Research Article Existence and Boundedness of Solutions for Nonlinear Volterra Difference Equations in Banach Spaces

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1 Abstract ad Applied Aalysis Volume 2016, Article ID , 6 pages Research Article Existece ad Boudedess of Solutios for Noliear Volterra Differece Equatios i Baach Spaces Rigoberto Media Departameto de Ciecias Exactas, Uiversidad de Los Lagos, Casilla 933, Osoro, Chile Correspodece should be addressed to Rigoberto Media; rmedia@ulagos.cl Received 25 Jue 2016; Accepted 30 October 2016 Academic Editor: Jozef Baas Copyright 2016 Rigoberto Media. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We cosider a class of oliear discrete-time Volterra equatios i Baach spaces. Estimates for the orm of operator-valued fuctios ad the resolvets of quasi-ilpotet operators are used to fid sufficiet coditios that all solutios of such equatios are elemets of a appropriate Baach space. These estimates give us explicit boudedess coditios. The boudedess of solutios to Volterra equatios with ifiite delay is also ivestigated. 1. Itroductio I may pheomea of the real world, ot oly does their evolutio prove to be depedet o the preset state, but it is essetially specified by the etire previous history. These processes are ecoutered i the theory of viscoelasticity [1, 2], optimal cotrol problems [3], ad also descriptio of the motio of bodies with referece to hereditary [3 5]. The mathematical descriptio of these processes ca be carried outwiththeaidofequatioswiththeaftereffect,itegral, ad itegrodifferetial equatios. A sigificat cotributio to the developmet of this directio was made by V. Volterra, V. B. Kolmaovskii, N. N. Krasovskii, S. M. V. Luel, A. D. Myshkis, ad J. K. Hale. The aim of this article is to develop a techique for ivestigatig stability ad boudedess of oliear implicit Volterra differece systems described by Volterra operator equatios. Oly a few papers deal with the theory of geeral Volterra equatios ad most of them are devoted to the stability aalysis of explicit Volterra liear differece equatios with costat coefficiets or of covolutio type. For example, i Mih [6], some results o asymptotic stability ad almost periodicity are stated i terms of spectral properties of the equatios ad their solutios, which are liear Volterra equatios of covolutio type. I Mih [7], the asymptotic behavior of idividual orbits of liear fuctioal operator equatios are established usig a extesio of the Katzelso-Tzafriri s theorem [8]. I Nagabuchi [9], usig the decompositio of the phase space, together with the variatio of costats formula i the phase space, the existece of almost periodic solutios for forced liear Volterra differece equatios i Baach spaces is derived. The cosidered equatios are of covolutio type. I Murakami ad Nagabuchi [10], sufficiet stability properties ad the asymptotic almost periodicity for liear Volterra differece equatios i Baach spaces are derived. Gozález et al. [11] cosider a implicit oliear Volterra differece equatio i a Hilbert space ad obtaied sufficiet coditios so that the solutios exist ad have a bouded behavior. The coefficiets of the cosidered equatios are sequeces of real umbers. I Sog ad Baker [12], several ecessary ad sufficiet coditios for stability are obtaied for solutios of the liear Volterra differece equatios by cosiderig the equatios i various choices of Baach spaces. However, the mai results of this article are established essetially to implicit liear Volterra differece equatios. I Muroya ad Ishiwata [13], cosiderig a oliear Volterra differece equatio with ubouded delay, sufficiet coditios for the zero solutio to be globally asymptotically stable are derived. However, oly a explicit scalar Volterra differece is cosidered. I Győri ad Horváth

2 2 Abstract ad Applied Aalysis [14], sufficiet coditios are preseted uder which the solutios to a liear ocovolutio Volterra differece equatio coverge to limits, which are give by a limit formula. I Kolmaovskii et al. [15], stability ad boudedess problems of some class of scalar Volterra oliear differece equatios are ivestigated. Stability coditios ad boudedess are formulated i terms of the characteristics equatios. I Sog ad Baker [16], the fixed poit theory is used to establish sufficiet coditios to esure the stability of the zero solutio of a implicit oliear Volterra differece equatio. Besides, the existece of asymptotically periodic solutios is established. However, i this article equatios with liear kerel are maily cosidered. Oe of the basic methods i the theory of stability ad boudedess of Volterra differece equatios is the direct Lyapuovmethod(see[1,3,17,18]).ButfidigtheLyapuov fuctioals for Volterra differece equatios is still a difficult task. I this paper, to establish boudedess coditios of solutios, we will iterpret the Volterra differece equatios with oliear kerels as operator equatios i appropriate spaces. Such a approach for liear Volterra differece equatios has bee used by Myshkis [5], Kolmaovskii et al. [15], Kwapisz[19],adMedia[20,21].Thekowledgeofthese bouds is importat because they represet the error betwee the exact solutio of the origial problem ad its differece approximatio. Existece ad uiqueess problems for the Volterra differece equatios were discussed by some authors. Usually the solutios were sought i the phase space l ω p (Z+, X), ω > 0 ad Z + = 0, 1, 2,...} (e.g., see [15, 19]). I this paper, formulatig the Volterra discrete equatios i the phase space l p (Z +,X),whereX is a appropriate Hilbert space, ad assumig that the kerel operator has the Volterra property, we obtai sufficiet coditios for the existece ad uiqueess problem. Our results compare favorably with the above-metioed works i the followig sese: (a) Sufficiet coditios for the existece ad uiqueess of solutios of implicit oliear Volterra differece equatios are obtaied. (b) We established a theory o the asymptotic behavior of implicit oliear Volterra differece systems which are described by Volterra operator equatios. (c) Explicit estimates for the solutios of oliear Volterra operator equatios i Hilbert spaces are derived. The remaider of this article is orgaized as follows: i Sectio 2, we itroduce some otatios, prelimiary results, ad the statemet of the problem. I Sectio 3, the boudedess of solutios is derived usig orm-estimates for the resolvets of completely cotiuous quasi-ilpotet operators. I Sectio 4, we discuss the boudedess of solutios of ifiite-delay Volterra differece equatios. Fially, Sectio 5 is devoted to the discussio of our results: we highlight the mai coclusios. 2. Statemet of the Problem Let X be a complex Hilbert space with orm X fl.let l p =l p (Z +,X)ad l =l (Z +,X)be the spaces of sequeces h = h(j)} such that l p = h:z + X : h (j) p < } }, } l =h:z + X : sup j Z + h (j) <}. 1 p<, The spaces l p, 1 p, equipped with the stadard orms h p =( h (j) p ) h = sup j Z + h (j), are Baach spaces. Deote 1/p, for 1 p<, (1) (2) Ω r =h l : h r}, for 0<r. (3) We cosider Volterra differece equatios o a separable Hilbert space X where x () =f() + K (, j, x (j)), 0, (4) K:Z + Z + ξ X: ξ <r} X (5) is the kerel, Z + deotes the set of oegative itegers, ad f:z + Xis a give sequece. Equatio (4) ca be regarded as the discrete-time aalog of the classical itegral Volterra equatio of the secod kid t x (t) =f(t) + K (t, s, x (s)) ds, (t 0). (6) 0 We poit out the distictio betwee (4) ad the explicit Volterra equatio 1 x () =f() + K (, j, x (j)), 0. (7) The first step is to establish the solvability of (4). I the liear case, i which we write (4) as x () =f() + K (, j) x (j), 0, (8) with K(, j) beig a fiite-dimesioal matrix, the existece of x() for each follows from the property that (I K(, ))

3 Abstract ad Applied Aalysis 3 is ivertible for each 0.SogadBaker[16]dealwith the existece of solutios of (4), defied o d-dimesioal Euclidea spaces. I fact, cosiderig implicit equatios of the form x (+1) =ψ (x (0),x(1),...,x(),x(+1)) (9) ad assumig appropriate coditios o ψ },theycoclude that solutios of equatios x=ψ (u 0,u 1,...,u,x) (10) areuiqueadcabeexpresseditheform x=φ (u 0,u 1,...,u ). (11) We preset a differet ad more geeral approach to the problem of existece ad uiqueess discussed i [16]. Namely, for a give geeral discrete operator T:S(Z +,X) S(Z +,X), cosider the equatio u () = (Tu)(), Z +, (12) where S(Z +,X)deotes the liear space of all sequeces s: Z + X. We ask whe the solutios of this equatio belog to the space l p (Z +,X) of all fuctios x S(Z +,X) satisfyig the coditio ( x (j) p ) 1/p <, for 1 p. (13) Assumptio A. The operator T is a causal operator;thatis,for ay Z +, (Tu)() = (TV)(), if u (j) = V (j) for,1,..., 1. (14) That is, the value of (Tu) at is determied by the values of u(j) for,1,..., 1(e.g., see Cordueau [22]). Remark 1. The causal property of the operator T guaratees the existece ad uiqueess of the solutios of (12). Cosequetly, whe the operator T of (12) has, for istace, the form (Tx)() =f() + K (, j, x (j)), (15) we ca establish existece ad uiqueess results for the solutios of (4). Theorem 2. If the operator T defied i (15) is a causal operator, the there exists a solutio of (4). Fially, we will determie sufficiet coditios o the coefficiets of (4) such that its solutios belog to the space l p (Z +,X), 1 p. I the fiite-dimesioal case, the spectrum of a liear operator cosists of its eigevalues. The spectral theory of bouded liear operators o ifiite-dimesioal spaces is a importat but challegig area of research. For example, a operator may have a cotiuous spectrum i additio to, or istead of, a poit spectrum of eigevalues. A particularly simple ad importat case is that of compact, self-adjoit operators. Compact operators may be approximated by fiitedimesioal operators, ad their spectral theory is close to that of fiite-dimesioal operators. Toformulatetheextresult,letusitroducethefollowig otatios ad defiitios: let H be a separable Hilbert space ad A a liear compact operator i H. If e k } k=1 is a orthogoal basis i H ad the series k=1 (Ae k,e k ) coverges, the the sum of the series is called the trace of the operator A ad is deoted by trace (A) = tr (A) = k=1 (Ae k,e k ). (16) Defiitio 3. A operator A satisfyig the relatio tr(a A) < is said to be a Hilbert-Schmidt operator, where A is the adjoit operator of A. The orm N 2 (A) =N(A) = tr (A A) (17) is called the Hilbert-Schmidt orm of A. Defiitio 4. A bouded liear operator A is said to be quasi- Hermitia if its imagiary compoet A I = A A 2i is a Hilbert-Schmidt operator, where A operator of A. (18) is the adjoit Theorem 5 (see [23, Lemma 2.3.1]). Let V be a Hilbert- Schmidt completely cotiuous quasi-ilpotet (Volterra) operator actig i a separable Hilbert space H.Thetheiequality is true. 3. Mai Results Vk Nk (V), foray atural k, (19) k! Let A r deote the ifiite matrix with compoets A r,,j fl Assume that sup z X: z r N p (A r )=[ ( =1 [ which implies K (, j, z) ; for, j = 1, 2,.... (20) z β p (A r )= sup where 1/p + 1/q = 1. =1,2,... A p/q] q r,,j ) ] [ A q ] r,,j [ ] 1/q 1/q <, (21) <, (22)

4 4 Abstract ad Applied Aalysis Deote m p (A r )= N j p (A r ). (23) p j! Theorem 6. Assume that f l p (Z +,X) ad coditio (21) holds. The ay solutio x = (x 1,x 2,...) of (4) belogs to l p (Z +,X)ad satisfies the iequalities provided that x p m p (A r ) f p, (24) x f +β p (A r )m p (A r ) f p, (25) f +β p (A r )m p (A r ) f p <r. (26) Proof of Theorem 6. We will decompose the proof of Theorem 6 i the followig lemmas. Lemma 7. Assume that f l p (Z +, X), 1 p <, ad coditio (21) holds with r=,thatis,a with compoets A (, j) = sup K (, j, z),,,2,..., (27) z X z is a Hilbert-Schmidt kerel. The, ay solutio x=(x 1,x 2,...) of (4) belogs to l p (Z +,X)ad satisfies the iequality Proof. From (4) we have x () x p m p (A ) f p. (28) A (, j) x (j) + f (). (29) Let L be the liear operator defied o l p (Z +,X)by (Lh)() = Rewrite equatio (4) as A (, j) h (j), h l p (Z +,X). (30) x=f+lx. (31) Thus, by (32), x p Lj p f p. (35) Sice L is a quasi-ilpotet Hilbert-Schmidt operator, it follows, by Gil [23, Lemma 2.3.1], that Lk Nk p (A ). (36) p p k! This ad (35) yield the required result. Remark 8. By Hölder s iequality, we have m p (A )=(1 1 1/q p ) exp [N p q/p p (A )]. (37) Cosequetly, from Lemma 7, where x p γ p exp [N p p (A )] f p, (38) γ p =(1 1 1/q p ). (39) q/p Lemma 9. Assume that f l (Z +,X) ad coditio (21) holds with r=,adtheaysolutiox=(x 1,x 2,...)of (4) belogs to l (Z +,X)ad satisfies the iequality x f +β p (A ) m p (A ) f p. (40) Proof. From (31), it follows that But due to Hölder s iequality x f + Lx. (41) Lx sup =1,2,... Hece, (28) ad (42) yield k=1 [ =β p (A ) x p. A q (, k)] 1/q x p (42) x f +β p (A )m p (A ) f p. (43) Hece, Sice we have x=(i L) 1 f. (32) (I L) 1 = L j, (33) (I L) 1 p Lj. p (34) Now, we will complete the Proof of Theorem 6, cosiderig the case 0<r<. By Urysoh s Lemma [24, p. 15], there are scalar-valued fuctios λ r ad μ r defied o l p (Z +,X)ad X,respectively, such that λ r (h) = 1, h r, 0, h >r, (44) μ r (h) = 1, h r, 0, h >r.

5 Abstract ad Applied Aalysis 5 Defie K r (,j,z)=λ r (h) K (, j, z), f r (h) =μ r (h) f (h). Cosider the Volterra equatio x () =f r () + (45) K r (, j, x (j)), 0. (46) Due to Lemma 9 ad coditio (26), ay solutio of (46) satisfies (25) ad therefore belogs to Ω r.butk r =Kad f r = f o Ω r. This proves estimates (25) for a solutio of (4). Estimates (24) follows from Lemma 7. This completes the proof of Theorem Volterra Differece Equatios with Ifiite Delay Cosider the Volterra differece equatio of the form x () =f() + j= K (, j, x (j)), 0, (47) which ca be regarded as a retarded equatio whose delay is ifiite. I geeral this problem requires that oe give a iitial fuctio o (, 0], i order to obtai a uique solutio, after which the equatio ca be treated with the techiques of stadard Volterra equatios. The ouiqueess of solutios of (47) is a itrisic feature (e.g., see [25]). If φ:(,0] Xis a iitial fuctio to (47), the we write (47) as x () =(f() + + = f () + 0 j= K (, j, x (j)) K (, j, φ (j))) K (, j, x (j)), (48) so that the iitial fuctio φ becomespartofthesequece f()}. Uder these coditios, we ca apply Theorem 5 to (48), so that the ext result is obtaied. Theorem 10. Assume that f l p (Z +, X), 1 p, ad coditio (21) holds. The the solutio x() = x(, φ) of (48) belogs to l p (Z +,X)ad satisfies the iequalities x p m p (A r ) f p, x f +β p (A r )m p (A r ) f p. (49) Proof. Proceedig i a similar way to the proof of Theorem 6, with f () =f() + 0 j= K (, j, φ (j)), (50) istead of f, ad choosig a appropriate iitial fuctio φ such that f l p (Z +,X), the required result follows. Remark 11. Cosider 5. Coclusios x () =x(,φ) x (j, φ) = φ (j) 0, for j<0. (51) New coditios for the existece, uiqueess, ad boudedess of solutios of ifiite-dimesioal oliear Volterra differece systems are derived. Ulike the classic method of stability aalysis, we do ot use the techique of the Lyapuov fuctios i the process of costructio of the estimates for thesolutios.theproofsarecarriedoutusigestimatesfor the orm of powers of quasi-ilpotet operators. That is, we iterpreted the Volterra differece equatios, with oliear kerels, as operator equatios defied i the Baach spaces l p (Z +,X).Wewattopoitoutthattheresultsofthispaper cabeusefulfordiscussiosofl p (Z +,X)stability of the zero solutio of the homogeeous Volterra equatio correspodig to (4). I coectio with the above ivestigatios, some ope problems arise. The richess of the spectral properties of operators actig o ifiite-dimesioal Hilbert spaces will eed ew stability formulatios. Cosequetly, atural directios for future research is the geeralizatio of our results to local expoetial stabilizability of oliear Volterra differece equatios or ivestigatig the feedback stabilizatio of implicit oliear Volterra systems defied by operator Volterra equatios. Competig Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets This research was supported by Direcció de Ivestigació uder Grat NU06/16. Refereces [1] A. D. Drozdov ad V. B. Kolmaovskii, Stability i Viscoelasticity,vol.38,Elsevier,Amsterdam,TheNetherlads,1994. [2] N. Leviso, A oliear Volterra equatio arisig i the theory of superfluidity, Joural of Mathematical Aalysis ad Applicatios, vol. 1, o. 1, pp. 1 11, [3] V. B. Kolmaovskii ad A. I. Matasov, A approximate method for solvig optimal cotrol problem i hereditary systems, Doklady Mathematics,vol.55,o.3,pp ,1997. [4]P.Bore,V.Kolmaovskii,adL.Shaikhet, Stabilizatioof iverted pedulum by cotrol with delay, Dyamic Systems ad Applicatios,vol.9,o.4,pp ,2000. [5] A.D.Myshkis, GeeraltheoryofDifferetialequatioswith delay, America Mathematical Society Traslatios,vol.55,pp. 1 62, 1951.

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