, the minimization problem

f (x)→ infx∈X

,

where the functional f : F∗ →R is lower semi-continuous in the topology σ(F∗,F),has solutions. But sometimes it is necessary to consider extremal problems on con-vex and bounded subsets of Banach spaces, which are not isomorphous to conju-gate spaces (for example, in the space L1(0,1)). As a rule, these problems have notsolutions.

In this section we will show that any convex continuous functional defined on aconvex, closed, and bounded subset of a Banach space has a generalized extremeelement. In the previous section we studied generalized solutions compactly em-bedding a Banach space into another Banach space. Now, we will construct an iso-metrical and dense in a weak topology embedding of of the original Banach spaceinto an conjugate Banach space. This space will depend on elements of extremeproblems – functional and feasible set.

Let us cite some auxiliary results which provide a basis for the construction ofgeneralized solutions of convex extreme problems proposed below.

Theorem 8.6. Let X be a convex, bounded, and closed subset of a Banach space(E,‖×‖E). Then there exists such a linear subspace F ⊂ E∗ such that:

(1) (F,‖×‖E∗) is a separable linear normed space.(2) F is a subspace of characteristic one, i.e.

∀x ∈ E : ‖x‖E = supy∈F∩S1(E∗) |〈y,x〉E∗,E | .


(3) an arbitrary point x ∈ X \E is strictly Hausdorff from the set X by an elementof the subspace F, i.e.

∀x ∈ E \X ∃y ∈ F : 〈y,x〉E∗,E > supx∈X〈y, x〉E∗,E .

Proof. Let us consider a set M = {xn : n ∈ N} which is countable and everywheredense in E . Let us construct a system of functionals {yn} ⊂ E∗ such that

∀n ∈N : ‖yn‖E∗ = 1, 〈yn,xn〉E∗,E = ‖xn‖E .

Denote by F0 a linear span of the set {yn : n∈N}. Let x ∈ E . For any ε > 0 considersuch an element xn ∈M that ‖x− xn‖E < ε. Then,

〈yn,x〉E∗,E = 〈yn,xn + x− xn〉E∗,E = ‖xn‖E+

+〈yn,x− xn〉E∗,E > ‖xn‖E − ε ≥ ‖x‖E−2ε.

Therefore, F0 is a subset E∗ of characteristics one.Without loss of generality, we can consider that θ ∈ X and X ⊂ Sr(E), where

r > 0. Let us consider a set M \X = {zn}. Note that

δn = ρ(zn,X) = infx∈X‖zn− x‖E > 0.

Now, let us construct a set for every n ∈N:

Xn =⋃

x∈X

(

x+Sδn/2(E))

.

Let μn be a Minkowski functional of the set Xn. Let us define the followingfunctional on the linear subspace Ln = {λ zn : λ ∈R}:

y′n(λ zn) = λ ∀λ ∈R.

From the fact that y′n(zn) = 1 and zn ∈ E \Xn it follows that y′n(zn) < μn(zn). Bythe Hahn–Banach Theorem there exists a functional y′′n ∈ E∗ such that:

〈y′′n,x〉E∗,E = y′n(x) ∀x ∈ Ln,

〈y′′n,x〉E∗,E ≤ μn(x) ∀x ∈ E,

Let us show that ‖y′′n‖E∗ ≤ 2/δn. To do that let us majorize values μn on the ballS1(E). Since Sδn/2(E)⊂ Xn, then

μn(x)≤ μSδn/2(E)(x) ∀x ∈ E,

where μSδn/2(E)is a Minkowski functional of the set Sδn/2(E). By the definition of

a Minkowski functional the inequality μSδn/2(E)(x) ≤ δn/2 holds for x ∈ S1(E). It

implies the estimation ‖y′′n‖E∗ ≤ 2/δn.


Let us show that functionals from the set {y′′n : n ∈ N} strictly separate points ofthe set E \X from the set X . Indeed, let x ∈ E \X and α = ρ(x,X) = infx′∈X‖x−x′‖E > 0. Let us consider a point zn ∈M \X such that ‖zn− x‖E < ε for ε ∈ (0,α).Then for the functional y′′n ∈ E∗ the following inequality holds

〈y′′n ,x〉E∗,E = 〈y′′n,zn〉E∗,E + 〈y′′n,x− zn〉E∗,E ≥ 1−‖y′′n‖E∗‖x− zn‖E > 1− 2δn

ε.

Since− ε < α− δn < ε, (8.6)

we get the estimation

〈y′′n ,x〉E∗,E > 1− 2εα− ε

. (8.7)

Let us prove the following inequality:

supx′∈X 〈y′′n,x′〉E∗,E ≤ 1−Cn, (8.8)

where Cn = C(r,δn) is a number from the interval (0,1). Let us select a numberCn ∈ (0,1) satisfying the condition

(1−Cn)

(

1+δn

2r

)

> 1,

i.e.

Cn ∈(

0,δn

2r+ δn

)

. (8.9)

Let us show that (8.9) implies (8.8). Indeed, if the inequality 〈y′′n,x′〉E∗,E > 1−Cn

holds for some point x′ ∈ X , then x′′ = x′+ δn2r x′ ∈ Xn. Therefore, 〈y′′n,x′′〉E∗,E ≤ 1.

From the other hand, we have

〈y′′n,x′′〉E∗,E = 〈y′′n ,x′〉E∗,E(

1+δn

2r

)

> (1−Cn)

(

1+δn

2r

)

> 1.

So, we have a contradiction.Taking into account (8.6), (8.8), and (8.9), we obtain the inequality

supx′∈X 〈y′′n,x′〉E∗,E ≤ 1− α− ε2r+α + ε

. (8.10)

The inequalities (8.7) and (8.10) allow to conclude that selecting ε > 0 smallenough for some y′′n we can get the inequality

〈y′′n ,x〉E∗,E > supx′∈X〈y,x′〉E∗,E ,

which required to be proved.


Let us denote by F a linear span of the set F0∪{y′′n : n ∈ N}. It is clear that thelinear normed space (F,‖× ‖E∗) is separable and other statements of the theoremare also true. ��Remark 8.50. If the Banach space (E,‖×‖E) is reflexive, then the closure of the lin-ear subset F ⊂ E∗ constructed in Theorem 8.6 coincides with E∗, as a Banach spacetopologically conjugate to a reflexive Banach space does not contain any proper,closed and total linear subspaces.

Theorem 8.7. Let X be a non-empty subset of a separable Banach space (E,‖ ·‖E),f : E → R be a continuous convex functional. Then there exist a linear subspaceF ⊂ E∗ such that:

(1) (F,‖ · ‖E∗) is a separable linear normed space.(2) If for any y ∈ F 〈y,xn〉E∗,E → 〈y,x〉E∗,E as n→ ∞ (x ∈ X, xn ∈ X), then

f (x) ≤ limn→∞

f (xn).

Proof. It is known that a continuous convex functional is locally Lipschitz; in addi-tion, it has a subdifferential in the sense of convex analysis [15].

Let {x′n : n∈N} be a countable and dense subset of X . For any n∈N let us selectan arbitrary functional y′n ∈ ∂ f (xn) and consider a set F , which is a linear span ofthe set {y′n : n ∈N}.

Let us show that the theorem is true for F . Of course, it is necessary to prove thesecond statement only. Let x ∈ X and the sequence xn ∈ X be such that

∀y ∈ F : 〈y,xn〉E∗,E → 〈y,x〉E∗,E .

Let us take x′k and consider the difference

f (xn)− f (x) = f (xn)− f (x′k)+ f (x′k)− f (x)

≥ 〈y′k,xn− x′k〉E∗,E + f (x′k)− f (x)

= 〈y′k,xn− x〉E∗,E + 〈y′k,x− x′k〉E∗,E + f (x′k)− f (x).

Passing to the lower limit as n→ ∞, we obtain

limn→∞

f (xn)− f (x)≥ 〈y′k,x− x′k〉E∗,E + f (x′k)− f (x). (8.11)

Let us show that selecting x′k properly we can make the right-hand side of (8.11)arbitrarily small, and the theorem will be proved. Let the functional f on the ballx+Sδ (E) satisfy the Lipschitz condition with constant L = L(x,δ ) ≥ 0. Select x′n ∈x+Sδ(E) in a such way that ‖x′k− x‖E → 0. Then we get

limn→∞

f (xn)− f (x)≥−(‖y′k‖E∗+L)‖x′k− x‖E . (8.12)


The fact that the functional f is Lipschitz continuous on x+Sδ (E) implies that theset {‖y′k‖E∗} is bounded, and therefore, the right-hand side of (8.12) tends to zero.

��The following theorem is an direct corollary of Theorems 8.6 and 8.7.

Theorem 8.8. Let X be a non-empty, convex, bounded, and closed subset of a sepa-rable Banach space (E,‖·‖E), fk : E→R be continuous convex functionals (k∈N).Then there exists a linear subspace F ⊂ E∗ such that:

(1) (F,‖ · ‖E∗) is a separable Banach space.(2) F is a subspace E∗ of characteristics one.(3) an arbitrary point x ∈ X \E is strictly Hausdorff from the set X by an element

of the subspace F.(4) if for all y ∈ F 〈y,xn〉E∗,E → 〈y,x〉E∗,E as n→ ∞ (x ∈ X, xn ∈ X), then

∀k ∈N f (x)≤ limn→∞

f (xn).

Proof. The theorem holds for a set F , which is a closed linear span of the set∞⋃

k=0Fk,

where F0 is a linear subspace E∗ satisfying assumptions of Theorem 8.6, Fk is alinear subspace, which is constructed for the kth functional and satisfies conditionsof Theorem 8.7. ��

Let us consider generalized extreme elements of convex functionals. Let(E,‖ · ‖E) be a separable Banach space. Let us consider the minimization problem

f (x)→ infx∈X

, (8.13)

where X = ∅ is a convex, bounded, and closed subset of the space E , functionalf is continuous and convex on the set X . Denote by infX f the infimum of f on X ,and denote by arginfX f = {x ∈ X : f (x) = infX f} the set of classical solutions ofProblem (8.13).

If there are no additional propositions, Problem (8.13) can have no solutions.Our aim is to construct a generalized statement of the problem (8.13) which has asolution.

Let us construct a Banach space (F,‖ · ‖E∗) satisfying Theorem 8.8 for the setX and functional f . Denote by ‖ · ‖F the narrowing of the norm ‖ · ‖E∗ onto F .The linear subspace F ⊂ E∗ is total. Therefore, we can consider in E a Hausdorffand locally convex topology σ(E,F). The fundamental system of neighbors of thistopology is the following collection of sets:

W (y1, ...,yn;ε) = {x ∈ E : |〈yk,x〉E∗,E |< ε, yk ∈ F, 1≤ k ≤ n} .

Theorem 8.8 implies that the functional f is lower continuous on X with respectto the topology σ(E,F), and the set X is closed in the topology σ(E,F).


Let us consider a Banach space (F∗,‖ · ‖F∗) conjugate to the space (F,‖ · ‖F).This space can be used as an extension of the space (E,‖ · ‖E). If the Banachspace (E,‖ · ‖E) is reflexive (in this case Problem (8.13) is solvable), then (seeRemark 8.50) the space (F∗,‖ · ‖F∗) coincides with (E,‖ · ‖E).

The following simple statement describes the relation between the spaces(E,‖ · ‖E) and (F∗,‖ · ‖F∗).

Statement 8.4. The space (E,‖ · ‖E) is linearly and isometrically embedded intothe space (F∗,‖ · ‖F∗).

Proof. A point x ∈ E induces a linear functional jx on F:

( jx)(y) = 〈y,x〉E∗,E ∀y ∈ F.

It is clear that jx ∈ F∗. The operator j : E → F∗ is linear. The fact that it isisometrical follows from the following equalities:

‖ jx‖F∗ = sup‖y‖F=1|( jx)(y)| = supy∈F∩S1(E∗)|〈y,x〉E∗,E |= ‖x‖E ∀x ∈ E.

Here we have taken into account that F is a subspace in E∗ of characteristics one.��

Remark 8.51. It is easy to see that for all x ∈ E jx = πx|F , where π : E → E∗∗ is acanonical embedding of E into the second conjugate space E∗∗.

Statement 8.5. The set j (S1(E)) is sequentially dense in S1(F∗) with respect to thetopology σ(F∗,F).

Proof. Denote by Ms a sequential closure of the set M ⊂ F∗ in topology σ(F∗,F).Let us show that j(S1(E))s = S1(F∗). The closure and sequential closure of a convexbounded subset of a conjugate space to a separable Banach space coincide in thetopology σ(F∗,F) [86]. Therefore, it is sufficient to prove that S = S1(F∗), whereS is a σ(F∗,F)-closure of the set j(S1(E)).

The fact that the ball S1(F∗) is σ(F∗,F)-closed implies that S⊂ S1(F∗). The setS is convex. Let us show that S ⊃ S1(F∗). Suppose that there exists such a pointthat x0 ∈ S1(F∗)\S. Then there exist a σ(F∗,F)-continuous linear functional y andpositive numbers c and ε such that

c+ ε ≤ y(x0), supx∈Sy(x)≤ c.

By the Banach theorem on weakly continuous linear functional [7] for y there exista unique y ∈ F such that y(x) = 〈x,y〉F∗,F for all x ∈ F∗. Since j(S1(E)) ⊂ S1(F∗),then 〈y,x〉E∗,E ≤ c for all x ∈ S1(E). The central symmetry of the ball S1(E) im-plies that |〈y,x〉E∗ ,E | ≤ c for any x ∈ S1(E), i.e. ‖y‖F ≤ c. Then, we have |y(x0)| =|〈x0,y〉F∗,F | ≤ ‖x0‖F∗‖y‖F ≤ c and that contradicts the inequality y(x0) � c+ ε.Therefore, S⊃ S1(F∗) and hence S = S1(F∗) ��Statement 8.6. The set j(E) is sequentially dense in F∗ with respect to the topologyσ(F∗,F).


Proof. Since j(E)s is a linear subspace in F∗, there exists the embeddingj(S1(E))s ⊂ j(E)s and, by Statement 8.5, j(S1(E))s = S1(F∗), then j(E)s co-incides with the entire space F∗. ��Statement 8.7. The bounded set M ⊂ F∗ is relatively sequentially compact in thetopology σ(F∗,F).

Proof. It follows from the Banach–Alaoglu Theorem and the fact that the space F∗is conjugate to a separable Banach space (F,‖ · ‖F). ��

Let us define the generalization of problem (8.13).Let us consider the set ˜X , which is a sequential closure of the set j(X) in the

topology σ(F∗,F). The set ˜X ⊂ F∗ is convex and σ(F∗,F)-compact.Note that the elements of E , which does not belong to X , cannot be elements of

the set ˜X . More precisely, if x ∈ E \X , then jx ∈ F∗ \ ˜X . Let us suppose the contrary.Then there exists a sequence xn ∈ X such that jxn → jx in the topology σ(F∗,F).But the fact that x ∈ E \ X implies that there exists a functional y ∈ F such that〈y,x〉E∗,E > supx′∈X 〈y,x′〉E∗,E , in particular, 〈 jx,y〉F∗,F > supn∈N〈 jxn,y〉F∗,F .

Let us construct the continuation of the functional f onto the set ˜X . For all x ∈ ˜Xput

˜f (x) = inf

{

limn→∞

f (xn) : xn ∈ X , jxn→ x in the topology σ(F∗,F)

}

. (8.14)

Lemma 8.2. The following statements hold:

(1) The functional ˜f is convex on ˜X.(2) ˜f | j(X) = f .

(3) infX f = in f˜X˜f .

(4) The functional ˜f is lower σ(F∗,F)-semicontinuous on ˜X.

Proof. Statement 1) immediately follows from the convexity of f and the definitionof ˜f . Indeed, let us consider points x′ ∈ ˜X , x′′ ∈ ˜X . Let us fix an arbitrary ε > 0.It follows from (8.14) (the definition of ˜f ) that there exist such sequences x′n ∈ Xx′′n ∈ X , that

jx′n→ x′ and jx′′n → x′′ in the topology σ(F∗,F),

f (x′n)< ˜f (x′)+ ε and f (x′′n)< ˜f (x′′)+ ε.

Let us take λ ∈ [0,1] and pass to the lower limit in the inequality

f (λ x′n +(1−λ )x′′n)≤ λ f (x′n)+ (1−λ ) f (x′′n)< λ ˜f (x′)+ (1−λ )˜f (x′′)+ ε

˜f (λ x′+(1−λ )x′′)≤ limn→∞

f (λx′n +(1−λ )x′′n)≤ λ ˜f (x′)+ (1−λ )˜f (x′′)+ ε.


Since ε > 0 is arbitrary, we have:

˜f (λ x′+(1−λ )x′′)≤ λ ˜f (x′)+ (1−λ )˜f(x′′).

Let us prove Statement 2). Let x ∈ X . Consider a sequence xn ∈ X such thatjxn→ jx in the topology σ(F∗,F). The fact that f is lower σ(E,F)-semicontinuouson X implies that f (x) ≤ lim

n→∞f (xn). Passing to infimum in the inequality, we have

f (x) ≤ ˜f ( jx). Taking a stationary sequence xn = x, we get the opposite inequalityf (x)≥ ˜f ( jx). Therefore, ˜f | j(X) = f .

Statement 2) implies the inequality infX f ≥ inf˜X˜f . Suppose that infX f > inf

˜X˜f .

Then for some ε > 0 there exists a point x ∈ ˜X such that infX f > ˜f (x)+ε . The defi-nition ˜f implies that there exists a point x ∈ X such that ˜f (x)+ε > f (x). Therefore,infX f > f (x). This contradiction proves Statement 3).

Let us prove Statement 4). Let x ∈ ˜X . Consider a sequence xn ∈ ˜X such thatxn→ x in the topology σ(F∗,F). Let us show that

˜f (x)≤ limn→∞

˜f (xn). (8.15)

If xn ∈ j(X) for all n ∈ N, then (8.15) immediately follows from (8.14). Supposethat xn ∈ ˜X \ j(X) for all n∈N. It is known that the topology σ(F∗,F) is metrized bysome metric d on the set ˜X [7]. Let us take an arbitrary ε > 0. The fact that j(X) isσ(F∗,F)-dense in ˜X and (8.14) implies that there exists a sequence of points xn ∈ Xsuch that

d( jxn, xn)≤ d(xn, x), (8.16)

f (xn)< ˜f (xn)+ ε. (8.17)

Inequality (8.16) implies that jxn converges to x ∈ ˜X in the topology σ(F∗,F).Taking into account (8.14), let us pass to the lower limit in (8.17):

˜f (x)≤ limn→∞

f (xn)≤ limn→∞

˜f (xn)+ ε.

Thus, inequality (8.15) holds. ��Let us set up a correspondence between Problem (8.13) and the following mini-

mization problem˜f (x)→ inf

x∈˜X, (8.18)

We will call it a generalized definition of Problem (8.13) or F∗-extension of Prob-lem (8.13).

Remark 8.52. It should be stressed that if the space (E,‖ ·‖E ) is reflexive, then F∗-extension of Problem (8.13), i.e. (8.18), coincides with the original problem (8.13).


Remark 8.53. Using Theorem 8.8, we can construct an extension of the problemdefinition with the help of a general space F∗ for an arbitrary countable family ofconvex extreme problems

fk(x)→ infx∈X

, k ∈ N.

Definition 8.40. An element x ∈ ˜X is called a generalized solution of Prob-lem (8.13), if ˜f (x) = inf

˜X˜f . Denote by arginf

˜X˜f the set of all generalized solution

of Problem (8.13).

The following theorem holds.

Theorem 8.9. The set arginf˜X˜f of generalized solution of Problem (8.13) is non-

empty, convex and σ(F∗,F)-compact.

Proof. The convexity of the set arginf˜X˜f follows from the convexity of ˜X and ˜f .

Let us show that arginf˜X˜f =∅. Consider an arbitrary sequence of points xn ∈ ˜X

minimizing the functional ˜f for F∗-extension of Problem (8.13), i.e.

˜f (xn)→ inf˜X˜f .

The set ˜X is compact in the topology σ(F∗,F), therefore, there exist such a subse-quence {xnk} and a point x ∈ ˜X that

xnk → x in the topology σ(F∗,F).

The fact that the functional ˜f is lower σ(F∗,F)-semicontinuous implies that

˜f (x)≤ limk→∞

˜f (xnk) = inf˜X˜f ;

hence, x ∈ arginf˜X˜f .

The compactness in the topology σ(F∗,F) can be proved in a similar way. ��Let us ascertain the relation between generalized solutions, classical solutions

and minimizing sequences of Problem (8.13).

Theorem 8.10. The following statements hold:

(1) j(arg infX f ) = j(X)⋂

arginf˜X˜f .

(2) If {xn} is a minimizing sequence of Problem (8.13), then there exist such anelement x ∈ arginf

˜X˜f and subsequence {xnk} that jxnk → x in the topology

σ(F∗,F).(3) If x ∈ arginf

˜X˜f , then there exists such a minimizing sequence of Problem (8.13)

{xn} that jxn→ x in the topology σ(F∗,F).

Proof. The first statement follows from Statements 2) and 3) of Lemma 8.2. Letus prove Statement 2). Let xn ∈ X and f (xn)→ infX f . The fact that the set ˜X isσ(F∗,F)-compact implies that there exists a subsequence {xnk} such that jxnk → x


in the topology σ(F∗,F), x ∈ ˜X . It follows from the fact that the functional ˜f islower σ(F∗,F)-semicontinuous that

˜f (x)≤ limk→∞

˜f ( jxnk) = limk→∞

f (xnk) = infX f = inf˜X˜f ,

i.e. x ∈ arginf˜X˜f .

Finally, let us prove Statement 3). Let x ∈ arginf˜X˜f . The fact that j(X) in ˜X ⊂ F∗

is σ(F∗,F)-dense and the functional ˜f is lower σ(F∗,F)-semicontinuous impliesthat there exists a sequence of points xn ∈ X such that jxn → x in the topologyσ(F∗,F) and

inf˜X˜f = ˜f (x)≤ lim

n→∞˜f ( jxn) = lim

n→∞f (xn).

Passing to a subsequence (when it is necessary) and conserving the old denotations,we get:

jxn→ x in the topology σ(F∗,F),

limn→∞

f (xn) = infX f ,

which required to be proved. ��Let us formulate a sequential analogue of the classical condition of extreme for

Problem (8.13). Suppose that the functional f is differentiable by Gateau on theset X .

Theorem 8.11. Let x ∈ arginf˜X˜f . Then there exists a sequence {xn}, xn ∈ X such

that:

jxn→ x in the topology σ(F∗,F), (8.19)

f (xn)→ infX f = ˜f (x), (8.20)

limn→∞〈 f ′(xn),x− xn〉E∗,E ≥ 0 ∀x ∈ X . (8.21)

Proof. Theorem 8.10 implies that there exists a sequence x′n ∈ X such that

jx′n→ x in the topology σ(F∗,F), (8.22)

0≤ f (x′n)− infX f < 1/n. (8.23)

According to the Ekeland’s variational principle [15] for an arbitrary n ∈ N thereexists such a point xn ∈ X , that

f (xn)≤ f (x′n), (8.24)

‖xn− x′n‖E ≤√

1/n, (8.25)

f (xn)< f (x)+√

1/n‖x− xn‖E ∀x ∈ X \ {xn}. (8.26)

8.5 Some Remarks 187

From (8.22) and (8.25) we get (8.19), and from (8.23) and (8.24) we get (8.20).Let us substitute the point x with the point xn + τ(x− xn) ∈ X (x ∈ X , τ ∈ (0,1))in (8.26):

f (xn + τ(x− xn))− f (xn)>−τ√

1/n‖x− xn‖E .

Dividing the latter inequality by τ and passing to the limit as τ→ 0, we have:

〈 f ′(xn),x− xn〉E∗,E ≥−√

1/n‖x− xn‖E ∀x ∈ X .

Tending n to infinity and taking into account the fact that the set X is bounded, weget inequality (8.21). ��Remark 8.54. If we do not demand the smoothness of f , then the inequality (8.21)in Theorem 8.11 should be replaced either by

limn→∞

f ′(xn;x− xn)≥ 0 ∀x ∈ X ,

where f ′(x; p) is a directional derivative of f in direction p at the point x, either bythe inequality

limn→∞

{

supy∈∂ f (xn)〈y,x− xn〉E∗,E}

≥ 0 ∀x ∈ X .

Remark 8.55. Because of the reasonings above it becomes clear that we couldembed the space E into the second conjugate space E∗∗ and, repeating the corre-sponding closuring and extensions we could get the generalized solutions of Prob-lem (8.13) from the space E∗∗. However, as a rule the elements of the space E∗∗have non-constructive description only. That is why we have selected that way thatleads to embedding E into F∗.

8.5 Some Remarks

At first, let us discuss the possibility to apply the scheme described above to theconvex maximization problem.

Let us consider an extreme problem

f (x)→ supx∈X

, (8.27)

where X = ∅ is a convex, bounded and closed subset of a separable Banach spaceE , the functional f : E → R is convex and Lipschitz on bounded subsets of E .

Let us note that the Lipschitz property of a convex functional on bounded sets isequivalent to the boundedness of this functional on bounded sets. Indeed, let M > 0and K = supx∈S2M(E) | f (x)|, then 2K/M is a Lipschitz constant for f on the ballSM(E) [15].


If the functional f is not sequentially continuous in the topology σ(E,E∗), thenthere exists a non-empty convex and bounded set X ⊂ E where the functional f doesnot attain its supremum [65].

Theorem 8.12. Let (E,‖·‖E) be a linear normed space , f : E→R be a continuousconvex functional. If the functional f is not sequentially continuous in the topologyσ (E,E∗), then there exists a bounded, closed and convex set X ⊂ E, where thefunctional f does not attain its supremum.

Proof. Let x∗ ∈E be a point, at which the functional f is not sequentially continuousin the topology σ (E,E∗), i.e. there exist such a number ε > 0 and sequence xn ∈ E ,that xn→ x∗ weakly in E and | f (xn)− f (x∗)| ≥ ε for every n ∈ N. Without loss ofgenerality, we can consider that f (x∗) = 0.

There exists such a number n∗ ∈ N that the inequality f (xn) ≥ ε holds for alln≥ n∗. Indeed, otherwise there exists such a subsequence {xnk}, that f

(

xnk

)

<−ε,and, respectively, lim

n→∞f (xn)≤−ε and that contradicts to the fact that the functional

f is lower semicontinuity.For an arbitrary number n≥ n∗ there exists such a point

yn ∈ [xn,x∗] = {y ∈ X : y = λxn +(1−λ )x∗, λ ∈ [0,1]} ,

that f (yn) = αnε, where {αn} is a sequence of real numbers from the interval (0,1),which steadily converges to 1. It is clear that yn→ x∗ weakly in E . Let us considera closed, convex and bounded set X = conv{yn : n≥ n∗}. The fact that f is convexand continuous implies that X ⊂ {x ∈ E : f (x)≤ ε}.

Let us show that the functional f does not attain its supremum on the set X .Suppose that there exists x ∈ X : f (x) = supx∈X f (x). In particular, f (x)≥ f (yn) =αnε for all n≥ n∗, whence,

f (x)≥ supn≥n∗αnε = ε. (8.28)

Let us consider {zp} such that zp ∈ conv{yn : n≥ n∗}, zp→ x strongly in E . Thefact that the functional f is continuous and (8.28) imply

ε ≤ f (x) = limp→∞

f (zp) .

If we show that zp−−−→p→∞

x∗ weakly in E , then we get x = x∗ and an absurd inequality

f (x∗) = 0 < ε ≤ f (x) = f (x∗) = 0.

For an arbitrary p∈N the vector zp has the form zp = ∑∞n=n∗ λp,nyn, where λp,n ∈

[0,1], ∑∞n=n∗ λp,n = 1 and λp,n = 0 for a fixed p starting from some n.


Let us prove that λp,n→ 0 as p→ ∞ for an arbitrary n≥ n∗. Taking into accountthe convexity of the functional f for every m≥ n∗, we can write the inequality

f (zp) = f( ∞

∑n=n∗

λp,nyn

)

≤∞

∑n=n∗

λp,n f (yn) =∞

∑n=n∗

λp,nαnε

≤ λp,mαmε +∞

∑n=n∗,n =m

λp,nε

=( ∞

∑n=n∗

λp,n+(αm−1)λp,m

)

ε

= (1+(αm−1)λp,m)ε.

Passing to the lower limit as p and fixing the number m≥ n∗, we have:

limp→∞

f (zp)≤ limp→∞

(1+(αm−1)λp,m)ε =(

1+(αm−1) limp→∞

λp,m

)

ε.

Hence, limp→∞

λp,m ≤ 0 and, taking into account the fact that λp,m is non-negative, we

getlimp→∞

λp,m = 0. (8.29)

Let us take a functional l ∈ E∗ and introduce a denotation C = supx∈X

∣

∣〈l,x〉E∗,E∣

∣.Then for all n≥ n∗:

∣

∣〈l,x∗〉E∗,E∣

∣≤C∣

∣〈l,yn〉E∗,E∣

∣≤C.

The weak convergency of the sequence {yn} to x∗ implies that for an arbitraryε ′ > 0 there exists n′ ≥ n∗, such that

∣

∣〈l,yn− x∗〉E∗,E∣

∣< ε ′/2 ∀n≥ n′.

For all p ∈ N

∣

∣

∣

∞

∑n=n′

λp,n 〈l,yn− x∗〉E∗,E∣

∣

∣≤∞

∑n=n′

λp,n∣

∣〈l,yn− x∗〉E∗,E∣

∣<( ∞

∑n=n′

λp,n

)ε ′

2≤ ε ′

2.

Taking into account (8.29), we conclude that ∃p′ ∈ N: 0 ≤ λp,n < ε ′4Cn′ for all

p≥ p′, n∗ ≤ n < n′. Therefore, for all p≥ p′:

∣

∣

∣

⟨

l,zp− x∗⟩

E∗,E

∣

∣

∣=∣

∣

∣

∞

∑n=n∗

λp,n 〈l,yn− x∗〉E∗,E∣

∣

∣

≤n′−1

∑n=n∗

λp,n

∣

∣

∣〈l,yn− x∗〉E∗,E∣

∣

∣+∞

∑n=n′

λp,n

∣

∣

∣〈l,yn− x∗〉E∗,E∣

∣

∣

<n′−1

∑n=n∗

ε ′

4Cn′2C+

ε ′

2< ε ′.

Thus, we proved that zp→ x∗ in the topology σ(E,E∗). ��


The analysis of typical problem definitions in optimal control theory and estima-tion theory shows that the condition of weak sequential continuity of the functionalf , as a rule, does not hold and Problem (8.27) frequently has not solutions.

Let us use the approach described above to get some “regularization” of Prob-lem 8.27.

Let us construct spaces (F,‖ ·‖F) and (F∗,‖ ·‖F∗). Then, we will enclose the setj(X) in the space F∗ up to the set ˜X and we will extend the functional f onto F∗ inthe following way:

˜f (x) = inf{

limn→∞

f (xn) : xn ∈ X , jxn→ x in σ(F∗,F), {xn} is bounded}

.

The extended functional ˜f is convex and lower semi-continuous with respect to thetopology σ(F∗,F). The following inequality holds:

˜f (x)≤ sup{

f (x) : x ∈ S‖x‖F∗ (E)} ∀x ∈ F∗.

Thus, the functional ˜f is bounded on bounded subsets of the space F∗, therefore, ithas Lipschitz property on the set ˜X .

Let us consider the problem

˜f (x)→ supx∈˜X

.

This problem can have not solutions. However, it is posed in a conjugate Banachspace and the feasible set ˜X is compact in the topology σ(F∗,F).

The results of the work [103] imply that for any ε > 0 there exists such a func-tional y ∈ F that ‖y‖F < ε and the problem

˜f (x)+ 〈x,y〉F∗,F → supx∈˜X

.

has non-empty set of solutions.Let F be a subspace of the space E∗ of characteristics one. Then, the functional

E � x �→ ‖x‖E is lower semi-continuous with respect to the topology σ(E,F). In-deed, let x ∈ E . For any ε > 0 there exists y ∈ F ∩S1(E∗) such that

‖x‖E − ε ≤ 〈y,x〉E∗,E .

If xn→ x in the topology σ(E,F), then

limn→∞‖xn‖E−‖x‖E = lim

n→∞

(‖xn‖E−‖x‖E−〈y,xn− x〉E∗,E)

= limn→∞

(‖xn‖E−〈y,xn〉E∗,E −‖x‖E + 〈y,x〉E∗,E)≥−ε,


as ‖xn‖E −〈y,xn〉E∗,E ≥ 0. The fact that ε > 0 is arbitrary implies that

‖x‖E ≤ limn→∞‖xn‖E .

Consider the best approximation problem

‖x− y‖E→ infx∈X

,

where X = ∅ is convex, bounded and closed subset of the space E , y ∈ E \X . Thismeans that the construction of the generalized definitions of the best approximationproblems depend only on the properties of the set X .

Let us consider in detail the example of the application of generalized extremeelements. Let us formulate an extreme problem in the space L1(−1,1):

f (x) =∫ 1

−1|x(t)|dt−2

∫ 1

−1(1− t2)x(t)dt→ inf, (8.30)

x ∈ S1(L1(−1,1)) ={

x ∈ L1(−1,1) : ‖x‖L1(−1,1) =

∫ 1

−1|x(t)|dt ≤ 1

}

. (8.31)

Let us show that problem (8.30), (8.31) has no solutions in the space L1(−1,1).Indeed, from the one hand, it is clear that

∀x ∈ S1(L1(−1,1)) : f (x)≥ ‖x‖L1(−1,1)−2‖x‖L1(−1,1) =−‖x‖L1(−1,1) ≥−1

and the values of the functional f on the elements of the sequence xn = nχ[0,1/n] ∈S1(L1(−1,1)) satisfy the relations

f (xn) =−1+2

3n2 →−1 as n→ ∞.

Thus,inf

S1(L1(−1,1))f =−1.

From the other hand, the functional f does not attain the value −1 on the ballS1(L1(−1,1)). Indeed, if we suppose that f (x) = −1 for some x ∈ S1(L1(−1,1)),then

∫ 1

−1|x(t)|dt =

∫ 1

−1(1− t2)x(t)dt = 1. (8.32)

However, there exists δ > 0 such that

∫ −δ

−1|x(t)|dt > 0 or

∫ 1

δ|x(t)|dt > 0,


otherwise x = 0 almost everywhere on [−1,1]. We get the chain of the inequalities,that contradicts to (8.32)

∫ 1

−1(1− t2)x(t)dt =

∫ −δ

−1(1− t2)x(t)dt +

∫ δ

−δ(1− t2)x(t)dt +

∫ 1

δ(1− t2)x(t)dt

≤∫ −δ

−1(1− δ 2)|x(t)|dt +

∫ δ

−δ|x(t)|dt +

∫ 1

δ(1− δ 2)|x(t)|dt

=

∫ 1

−1|x(t)|dt− δ 2

∫ −δ

−1|x(t)|dt− δ 2

∫ 1

δ|x(t)|dt < ‖x‖L1(−1,1).

Let us consider problems (8.30) and (8.31) in a generalized definition. As a sub-space F ⊂ (L1(−1,1))∗, which satisfies Theorems 8.6 and 8.7, we can choose theset of the functionals induced by the elements of C([−1,1]), i.e.

y ∈ F ⇔ ∃y ∈C([−1,1]) : 〈y,x〉L∞ ,L1 =

∫ 1

−1y(t)x(t)dt ∀x ∈ L1(−1,1).

The mapping F � y �→ y ∈ C([−1,1]) is a linear isometrical isomorphism betweenthe spaces (F,‖ · ‖L∞) and

(

C([−1,1]),‖ · ‖C([−1,1]))

. By the Riesz theorem [31] theconjugate space (F∗,‖ · ‖F∗) is isometrically isomorphous to the space M(−1,1)of all Borel measures of finite variate defined on the segment [−1,1]. The setS1(L1(−1,1)) can be closed in the Banach space M(−1,1) up to the set

S1(M(−1,1)) ={

μ ∈M(−1,1) : ‖μ‖M(−1,1) = var(μ)≤ 1}

.

The functional f can be extended onto the entire space M(−1,1).So, we get the generalized problem

˜f (μ) =∫ 1

−1d|μ |(t)−2

∫ 1

−1(1− t2)dμ(t)→ inf,

μ ∈M(−1,1), var(μ)≤ 1.

for which Theorems 8.9 and 8.10 hold. By the way, the generalized solution of thisproblem is a Dirac δ -measure, lumped at the point 0.

The requirement that the Banach space (E,‖ · ‖E) must be separable is not aprincipal restriction. The construction of generalized solutions of convex extremeproblems can be used without this condition. Of course, we can use more universaltopological mathematical tools – nets and filters. The extension of the functional fonto the set ˜X (now, it is a closure of j(X) in the topology σ(F∗,F)) can be con-structed in the following way:

˜f (x) = inf{

limα

f (xα) : xα ∈ X , jxα → x in the topology σ(F∗,F)}

,


or

˜f (x) = supV∈V (x)

(

infx∈ j−1(V∩˜X)

f (x))

,

where V (x) is a fundamental system of neighborhoods of the point x ∈ F∗ in thetopology σ(F∗,F). The functional ˜f is called lower semi-continuous regularizationof the functional f [53].

Let us consider the possibility of using the scheme of F∗-extension in the prob-lems of minimization of non-convex functionals. The extreme problem has the fol-lowing form:

f (x)→ infx∈X

, (8.33)

where, in contrast to (8.13), the convexity of the functional f is not supposed. Let usconsider the subspace F ⊂ E∗ satisfying Theorem 8.6. If the functional f is lowersemi-continuous on X in the topology σ(E,F), then it is possible to construct a gen-eralized definition of Problem (8.33), which is correct in the space F∗. Therefore,the key problem consists in the investigation of the conditions of lower σ(E,F)-semicontinuity of the non-convex functional f on the set X for the given total sub-space F ⊂ E∗.

A.Ioffe and V.Tikhomirov [28] considered the problem of minimization of real-valued functional f , defined on a Hausdorff topological space X , and formalizedthe concept of the variational problem extension. Namely, the problem f → infX

matches to a pair (X , f ), called by the authors a variational [28]. A variational pair(Y,g) is called an extension of (X , f ), if there exists a continuous mapping i : X→Ysuch that:

(1) The set i(X) is dense in Y .(2) f (x) ≥ g(i(x)) ∀x ∈ X .(3) ∀y∈Y ∀V ∈V (y) infx∈i−1(V ) f (x)≤ g(y), where V (y) is a fundamental system

of neighborhoods of the point y ∈ Y .

If the functional g is lower semi-continuous and the space Y is compact, then theextension (Y,g) is called a regular extension of the variational pair (X , f ).

The extensions of classic variational problems were studied in [28]. However, theauthors systematically used an approach, which differs from the approach describedabove. The set of feasible solutions did not change, but the functional was conversedinto a convex one in a special way

The definition of extension implies that infX f = infY g. It is easy to show that ify = i(x) is a locally optimal point of a pair (Y,g), then x is a locally optimal point ofa pair (X , f ). From the other hand, in general we can say nothing about the existenceof minX f , if there exists minY g, but frequently it is possible to formulate sufficientconditions of the existence of minX f , based on the existence of minY g. For example,the theorems on existence of classical solutions of elliptic boundary value problemswere proved. Their proofs were based on increasing smoothness of weak solutionsfrom Sobolev spaces.


We constructed a regular extension of Problem (8.13) in the sense of Ioffe-Tikhomirov. The resulting problem (˜P) is a relaxation of problem (8.13) in the senseof [15], as their infimums coincide, and all solutions (˜P) are limits of minimizingsequences of Problem (8.13).

Also, it is interesting to study the scheme of generalization based on the F∗-extension on the game search saddle point problems, problems of Nash and Paretoequilibria, and problems of hierarchical optimization (the Stackelberg problem).

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Index

BBasis

orthonormal, 113Schauder, 103

Bilinear form, 2

CCondition number, 109

Todd, 109, 111Turing, 109, 111

DDual pair, 2

EEmbedding, 2

natural, 4Equation

differentialelliptic, 97parabolic, 49

integralFredholm 1st kind, 48Volterra 1st kind, 44Volterra 2nd kind, 46

linear, 18nonlinear, 137

Extreme element, 163generalized, 164

FFilter, 128

Cauchy, 129minimal, 127

Functionalcontinuous, 3convex, 2

linear, 1Lipschitz continuous, 181locally Lipschitz, 180Minkowski, 178

IInequality

a2 +b2 � 2ab, 93a priori, 17Bessel, 32Cauchy-Buniakovsky, 86Cauchy-Schwarz, 71coercivity, 86Friedrichs, 73Holder, 5Hirsch-Bendixon, 105Schwarz, 52triangle, 141

Isometry, 23Isomorphism, 23

LLimit element

strong, 10

MMatrix

finite, 105ill-posed, 109infinite, 37

MethodHotteling, 111Neumann, 106

NNear-solution, 10

strong, 10weak, 12

D.A. Klyushin et al., Generalized Solutions of Operator Equations and Extreme Elements, 201Springer Optimization and Its Applications 55, DOI 10.1007/978-1-4614-0619-8,© Springer Science+Business Media, LLC 2012

202 Index

Neighborhood, 2Normalization, 113

OOperator

adjoint, 2, 3bijective, 3closable, 9coercive, 9completely regular, 41continuous, 3elliptic, 86Fredholm, 34Hammerstein, 149Hermitian, 31Hilbert-Schmidt, 29injective, 3integral, 32linear, 2Nemytskii, 148of embedding, 3of natural embedding, 4surjective, 3Uryson, 148Volterra, 44

PPolar, 2Principle

Ekeland, 186

SSet

adjoint, 3bounded, 3embedded densely, 4total, 2

Solutionclassical, 8generalized

strong, 8, 18weak, 11

SpaceC(0,1), 4C1(0,1), 4Lp(D), 5

W (l)p (D), 5

Banach, 4conjugate, 1dual, 2embedded, 3Frechet, 99Hilbert, 29intermediate, 3linear, 1normed, 3perfectly complete, 98reflexive, 10second conjugate, 1Sobolev, 5topological, 3

locally convex, 22uniform, 152vector, 1

Subdifferential, 3System

completely regular, 40normed, 113regular, 40

TTheorem

Arzela, 5Banach-Alaoglu, 177, 183Banach-Steinhaus, 22, 121Eberlein-Schmulian, 172Hahn-Banach, 24, 25, 67, 133, 178Hirsch-Bendixon, 105Krasnoselskii, 124Lebesque, 148Lions, 97Luzin, 5Mackey-Arens, 129Mercer, 33Riesz Representation, 67Vishik-Lax-Milgram, 97

Topology, 2normed, 8weak, 3

UUniform structure, 126

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