research article existence and boundedness of solutions...

Research ArticleExistence and Boundedness of Solutions forNonlinear Volterra Difference Equations in Banach Spaces

Rigoberto Medina

Departamento de Ciencias Exactas Universidad de Los Lagos Casilla 933 Osorno Chile

Correspondence should be addressed to Rigoberto Medina rmedinaulagoscl

Received 25 June 2016 Accepted 30 October 2016

Academic Editor Jozef Banas

Copyright copy 2016 Rigoberto Medina This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider a class of nonlinear discrete-time Volterra equations in Banach spaces Estimates for the norm of operator-valuedfunctions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equationsare elements of an appropriate Banach spaceThese estimates give us explicit boundedness conditionsThe boundedness of solutionsto Volterra equations with infinite delay is also investigated

1 Introduction

In many phenomena of the real world not only does theirevolution prove to be dependent on the present state but itis essentially specified by the entire previous history Theseprocesses are encountered in the theory of viscoelasticity[1 2] optimal control problems [3] and also description ofthe motion of bodies with reference to hereditary [3ndash5] Themathematical description of these processes can be carriedout with the aid of equations with the aftereffect integraland integrodifferential equations A significant contributionto the development of this direction was made by V VolterraV B Kolmanovskii N N Krasovskii S M V Lunel A DMyshkis and J K Hale

The aim of this article is to develop a technique forinvestigating stability and boundedness of nonlinear implicitVolterra difference systems described by Volterra operatorequations Only a few papers deal with the theory of gen-eral Volterra equations and most of them are devoted tothe stability analysis of explicit Volterra linear differenceequations with constant coefficients or of convolution typeFor example in Minh [6] some results on asymptoticstability and almost periodicity are stated in terms of spectralproperties of the equations and their solutions which arelinearVolterra equations of convolution type InMinh [7] the

asymptotic behavior of individual orbits of linear functionaloperator equations are established using an extension of theKatznelson-Tzafririrsquos theorem [8] InNagabuchi [9] using thedecomposition of the phase space together with the variationof constants formula in the phase space the existence ofalmost periodic solutions for forced linear Volterra differenceequations in Banach spaces is derived The considered equa-tions are of convolution type In Murakami and Nagabuchi[10] sufficient stability properties and the asymptotic almostperiodicity for linear Volterra difference equations in Banachspaces are derived Gonzalez et al [11] consider an implicitnonlinear Volterra difference equation in a Hilbert space andobtained sufficient conditions so that the solutions exist andhave a bounded behavior The coefficients of the consideredequations are sequences of real numbers In Song and Baker[12] several necessary and sufficient conditions for stabilityare obtained for solutions of the linear Volterra differenceequations by considering the equations in various choicesof Banach spaces However the main results of this articleare established essentially to implicit linear Volterra differ-ence equations In Muroya and Ishiwata [13] consideringa nonlinear Volterra difference equation with unboundeddelay sufficient conditions for the zero solution to be globallyasymptotically stable are derived However only an explicitscalarVolterra difference is considered InGyori andHorvath

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 1319049 6 pageshttpdxdoiorg10115520161319049

2 Abstract and Applied Analysis

[14] sufficient conditions are presented underwhich the solu-tions to a linear nonconvolution Volterra difference equationconverge to limits which are given by a limit formula InKolmanovskii et al [15] stability and boundedness problemsof some class of scalarVolterra nonlinear difference equationsare investigated Stability conditions and boundedness areformulated in terms of the characteristics equations In Songand Baker [16] the fixed point theory is used to establishsufficient conditions to ensure the stability of the zero solu-tion of an implicit nonlinear Volterra difference equationBesides the existence of asymptotically periodic solutionsis established However in this article equations with linearkernel are mainly considered

One of the basic methods in the theory of stability andboundedness of Volterra difference equations is the directLyapunovmethod (see [1 3 17 18]) But finding the Lyapunovfunctionals for Volterra difference equations is still a difficulttask

In this paper to establish boundedness conditions ofsolutions we will interpret the Volterra difference equationswith nonlinear kernels as operator equations in appropriatespaces Such an approach for linear Volterra difference equa-tions has been used by Myshkis [5] Kolmanovskii et al [15]Kwapisz [19] and Medina [20 21] The knowledge of thesebounds is important because they represent the error betweenthe exact solution of the original problem and its differ-ence approximation

Existence and uniqueness problems for the Volterradifference equations were discussed by some authors Usuallythe solutionswere sought in the phase space 119897120596119901 (119885+ 119883) 120596 gt 0and 119885+ = 0 1 2 (eg see [15 19]) In this paper formu-lating the Volterra discrete equations in the phase space119897119901(119885+ 119883) where 119883 is an appropriate Hilbert space andassuming that the kernel operator has the Volterra propertywe obtain sufficient conditions for the existence and unique-ness problem

Our results compare favorably with the above-mentionedworks in the following sense

(a) Sufficient conditions for the existence and uniquenessof solutions of implicit nonlinear Volterra differenceequations are obtained

(b) We established a theory on the asymptotic behavior ofimplicit nonlinear Volterra difference systems whichare described by Volterra operator equations

(c) Explicit estimates for the solutions of nonlinearVolterra operator equations in Hilbert spaces arederived

The remainder of this article is organized as followsin Section 2 we introduce some notations preliminaryresults and the statement of the problem In Section 3 theboundedness of solutions is derived using norm-estimates forthe resolvents of completely continuous quasi-nilpotent oper-ators In Section 4 we discuss the boundedness of solutions ofinfinite-delay Volterra difference equations Finally Section 5is devoted to the discussion of our results we highlight themain conclusions

2 Statement of the Problem

Let119883 be a complex Hilbert space with norm sdot 119883 fl sdot Let119897119901 = 119897119901(119885+ 119883) and 119897infin = 119897infin(119885+ 119883) be the spaces of sequencesℎ = ℎ(119895)infin119895=1 such that

119897119901 = ℎ 119885+ 997888rarr 119883 infinsum119895=1

1003817100381710038171003817ℎ (119895)1003817100381710038171003817119901 lt infin 1 le 119901 lt infin

119897infin = ℎ 119885+ 997888rarr 119883 sup119895isin119885+

1003817100381710038171003817ℎ (119895)1003817100381710038171003817 lt infin (1)

The spaces 119897119901 1 le 119901 le infin equippedwith the standard norms

ℎ119901 = (infinsum119895=1

1003817100381710038171003817ℎ (119895)1003817100381710038171003817119901)1119901

for 1 le 119901 lt infinℎinfin = sup

119895isinZ+

1003817100381710038171003817ℎ (119895)1003817100381710038171003817 (2)

are Banach spacesDenote

Ω119903 = ℎ isin 119897infin ℎinfin le 119903 for 0 lt 119903 le infin (3)

We consider Volterra difference equations on a separableHilbert space119883

119909 (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895 119909 (119895)) 119899 ge 0 (4)

where

119870 119885+ times 119885+ times 120585 isin 119883 10038171003817100381710038171205851003817100381710038171003817infin lt 119903 997888rarr 119883 (5)

is the kernel 119885+ denotes the set of nonnegative integers and119891 119885+ rarr 119883 is a given sequenceEquation (4) can be regarded as the discrete-time analog

of the classical integral Volterra equation of the second kind

119909 (119905) = 119891 (119905) + int1199050119870 (119905 119904 119909 (119904)) 119889119904 (119905 ge 0) (6)

We point out the distinction between (4) and the explicitVolterra equation

119909 (119899) = 119891 (119899) + 119899minus1sum119895=0

119870(119899 119895 119909 (119895)) 119899 ge 0 (7)

The first step is to establish the solvability of (4) In the linearcase in which we write (4) as

119909 (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895) 119909 (119895) 119899 ge 0 (8)

with 119870(119899 119895) being a finite-dimensional matrix the existenceof 119909(119899) for each 119899 follows from the property that (119868 minus119870(119899 119899))

Abstract and Applied Analysis 3

is invertible for each 119899 ge 0 Song and Baker [16] deal withthe existence of solutions of (4) defined on 119889-dimensionalEuclidean spaces In fact considering implicit equations ofthe form

119909 (119899 + 1) = 120595119899 (119909 (0) 119909 (1) 119909 (119899) 119909 (119899 + 1)) (9)

and assuming appropriate conditions on 120595119899 they concludethat solutions of equations

119909 = 120595119899 (1199060 1199061 119906119899 119909) (10)

are unique and can be expressed in the form

119909 = 120593119899 (1199060 1199061 119906119899) (11)

We present a different and more general approach to theproblem of existence and uniqueness discussed in [16]Namely for a given general discrete operator 119879 119878(119885+ 119883) rarr119878(119885+ 119883) consider the equation

119906 (119899) = (119879119906) (119899) 119899 isin 119885+ (12)

where 119878(119885+ 119883) denotes the linear space of all sequences 119904 119885+ rarr 119883We ask when the solutions of this equation belong to the

space 119897119901(119885+ 119883) of all functions 119909 isin 119878(119885+ 119883) satisfying thecondition

(infinsum119895=1

1003817100381710038171003817119909 (119895)1003817100381710038171003817119901)1119901

lt infin for 1 le 119901 le infin (13)

Assumption A The operator 119879 is a causal operator that is forany 119899 isin 119885+

(119879119906) (119899) = (119879V) (119899) if 119906 (119895) = V (119895) for 119895 = 0 1 119899 minus 1 (14)

That is the value of (119879119906) at 119899 is determined by the values of119906(119895) for 119895 = 0 1 119899 minus 1 (eg see Corduneanu [22])

Remark 1 The causal property of the operator 119879 guaranteesthe existence and uniqueness of the solutions of (12) Con-sequently when the operator 119879 of (12) has for instance theform

(119879119909) (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895 119909 (119895)) (15)

we can establish existence and uniqueness results for thesolutions of (4)

Theorem 2 If the operator 119879 defined in (15) is a causaloperator then there exists a solution of (4)

Finally we will determine sufficient conditions on thecoefficients of (4) such that its solutions belong to the space119897119901(119885+ 119883) 1 le 119901 le infin

In the finite-dimensional case the spectrum of a linearoperator consists of its eigenvalues The spectral theory of

bounded linear operators on infinite-dimensional spaces isan important but challenging area of research For examplean operator may have a continuous spectrum in addition toor instead of a point spectrum of eigenvalues A particularlysimple and important case is that of compact self-adjointoperators Compact operatorsmay be approximated by finite-dimensional operators and their spectral theory is close tothat of finite-dimensional operators

To formulate the next result let us introduce the followingnotations and definitions let 119867 be a separable Hilbert spaceand 119860 a linear compact operator in 119867 If 119890119896infin119896=1 is anorthogonal basis in119867 and the seriessuminfin119896=1(119860119890119896 119890119896) convergesthen the sum of the series is called the trace of the operator119860and is denoted by

trace (119860) = tr (119860) = infinsum119896=1

(119860119890119896 119890119896) (16)

Definition 3 An operator119860 satisfying the relation tr(119860lowast119860) ltinfin is said to be a Hilbert-Schmidt operator where 119860lowast is theadjoint operator of 119860

The norm

1198732 (119860) = 119873 (119860) = radictr (119860lowast119860) (17)

is called the Hilbert-Schmidt norm of 119860Definition 4 A bounded linear operator119860 is said to be quasi-Hermitian if its imaginary component

119860119868 = 119860 minus 119860lowast2119894 (18)

is a Hilbert-Schmidt operator where 119860lowast is the adjointoperator of 119860Theorem 5 (see [23 Lemma 231]) Let 119881 be a Hilbert-Schmidt completely continuous quasi-nilpotent (Volterra) oper-ator acting in a separable Hilbert space119867 Then the inequality

1003817100381710038171003817100381711988111989610038171003817100381710038171003817 le 119873119896 (119881)radic119896 119891119900119903 119886119899119910 119899119886119905119906119903119886119897 119896 (19)

is true

3 Main Results

Let 119860119903 denote the infinite matrix with components

119860119903119899119895 fl sup119911isin119883119911le119903

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 for 119899 119895 = 1 2 (20)

Assume that

119873119901 (119860119903) = [[infinsum119899=1

( 119899sum119895=1

119860119902119903119899119895)119901119902]

]1119902

lt infin (21)

which implies

120573119901 (119860119903) = sup119899=12

[[119899sum119895=1

119860119902119903119899119895]]1119902

lt infin (22)

where 1119901 + 1119902 = 1


Denote

119898119901 (119860119903) = infinsum119895=0

119873119895119901 (119860119903)119901radic119895 (23)

Theorem 6 Assume that 119891 isin 119897119901(119885+ 119883) and condition (21)holds Then any solution 119909 = (1199091 1199092 ) of (4) belongs to119897119901(119885+ 119883) and satisfies the inequalities

119909119901 le 119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (24)

119909infin le 10038171003817100381710038171198911003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (25)

provided that10038171003817100381710038171198911003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 lt 119903 (26)

Proof of Theorem 6 We will decompose the proof of Theo-rem 6 in the following lemmas

Lemma 7 Assume that 119891 isin 119897119901(119885+ 119883) 1 le 119901 lt infin andcondition (21) holds with 119903 = infin that is 119860infin with components

119860infin (119899 119895) = sup119911isin119883

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 119899 119895 = 1 2 (27)

is a Hilbert-Schmidt kernelThen any solution 119909 = (1199091 1199092 )of (4) belongs to 119897119901(119885+ 119883) and satisfies the inequality

119909119901 le 119898119901 (119860infin) 10038171003817100381710038171198911003817100381710038171003817119901 (28)

Proof From (4) we have

119909 (119899) le 119899sum119895=1

119860infin (119899 119895) 1003817100381710038171003817119909 (119895)1003817100381710038171003817 + 1003817100381710038171003817119891 (119899)1003817100381710038171003817 (29)

Let 119871 be the linear operator defined on 119897119901(119885+ 119883) by(119871ℎ) (119899) = 119899sum

119895=1

119860infin (119899 119895) ℎ (119895) ℎ isin 119897119901 (119885+ 119883) (30)

Rewrite equation (4) as

119909 = 119891 + 119871119909 (31)

Hence

119909 = (119868 minus 119871)minus1 119891 (32)

Since

(119868 minus 119871)minus1 = infinsum119895=0

119871119895 (33)

we have

10038171003817100381710038171003817(119868 minus 119871)minus110038171003817100381710038171003817119901 leinfinsum119895=0

1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 (34)

Thus by (32)

119909119901 le infinsum119895=0

1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 10038171003817100381710038171198911003817100381710038171003817119901 (35)

Since 119871 is a quasi-nilpotent Hilbert-Schmidt operator itfollows by Gilrsquo [23 Lemma 231] that

1003817100381710038171003817100381711987111989610038171003817100381710038171003817119901 le 119873119896119901 (119860infin)119901radic119896 (36)

This and (35) yield the required result

Remark 8 By Holderrsquos inequality we have

119898119901 (119860infin) = (1 minus 1119901119902119901)minus1119902

exp [119873119901119901 (119860infin)] (37)

Consequently from Lemma 7

119909119901 le 120574119901 exp [119873119901119901 (119860infin)] 10038171003817100381710038171198911003817100381710038171003817119901 (38)

where

120574119901 = (1 minus 1119901119902119901)minus1119902 (39)

Lemma 9 Assume that 119891 isin 119897infin(119885+ 119883) and condition (21)holds with 119903 = infin and then any solution 119909 = (1199091 1199092 ) of(4) belongs to 119897infin(119885+ 119883) and satisfies the inequality

119909infin le 10038171003817100381710038171198911003817100381710038171003817infin + 120573119901 (119860infin)119898119901 (119860infin) 10038171003817100381710038171198911003817100381710038171003817119901 (40)

Proof From (31) it follows that

119909infin le 10038171003817100381710038171198911003817100381710038171003817infin + 119871119909infin (41)

But due to Holderrsquos inequality

119871119909infin le sup119899=12

[ 119899sum119896=1

119860119902infin (119899 119896)]1119902 119909119901= 120573119901 (119860infin) 119909119901

(42)

Hence (28) and (42) yield


Now we will complete the Proof of Theorem 6 consider-ing the case 0 lt 119903 lt infin

By Urysohnrsquos Lemma [24 p 15] there are scalar-valuedfunctions 120582119903 and 120583119903 defined on 119897119901(119885+ 119883) and119883 respectivelysuch that

120582119903 (ℎ) = 1 ℎinfin le 1199030 ℎinfin gt 119903

120583119903 (ℎ) = 1 ℎ le 1199030 ℎ gt 119903

(44)


Define119870119903 (119899 119895 119911) = 120582119903 (ℎ)119870 (119899 119895 119911)

119891119903 (ℎ) = 120583119903 (ℎ) 119891 (ℎ) (45)

Consider the Volterra equation

119909 (119899) = 119891119903 (119899) + 119899sum119895=1

119870119903 (119899 119895 119909 (119895)) 119899 ge 0 (46)

Due to Lemma 9 and condition (26) any solution of (46)satisfies (25) and therefore belongs to Ω119903 But 119870119903 = 119870 and119891119903 = 119891 on Ω119903 This proves estimates (25) for a solution of(4) Estimates (24) follows from Lemma 7This completes theproof of Theorem 6

4 Volterra Difference Equations withInfinite Delay

Consider the Volterra difference equation of the form

119909 (119899) = 119891 (119899) + 119899sum119895=minusinfin

119870(119899 119895 119909 (119895)) 119899 ge 0 (47)

which can be regarded as a retarded equation whose delayis infinite In general this problem requires that one give anldquoinitial functionrdquo on (minusinfin 0] in order to obtain a uniquesolution after which the equation can be treated with thetechniques of standard Volterra equations The nonunique-ness of solutions of (47) is an intrinsic feature (eg see [25])If 120593 (minusinfin 0] rarr 119883 is an initial function to (47) then wewrite (47) as

119909 (119899) = (119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)))+ 119899sum119895=1

119870(119899 119895 119909 (119895))= (119899) + 119899sum

119895=1

119870(119899 119895 119909 (119895)) (48)

so that the initial function 120593 becomes part of the sequence(119899) Under these conditions we can apply Theorem 5 to(48) so that the next result is obtained

Theorem 10 Assume that isin 119897119901(119885+ 119883) 1 le 119901 le infin andcondition (21) holds Then the solution 119909(119899) = 119909(119899 120593) of (48)belongs to 119897119901(119885+ 119883) and satisfies the inequalities

119909119901 le 119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901 119909infin le 1003817100381710038171003817100381710038171003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901

(49)

Proof Proceeding in a similar way to the proof ofTheorem 6with

(119899) = 119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)) (50)

instead of 119891 and choosing an appropriate initial function 120593such that isin 119897119901(119885+ 119883) the required result follows

Remark 11 Consider

119909 (119899) = 119909 (119899 120593) forall119899 ge 0119909 (119895 120593) = 120593 (119895) for 119895 lt 0 (51)

5 Conclusions

New conditions for the existence uniqueness and bounded-ness of solutions of infinite-dimensional nonlinear Volterradifference systems are derived Unlike the classic method ofstability analysis we do not use the technique of the Lyapunovfunctions in the process of construction of the estimates forthe solutions The proofs are carried out using estimates forthe norm of powers of quasi-nilpotent operators That is weinterpreted the Volterra difference equations with nonlinearkernels as operator equations defined in the Banach spaces119897119901(119885+ 119883) We want to point out that the results of this papercan be useful for discussions of 119897119901(119885+ 119883) stability of the zerosolution of the homogeneous Volterra equation correspond-ing to (4)

In connection with the above investigations some openproblems arise The richness of the spectral properties ofoperators acting on infinite-dimensional Hilbert spaces willneed new stability formulations Consequently natural direc-tions for future research is the generalization of our resultsto local exponential stabilizability of nonlinear Volterra dif-ference equations or investigating the feedback stabilizationof implicit nonlinear Volterra systems defined by operatorVolterra equations

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by Direccion de Investigacionunder Grant NU0616

References

[1] A D Drozdov and V B Kolmanovskii Stability in Viscoelastic-ity vol 38 Elsevier Amsterdam The Netherlands 1994

[2] N Levinson ldquoA nonlinear Volterra equation arising in thetheory of superfluidityrdquo Journal of Mathematical Analysis andApplications vol 1 no 1 pp 1ndash11 1960

[3] V B Kolmanovskii and A I Matasov ldquoAn approximate methodfor solving optimal control problem in hereditary systemsrdquoDoklady Mathematics vol 55 no 3 pp 475ndash478 1997

[4] P Borne V Kolmanovskii and L Shaikhet ldquoStabilization ofinverted pendulumby control with delayrdquoDynamic Systems andApplications vol 9 no 4 pp 501ndash514 2000

[5] A D Myshkis ldquoGeneral theory of Differential equations withdelayrdquo American Mathematical Society Translations vol 55 pp1ndash62 1951


[6] NVMinh ldquoOn the asymptotic behaviour ofVolterra differenceequationsrdquo Journal of Difference Equations andApplications vol19 no 8 pp 1317ndash1330 2013

[7] NVMinh ldquoAsymptotic behavior of individual orbits of discretesystemsrdquo Proceedings of the AmericanMathematical Society vol137 no 9 pp 3025ndash3035 2009


[9] Y Nagabuchi ldquoDecomposition of phase space for linearVolterra difference equations in a Banach spacerdquo FunkcialajEkvacioj vol 49 no 2 pp 269ndash290 2006

[10] S Murakami and Y Nagabuchi ldquoStability properties andasymptotic almost periodicity for linear Volterra differenceequations in a Banach spacerdquo Japanese Journal of Mathematicsvol 31 no 2 pp 193ndash223 2005

[11] C Gonzalez A Jimenez-Melado and M Lorente ldquoExistenceand estimate of solutions of some nonlinear Volterra differenceequations in Hilbert spacesrdquo Journal of Mathematical Analysisand Applications vol 305 no 1 pp 63ndash71 2005

[12] Y Song and C T Baker ldquoLinearized stability analysis of dis-crete Volterra equationsrdquo Journal of Mathematical Analysis andApplications vol 294 no 1 pp 310ndash333 2004

[13] Y Muroya and E Ishiwata ldquoStability for a class of differenceequationsrdquo Journal of Computational and Applied Mathematicsvol 228 no 2 pp 561ndash570 2009

[14] I Gyori and LHorvath ldquoAsymptotic representation of the solu-tions of linear Volterra difference equationsrdquo Advances in Dif-ference Equations vol 2008 Article ID 932831 2008

[15] V B Kolmanovskii A D Myshkis and J-P Richard ldquoEstimateof solutions for some Volterra difference equationsrdquo NonlinearAnalysis Theory Methods amp Applications An InternationalMultidisciplinary Journal Series ATheory andMethods vol 40no 1ndash8 pp 345ndash363 2000

[16] Y Song and C T H Baker ldquoPerturbation theory for discreteVolterra equationsrdquo Journal of Difference Equations and Appli-cations vol 9 no 10 pp 969ndash987 2003

[17] M R Crisci V B Kolmanovskii E Russo andA Vecchio ldquoSta-bility of continuous and discrete Volterra integro-differentialequations by Liapunov approachrdquo Journal of Integral Equationsand Applications vol 7 no 4 pp 393ndash411 1995

[18] S N Elaydi An Introduction to Difference Equations Under-graduate Texts in Mathematics Springer New York NY USA1996

[19] M Kwapisz ldquoOn 119897P solutions of discrete Volterra equationsrdquoAequationes Mathematicae vol 43 no 2-3 pp 191ndash197 1992

[20] R Medina ldquoSolvability of discrete Volterra equations inweighted spacesrdquo Dynamic Systems and Applications vol 5 no3 pp 407ndash421 1996

[21] RMedina ldquoStability results for nonlinear difference equationsrdquoNonlinear Studies vol 6 no 1 pp 73ndash83 1999

[22] C Corduneanu Functional Equations with Causal Operatorsvol 16 Taylor and Francis London UK 2002

[23] M I Gilrsquo Norm Estimations for Operator-Valued Functions andApplications vol 192 ofMonographs and Textbooks in Pure andAppliedMathematicsMarcel Dekker NewYorkNYUSA 1995

[24] N Dunford and J T Schwarz Linear Operators Part I WileyInterscience New York NY USA 1966

[25] P Rejto and M Taboada ldquoUnique solvability of nonlinearVolterra equations in weighted spacesrdquo Journal of MathematicalAnalysis and Applications vol 167 no 2 pp 368ndash381 1992

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Stochastic AnalysisInternational Journal of


[14] sufficient conditions are presented underwhich the solu-tions to a linear nonconvolution Volterra difference equationconverge to limits which are given by a limit formula InKolmanovskii et al [15] stability and boundedness problemsof some class of scalarVolterra nonlinear difference equationsare investigated Stability conditions and boundedness areformulated in terms of the characteristics equations In Songand Baker [16] the fixed point theory is used to establishsufficient conditions to ensure the stability of the zero solu-tion of an implicit nonlinear Volterra difference equationBesides the existence of asymptotically periodic solutionsis established However in this article equations with linearkernel are mainly considered

One of the basic methods in the theory of stability andboundedness of Volterra difference equations is the directLyapunovmethod (see [1 3 17 18]) But finding the Lyapunovfunctionals for Volterra difference equations is still a difficulttask

In this paper to establish boundedness conditions ofsolutions we will interpret the Volterra difference equationswith nonlinear kernels as operator equations in appropriatespaces Such an approach for linear Volterra difference equa-tions has been used by Myshkis [5] Kolmanovskii et al [15]Kwapisz [19] and Medina [20 21] The knowledge of thesebounds is important because they represent the error betweenthe exact solution of the original problem and its differ-ence approximation

Existence and uniqueness problems for the Volterradifference equations were discussed by some authors Usuallythe solutionswere sought in the phase space 119897120596119901 (119885+ 119883) 120596 gt 0and 119885+ = 0 1 2 (eg see [15 19]) In this paper formu-lating the Volterra discrete equations in the phase space119897119901(119885+ 119883) where 119883 is an appropriate Hilbert space andassuming that the kernel operator has the Volterra propertywe obtain sufficient conditions for the existence and unique-ness problem

Our results compare favorably with the above-mentionedworks in the following sense

(a) Sufficient conditions for the existence and uniquenessof solutions of implicit nonlinear Volterra differenceequations are obtained

(b) We established a theory on the asymptotic behavior ofimplicit nonlinear Volterra difference systems whichare described by Volterra operator equations

(c) Explicit estimates for the solutions of nonlinearVolterra operator equations in Hilbert spaces arederived

The remainder of this article is organized as followsin Section 2 we introduce some notations preliminaryresults and the statement of the problem In Section 3 theboundedness of solutions is derived using norm-estimates forthe resolvents of completely continuous quasi-nilpotent oper-ators In Section 4 we discuss the boundedness of solutions ofinfinite-delay Volterra difference equations Finally Section 5is devoted to the discussion of our results we highlight themain conclusions

2 Statement of the Problem

Let119883 be a complex Hilbert space with norm sdot 119883 fl sdot Let119897119901 = 119897119901(119885+ 119883) and 119897infin = 119897infin(119885+ 119883) be the spaces of sequencesℎ = ℎ(119895)infin119895=1 such that

119897119901 = ℎ 119885+ 997888rarr 119883 infinsum119895=1

1003817100381710038171003817ℎ (119895)1003817100381710038171003817119901 lt infin 1 le 119901 lt infin

119897infin = ℎ 119885+ 997888rarr 119883 sup119895isin119885+

1003817100381710038171003817ℎ (119895)1003817100381710038171003817 lt infin (1)

The spaces 119897119901 1 le 119901 le infin equippedwith the standard norms

ℎ119901 = (infinsum119895=1

1003817100381710038171003817ℎ (119895)1003817100381710038171003817119901)1119901

for 1 le 119901 lt infinℎinfin = sup

119895isinZ+

1003817100381710038171003817ℎ (119895)1003817100381710038171003817 (2)

are Banach spacesDenote

Ω119903 = ℎ isin 119897infin ℎinfin le 119903 for 0 lt 119903 le infin (3)

We consider Volterra difference equations on a separableHilbert space119883

119909 (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895 119909 (119895)) 119899 ge 0 (4)

where

119870 119885+ times 119885+ times 120585 isin 119883 10038171003817100381710038171205851003817100381710038171003817infin lt 119903 997888rarr 119883 (5)

is the kernel 119885+ denotes the set of nonnegative integers and119891 119885+ rarr 119883 is a given sequenceEquation (4) can be regarded as the discrete-time analog

of the classical integral Volterra equation of the second kind

119909 (119905) = 119891 (119905) + int1199050119870 (119905 119904 119909 (119904)) 119889119904 (119905 ge 0) (6)

We point out the distinction between (4) and the explicitVolterra equation

119909 (119899) = 119891 (119899) + 119899minus1sum119895=0

119870(119899 119895 119909 (119895)) 119899 ge 0 (7)

The first step is to establish the solvability of (4) In the linearcase in which we write (4) as

119909 (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895) 119909 (119895) 119899 ge 0 (8)

with 119870(119899 119895) being a finite-dimensional matrix the existenceof 119909(119899) for each 119899 follows from the property that (119868 minus119870(119899 119899))



119909 (119899 + 1) = 120595119899 (119909 (0) 119909 (1) 119909 (119899) 119909 (119899 + 1)) (9)


119909 = 120595119899 (1199060 1199061 119906119899 119909) (10)


119909 = 120593119899 (1199060 1199061 119906119899) (11)


119906 (119899) = (119879119906) (119899) 119899 isin 119885+ (12)



(infinsum119895=1

1003817100381710038171003817119909 (119895)1003817100381710038171003817119901)1119901



(119879119906) (119899) = (119879V) (119899) if 119906 (119895) = V (119895) for 119895 = 0 1 119899 minus 1 (14)



(119879119909) (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895 119909 (119895)) (15)








(119860119890119896 119890119896) (16)


The norm

1198732 (119860) = 119873 (119860) = radictr (119860lowast119860) (17)


119860119868 = 119860 minus 119860lowast2119894 (18)


1003817100381710038171003817100381711988111989610038171003817100381710038171003817 le 119873119896 (119881)radic119896 119891119900119903 119886119899119910 119899119886119905119906119903119886119897 119896 (19)

is true

3 Main Results


119860119903119899119895 fl sup119911isin119883119911le119903

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 for 119899 119895 = 1 2 (20)

Assume that

119873119901 (119860119903) = [[infinsum119899=1

( 119899sum119895=1

119860119902119903119899119895)119901119902]

]1119902

lt infin (21)

which implies

120573119901 (119860119903) = sup119899=12

[[119899sum119895=1

119860119902119903119899119895]]1119902

lt infin (22)

where 1119901 + 1119902 = 1


Denote

119898119901 (119860119903) = infinsum119895=0

119873119895119901 (119860119903)119901radic119895 (23)


119909119901 le 119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (24)

119909infin le 10038171003817100381710038171198911003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (25)




119860infin (119899 119895) = sup119911isin119883

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 119899 119895 = 1 2 (27)


119909119901 le 119898119901 (119860infin) 10038171003817100381710038171198911003817100381710038171003817119901 (28)


119909 (119899) le 119899sum119895=1

119860infin (119899 119895) 1003817100381710038171003817119909 (119895)1003817100381710038171003817 + 1003817100381710038171003817119891 (119899)1003817100381710038171003817 (29)


119895=1

119860infin (119899 119895) ℎ (119895) ℎ isin 119897119901 (119885+ 119883) (30)


119909 = 119891 + 119871119909 (31)

Hence

119909 = (119868 minus 119871)minus1 119891 (32)

Since


119871119895 (33)

we have


1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 (34)

Thus by (32)

119909119901 le infinsum119895=0

1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 10038171003817100381710038171198911003817100381710038171003817119901 (35)


1003817100381710038171003817100381711987111989610038171003817100381710038171003817119901 le 119873119896119901 (119860infin)119901radic119896 (36)




exp [119873119901119901 (119860infin)] (37)


119909119901 le 120574119901 exp [119873119901119901 (119860infin)] 10038171003817100381710038171198911003817100381710038171003817119901 (38)

where

120574119901 = (1 minus 1119901119902119901)minus1119902 (39)






119871119909infin le sup119899=12

[ 119899sum119896=1

119860119902infin (119899 119896)]1119902 119909119901= 120573119901 (119860infin) 119909119901

(42)






120583119903 (ℎ) = 1 ℎ le 1199030 ℎ gt 119903

(44)


Define119870119903 (119899 119895 119911) = 120582119903 (ℎ)119870 (119899 119895 119911)

119891119903 (ℎ) = 120583119903 (ℎ) 119891 (ℎ) (45)


119909 (119899) = 119891119903 (119899) + 119899sum119895=1

119870119903 (119899 119895 119909 (119895)) 119899 ge 0 (46)




119909 (119899) = 119891 (119899) + 119899sum119895=minusinfin

119870(119899 119895 119909 (119895)) 119899 ge 0 (47)


119909 (119899) = (119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)))+ 119899sum119895=1

119870(119899 119895 119909 (119895))= (119899) + 119899sum

119895=1

119870(119899 119895 119909 (119895)) (48)



119909119901 le 119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901 119909infin le 1003817100381710038171003817100381710038171003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901

(49)


(119899) = 119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)) (50)


Remark 11 Consider

119909 (119899) = 119909 (119899 120593) forall119899 ge 0119909 (119895 120593) = 120593 (119895) for 119895 lt 0 (51)

5 Conclusions



Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 (119899 + 1) = 120595119899 (119909 (0) 119909 (1) 119909 (119899) 119909 (119899 + 1)) (9)


119909 = 120595119899 (1199060 1199061 119906119899 119909) (10)


119909 = 120593119899 (1199060 1199061 119906119899) (11)


119906 (119899) = (119879119906) (119899) 119899 isin 119885+ (12)



(infinsum119895=1

1003817100381710038171003817119909 (119895)1003817100381710038171003817119901)1119901



(119879119906) (119899) = (119879V) (119899) if 119906 (119895) = V (119895) for 119895 = 0 1 119899 minus 1 (14)



(119879119909) (119899) = 119891 (119899) + 119899sum119895=0

119870(119899 119895 119909 (119895)) (15)








(119860119890119896 119890119896) (16)


The norm

1198732 (119860) = 119873 (119860) = radictr (119860lowast119860) (17)


119860119868 = 119860 minus 119860lowast2119894 (18)


1003817100381710038171003817100381711988111989610038171003817100381710038171003817 le 119873119896 (119881)radic119896 119891119900119903 119886119899119910 119899119886119905119906119903119886119897 119896 (19)

is true

3 Main Results


119860119903119899119895 fl sup119911isin119883119911le119903

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 for 119899 119895 = 1 2 (20)

Assume that

119873119901 (119860119903) = [[infinsum119899=1

( 119899sum119895=1

119860119902119903119899119895)119901119902]

]1119902

lt infin (21)

which implies

120573119901 (119860119903) = sup119899=12

[[119899sum119895=1

119860119902119903119899119895]]1119902

lt infin (22)

where 1119901 + 1119902 = 1


Denote

119898119901 (119860119903) = infinsum119895=0

119873119895119901 (119860119903)119901radic119895 (23)


119909119901 le 119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (24)

119909infin le 10038171003817100381710038171198911003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (25)




119860infin (119899 119895) = sup119911isin119883

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 119899 119895 = 1 2 (27)


119909119901 le 119898119901 (119860infin) 10038171003817100381710038171198911003817100381710038171003817119901 (28)


119909 (119899) le 119899sum119895=1

119860infin (119899 119895) 1003817100381710038171003817119909 (119895)1003817100381710038171003817 + 1003817100381710038171003817119891 (119899)1003817100381710038171003817 (29)


119895=1

119860infin (119899 119895) ℎ (119895) ℎ isin 119897119901 (119885+ 119883) (30)


119909 = 119891 + 119871119909 (31)

Hence

119909 = (119868 minus 119871)minus1 119891 (32)

Since


119871119895 (33)

we have


1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 (34)

Thus by (32)

119909119901 le infinsum119895=0

1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 10038171003817100381710038171198911003817100381710038171003817119901 (35)


1003817100381710038171003817100381711987111989610038171003817100381710038171003817119901 le 119873119896119901 (119860infin)119901radic119896 (36)




exp [119873119901119901 (119860infin)] (37)


119909119901 le 120574119901 exp [119873119901119901 (119860infin)] 10038171003817100381710038171198911003817100381710038171003817119901 (38)

where

120574119901 = (1 minus 1119901119902119901)minus1119902 (39)






119871119909infin le sup119899=12

[ 119899sum119896=1

119860119902infin (119899 119896)]1119902 119909119901= 120573119901 (119860infin) 119909119901

(42)






120583119903 (ℎ) = 1 ℎ le 1199030 ℎ gt 119903

(44)


Define119870119903 (119899 119895 119911) = 120582119903 (ℎ)119870 (119899 119895 119911)

119891119903 (ℎ) = 120583119903 (ℎ) 119891 (ℎ) (45)


119909 (119899) = 119891119903 (119899) + 119899sum119895=1

119870119903 (119899 119895 119909 (119895)) 119899 ge 0 (46)




119909 (119899) = 119891 (119899) + 119899sum119895=minusinfin

119870(119899 119895 119909 (119895)) 119899 ge 0 (47)


119909 (119899) = (119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)))+ 119899sum119895=1

119870(119899 119895 119909 (119895))= (119899) + 119899sum

119895=1

119870(119899 119895 119909 (119895)) (48)



119909119901 le 119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901 119909infin le 1003817100381710038171003817100381710038171003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901

(49)


(119899) = 119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)) (50)


Remark 11 Consider

119909 (119899) = 119909 (119899 120593) forall119899 ge 0119909 (119895 120593) = 120593 (119895) for 119895 lt 0 (51)

5 Conclusions



Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Denote

119898119901 (119860119903) = infinsum119895=0

119873119895119901 (119860119903)119901radic119895 (23)


119909119901 le 119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (24)

119909infin le 10038171003817100381710038171198911003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 10038171003817100381710038171198911003817100381710038171003817119901 (25)




119860infin (119899 119895) = sup119911isin119883

1003817100381710038171003817119870 (119899 119895 119911)1003817100381710038171003817119911 119899 119895 = 1 2 (27)


119909119901 le 119898119901 (119860infin) 10038171003817100381710038171198911003817100381710038171003817119901 (28)


119909 (119899) le 119899sum119895=1

119860infin (119899 119895) 1003817100381710038171003817119909 (119895)1003817100381710038171003817 + 1003817100381710038171003817119891 (119899)1003817100381710038171003817 (29)


119895=1

119860infin (119899 119895) ℎ (119895) ℎ isin 119897119901 (119885+ 119883) (30)


119909 = 119891 + 119871119909 (31)

Hence

119909 = (119868 minus 119871)minus1 119891 (32)

Since


119871119895 (33)

we have


1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 (34)

Thus by (32)

119909119901 le infinsum119895=0

1003817100381710038171003817100381711987111989510038171003817100381710038171003817119901 10038171003817100381710038171198911003817100381710038171003817119901 (35)


1003817100381710038171003817100381711987111989610038171003817100381710038171003817119901 le 119873119896119901 (119860infin)119901radic119896 (36)




exp [119873119901119901 (119860infin)] (37)


119909119901 le 120574119901 exp [119873119901119901 (119860infin)] 10038171003817100381710038171198911003817100381710038171003817119901 (38)

where

120574119901 = (1 minus 1119901119902119901)minus1119902 (39)






119871119909infin le sup119899=12

[ 119899sum119896=1

119860119902infin (119899 119896)]1119902 119909119901= 120573119901 (119860infin) 119909119901

(42)






120583119903 (ℎ) = 1 ℎ le 1199030 ℎ gt 119903

(44)


Define119870119903 (119899 119895 119911) = 120582119903 (ℎ)119870 (119899 119895 119911)

119891119903 (ℎ) = 120583119903 (ℎ) 119891 (ℎ) (45)


119909 (119899) = 119891119903 (119899) + 119899sum119895=1

119870119903 (119899 119895 119909 (119895)) 119899 ge 0 (46)




119909 (119899) = 119891 (119899) + 119899sum119895=minusinfin

119870(119899 119895 119909 (119895)) 119899 ge 0 (47)


119909 (119899) = (119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)))+ 119899sum119895=1

119870(119899 119895 119909 (119895))= (119899) + 119899sum

119895=1

119870(119899 119895 119909 (119895)) (48)



119909119901 le 119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901 119909infin le 1003817100381710038171003817100381710038171003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901

(49)


(119899) = 119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)) (50)


Remark 11 Consider

119909 (119899) = 119909 (119899 120593) forall119899 ge 0119909 (119895 120593) = 120593 (119895) for 119895 lt 0 (51)

5 Conclusions



Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Define119870119903 (119899 119895 119911) = 120582119903 (ℎ)119870 (119899 119895 119911)

119891119903 (ℎ) = 120583119903 (ℎ) 119891 (ℎ) (45)


119909 (119899) = 119891119903 (119899) + 119899sum119895=1

119870119903 (119899 119895 119909 (119895)) 119899 ge 0 (46)




119909 (119899) = 119891 (119899) + 119899sum119895=minusinfin

119870(119899 119895 119909 (119895)) 119899 ge 0 (47)


119909 (119899) = (119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)))+ 119899sum119895=1

119870(119899 119895 119909 (119895))= (119899) + 119899sum

119895=1

119870(119899 119895 119909 (119895)) (48)



119909119901 le 119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901 119909infin le 1003817100381710038171003817100381710038171003817100381710038171003817infin + 120573119901 (119860119903)119898119901 (119860119903) 1003817100381710038171003817100381710038171003817100381710038171003817119901

(49)


(119899) = 119891 (119899) + 0sum119895=minusinfin

119870(119899 119895 120593 (119895)) (50)


Remark 11 Consider

119909 (119899) = 119909 (119899 120593) forall119899 ge 0119909 (119895 120593) = 120593 (119895) for 119895 lt 0 (51)

5 Conclusions



Competing Interests


Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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