Zhinan XIA

Abstract In this paper,the author studies the existence and uniqueness of discrete pseudo asymptotically periodic solutions for nonlinear Volterra difference equations of convolution type,where the nonlinear perturbation is considered as Lipschitz condition or non-Lipschitz case,respectively.The results are a consequence of application of different fixed point theorems,namely,the contraction mapping principle,the Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

Keywords Pseudo asymptotically periodic function,Volterra difference equations,Contraction mapping principle,Leray-Schauder alternative theorem

1 Introduction

Besides its theoretical interest,the study of asymptotic ω-periodicity is of great importance in applications.Many contributions have been made to the study of existence of asymptotically ω-periodic solutions for differential equations (see [3,7,22,31,33]for more details).On the other hand,the notion of S-asymptotic ω-periodicity,introduced by Henríquez et al.in [24–25],is related to and more general than that of asymptotic ω-periodicity.Since then,it has attracted the attention of many researchers (see [10,15,17,27]).Particularly,for discrete S-asymptotic ω-periodicity,the subject was studied in [2],where the authors discussed the existence of discrete S-asymptotically ω-periodic solutions of semilinear difference equations with infinite delay.Recently,the concept of (continuous)pseudo S-asymptotic ω-periodicity was introduced in[28]and some applications involving ordinary and partial differential equations were presented in[4,12,16,23,32].This paper is a continuation of this study,which introduces the concept of discrete pseudo S-asymptotic ω-periodicity and deals with its property.

In this paper,we study the existence and uniqueness of discrete pseudo S-asymptotically ω-periodic solutions of the Volterra difference equations of convolution type

where λ ∈C,a(·)is a summable function,A is a bounded linear operator on X and f :Z×X →X is a function bounded on bounded sets of X.

Volterra difference equations arise in numerical discrete approximations of Volterra integral or integro-differential equations.The Volterra difference equations can be used in the modelling of real phenomena in economy and ecology,the theory of viscoelasticity and the study of optimal control problems (see [19–20]).The asymptotic behaviour of solutions of (1.1)is a classical subject of dynamical systems and operator theory.Many researchers have made important contributions to this topics,for example,almost periodicity (see [8,30]),asymptotic almost periodicity (see [5,29]),almost automorphy (see [1,6,11]),lp-boundedness (see [9]),and Sasymptotic ω-periodicity(see [2]).To our knowledge,there is no work reported in literature on pseudo S-asymptotic ω-periodicity for (1.1).This is one of the key motivations of this study.

Motivated by the above mentioned papers,in this paper,we introduce a new class of functions called discrete pseudo S-asymptotically ω-periodic functions,which generalize the notation of discrete S-asymptotically ω-periodic functions.We systematically explore its properties in Banach spaces and discrete pseudo S-asymptotic ω-periodicity of (1.1)when the nonlinear perturbation function f is considered as Lipschitz condition or non-Lipschitz case,respectively.

The paper is organized as follows.In Section 2,some notations and preliminary results are presented,and the concept of discrete pseudo S-asymptotically ω-periodic functions is given.Sections 3 is divided into two parts.In Subsection 3.1,we investigate the existence and uniqueness of discrete pseudo S-asymptotically ω-periodic solution of (1.1)when f satisfies the Lipschitz condition.In Subsection 3.2,when f is non-Lipschitz,we explore the properties of solutions to the same equation.

2 Preliminaries and Basic Results

In order to facilitate the discussion below,we further introduce the following notations:

(2)C0(Z,X)=x ∈l∞(Z,X)

(3)Cω(Z,X)={x ∈l∞(Z,X)|x is ω-periodic},where ω ∈Z{0}.

(4)L(X)denotes the space of bounded linear operators from X to X endowed with the operator topology.

(5)UC(Z×X,X)denotes the set of all functions f : Z×X →X satisfying that ∀ε > 0,∃δ >0 such that

for all k ∈Z and x,y ∈X with

(6)UCk(Z×X,X)denotes the set of all functions f : Z×X →X satisfying that ∀ε > 0,∃δ >0 such that

for all k ∈Z and x,y ∈X withwhere Lf:Z →R+is a summable function.

First,we recall a useful compactness criterion.Let h : Z →R+be a function such that h(n)≥1 for all n ∈Z,and h(n)→∞as |n|→∞.Define

It is clear that C0h(Z,X)is a Banach space isometrically isomorphic with the space C0(Z,X).According to a compactness criterion due to Cuevas and Pinto[14],one has the following result.

Lemma 2.1(see [1])Let S be a subset of C0h(Z,X).Suppose that the following conditions are satisfied:

(i)The set Hn(S)=is relatively compact in X for all n ∈Z.

(ii)S is weighted equiconvergent at ±∞,that is for every ε > 0,there exists a T > 0 such that ||u(n)||<εh(n)for each |n|≥T for all u ∈S.Then S is relatively compact in C0h(Z,X).

Now,we recall the so-called Matkowski’s fixed point theorem (see [26])and the Leray-Schauder alternative theorem (see [21])which will be used in the sequel.

Theorem 2.1(Matkowski’s Fixed Point Theorem)Let (X,d)be a complete metric space and F :X →X be a map such that d(Fx,Fy)≤Φ(d(x,y))for all x,y ∈X,where Φ:[0,∞)→[0,∞)is a nondecreasing function such thatThen F has a unique fixed point z ∈X.

Theorem 2.2(Leray-Schauder Alternative Theorem)Let D be a closed convex subset of X such that 0 ∈D.Let Γ : D →D be a completely continuous map.Then the set{x ∈D : x=λΓ(x),0<λ<1} is unbounded or the map Γ has a fixed point in D.

Next,we give the concept of discrete pseudo S-asymptotically ω-periodic function.

Definition 2.1f ∈l∞(Z,X)is called discrete asymptotically ω-periodic if there exist g ∈Cω(Z,X)and ϕ ∈C0(Z,X)such that f=g+ϕ.The collection of those functions is denoted by APω(Z,X).

Definition 2.2f ∈l∞(Z,X)is called discrete S-asymptotically ω-periodic if there exists ω ∈Z{0} such thatThe collection of those functions is denoted by SAPω(Z,X).

Definition 2.3A sequence f ∈l∞(Z,X)is called discrete pseudo S-asymptotically ωperiodic if there exists ω ∈Z{0} such thatDenote by PSAPω(Z,X)the set of such functions.

It is easy to see that PSAPω(Z,X)is a Banach space when endowed with the norm ||f||d :=

Definition 2.4A sequence f ∈ l∞(Z × X,X)is called uniformly discrete pseudo Sasymptotically ω-periodic on bounded sets of X if for every bounded subset K ⊆X,

Denote by PSAPω(Z×X,X)the set of such functions.

Finally,we show some properties of PSAPω(Z,X).We have the following results.

Lemma 2.2If A ∈L(X)and u ∈PSAPω(Z,X),then Au ∈PSAPω(Z,X).

Lemma 2.3Let f ∈PSAPω(Z,X).Then f(·+τ)∈PSAPω(Z,X)for all τ ∈Z.

Lemma 2.4Let f ∈l∞(Z,X).Then f ∈PSAPω(Z,X)if and only if for any ε>0,

where Ef(n,ε)={k ∈[−n,n]∩Z| ||f(k+ω)−f(k)||≥ε}.

The proof of Lemma 2.4 is similar to that of [18,Lemma 2.9].Here we omit it.

Theorem 2.3Let f : Z×X →X be a function bounded on bounded sets of X.Assume that f ∈PSAPω(Z × X,X)∩UCk(Z × X,X).Then ψ(·)=f(·,u(·))∈PSAPω(Z,X)if u ∈PSAPω(Z,X).

ProofSince f ∈UCk(Z×X,X),for any ε>0,there exists δ >0 such that ||f(k,u(k+ω))−for allThen for the above ε>0,there exists N ∈N such that for n>N,

Denote

which implies that ψ(·)∈PSAPω(Z,X).

Corollary 2.1Let f : Z×X →X be a function bounded on bounded sets of X.Assume that f ∈PSAPω(Z×X,X)and satisfies the following Lipschitze type condition:

where Lf:Z →R+is a summable function.Then f(·,u(·))∈PSAPω(Z,X)if u ∈PSAPω(Z,X).

By making some revisions of the proof of Theorem 2.3,one get the following conclusions.

Theorem 2.4Let f :Z×X →X be a function bounded on bounded sets of X.Assume that f ∈PSAPω(Z×X,X)∩UC(Z×X,X).Then f(·,u(·))∈PSAPω(Z,X)if u ∈PSAPω(Z,X).

Corollary 2.2Let f : Z×X →X be a function bounded on bounded sets of X.Assume that f ∈PSAPω(Z×X,X)and there exists a constant Lf>0 such that

Then f(·,u(·))∈PSAPω(Z,X)if u ∈PSAPω(Z,X).

Lemma 2.5Let v : Z+→C be a summable function.Then Ψ(·)∈PSAPω(Z,X)if u ∈PSAPω(Z,X),where

ProofNote that

Then

By Lemma 2.3 and Lebesgue dominated convergence theorem,one has Ψ(·)∈PSAPω(Z,X).

3 Volterra Difference Equation

This section is devoted to establish some sufficient criteria for the existence and uniqueness of PSAPωsolutions of (1.1).

Consider the linear Volterra difference equations

where λ ∈C,a(·)is a summable function.

For a given λ ∈C,let s(λ,k)∈C be the solution of the difference equation

In this case,s(λ,k)is called the fundamental solution to (3.1)generated by a(·).We define the set

By [11],if λ ∈Ωs,the solution to (3.1)is given by

To establish our results,we introduce the following conditions:

(H1)λ ∈Ωs,A ∈L(X).

(H2)f ∈PSAPω(Z×X,X).

(H31)There exists a constant Lf>0 such that

(H32)There exists a linear nondecreasing function Φ:[0,∞)→[0,∞)and f satisfies

(H33)f satisfies the following Lipschitze type condition:

where Lf:Z →R+is a summable function.

(H34)f satisfies the locally Lipschitze condition,that is,for each σ >0,k ∈Z and u,v ∈X withone has

where Lf:R+→R+is a nondecreasing function.

(H4)f ∈UCk(Z×X,X)or f ∈UC(Z×X,X).

3.1 Lipschitz case

In this subsection,we study the existence and uniqueness of discrete pseudo S-asymptotically ω-periodic solution of (1.1)when the perturbation f satisfies the Lipschitz condition.

Theorem 3.1Assume that (H1),(H2),(H31)hold andThen (1.1)has a unique solution u ∈PSAPω(Z,X)which is given by

ProofSimilar as the proof in [11],it can be shown that u(·)given by (3.3)is the solution to (1.1).

Define the operator F :PSAPω(Z,X)→PSAPω(Z,X)by

Since u ∈PSAPω(Z,X)and (H31)holds,f(·,Au(·))∈PSAPω(Z,X)by Lemma 2.2 and Corollary 2.2.By Lemma 2.5,Fu ∈PSAPω(Z,X).Hence F is well defined.

For u,v ∈PSAPω(Z,X),

By the Banach contraction mapping principle,F has a unique fixed point u ∈PSAPω(Z,X),which is the unique PSAPωsolution to (1.1).

Example 3.1For a(k)=pk,where|p|<1,after a calculation using in(3.2)the unilateral-Z transform,we have s(λ,k)=λ(λ+p)k−1,k ≥1,and define

Consider the following difference equation:

where |p|< 1,λ ∈D(−p,1),g ∈PSAPω(Z,X).It is easy to see that (H1),(H2),(H31)hold with Lf=|µ| ||g||d.By Theorem 3.1,ifthen (3.5)has a unique solution u ∈PSAPω(Z,X).

Theorem 3.2Assume that (H1),(H2),(H32)hold.Then (1.1)has a unique solution u ∈PSAPω(Z,X)if (||A|| |s(λ,·)|1Φ)n(t)→0 as n →∞for each t>0.

ProofDefine the operator F as in (3.4),so F is well defined.For u,v ∈PSAPω(Z,X),one has

Since (||A|| |s(λ,·)|1Φ)n(t)→0 as n →∞for each t>0,by Theorem 2.1,F has a unique fixed point u ∈PSAPω(Z,X),which is the unique PSAPωsolution to (1.1).

Theorem 3.3Assume that (H1),(H2),(H33)hold.Then (1.1)has a unique solution u ∈PSAPω(Z,X).

ProofDefine the operator F as in(3.4),and F is well defined by Corollary 2.1 and Lemma 2.5.For u,v ∈PSAPω(Z,X),one has

Similarly,by [13,Lemma 3.2],one has

By the method of mathematical induction,we have

Moreover,since Lfis a summable function,definingone has

Next,consider with the local condition on the perturbation f,we have the following result.

Theorem 3.4Assume that (H1),(H2),(H34)hold,and if there exists r >0 such that

then (1.1)has a unique solution u ∈Br(PSAPω(Z,X)).

ProofLet u be in Br(PSAPω(Z,X))and define

by

Similar as the proof of Theorem 3.1,Fu ∈PSAPω(Z,X).Let u ∈Br(PSAPω(Z,X)).One has

Hence Fu ∈Br(PSAPω(Z,X))and F is well defined.

Moreover,for u,v ∈Br(PSAPω(Z,X)),

By (3.6),Lf(||A||r)||A|||s(λ,·)|1< 1.It follows that F is a contraction on Br(PSAPω(Z,X)).By the Banach contraction mapping principle,F has a unique fixed point in Br(PSAPω(Z,X)),which is the unique PSAPωsolution to (1.1).

3.2 Non-Lipschitz case

In this subsection,we study the existence of discrete pseudo S-asymptotically ω-periodic solution of (1.1)when the perturbation f is a non-Lipschitz nonlinearity.

Theorem 3.5Assume that(H1),(H2),(H4)hold and the following conditions are satisfied:

(A1)There are nondecreasing function W : R+→R+and a function M : Z →R+such that ||f(k,x)||≤M(k)W(||x||)for all k ∈Z,x ∈X.

(A3)For each ε > 0,there exists δ > 0 such that for every u,v ∈C0h(Z,X),||u−v||h ≤δ implies that

for all n ∈Z.

(A4)For all a,b ∈Z,a ≤b,σ > 0,the set {f(k,x)|a ≤k ≤b,||x||≤σ} is relatively compact in X.

Then (1.1)has a solution u ∈PSAPω(Z,X).

ProofDefine Γ:C0h(Z,X)→C0h(Z,X)by

Next,we will prove that Γ has a fixed point in PSAPω(Z,X).We divide the proof into several steps.

(i)For u ∈C0h(Z,X),by (A1),one has

whence

It follows from (A2)that Γ is well defined.

(ii)Γ is continuous.In fact,for each ε > 0,by (A3),there exists δ > 0,such that for u,v ∈C0h(Z,X),||u−v||h ≤δ,one has

Taking into account that h(n)≥1,by (A3),one haswhich implies thatHence Γ is continuous.

(iii)Γ is complete continuous.Let V=Γ(Br(C0h(Z,X)))and v=Γ(u)for u ∈Br(C0h(Z,X)).Initially,we prove thatis relatively compact in X for each n ∈Z.By(A2),for ε>0,we can choose l ∈Z+such that

Since v=Γ(u)for u ∈Br(C0h(Z,X)),we have

so

Note that

So

(iv)If uλ∈C0h(Z,X)is a solution of the equation uλ=λΓ(uλ)for some 0<λ<1,then

Hence,one has

and by (A5),we conclude that the set {uλ:uλ=λΓ(uλ),λ ∈(0,1)} is bounded.

(v)It follows from Theorems 2.3–2.4 and Lemma 2.5 that Γ(PSAPω(Z,X))⊆PSAPω(Z,X).Similar to the proof of (iv),we claim that there exists r0> 0 such that Γ(Br0(C0h(Z,X)))⊆Br0(C0h(Z,X)).Consequently,we infer that

Hence we derive the following conclusion:

By(i)–(iii),we see that Γ is completely continuous.Applying (iv)and Theorem 2.2,we deduce that Γ has a fixed point

Let unbe a sequence in Br0(C0h(Z,X))∩PSAPω(Z,X)such that it converges to u in the norm C0h(Z,X).For ε> 0,let δ > 0 be the constant in (A3).There exists n0∈Z+such thatFor n ≥n0,

which implies that (Γun)nconverges to Γu=u uniformly in Z.Whence u ∈PSAPω(Z,X).

Corollary 3.1Assume that (H1)–(H2)hold and the following conditions are satisfied

(a)f(k,0)=q(k)for k ∈Z.

(b)f satisfies the Hölder type condition

where 0<α<1,C1>0 is a constant.

(c)For all a,b ∈Z,a ≤b,σ >0,the set {f(k,x):a ≤k ≤b,||x||≤σ} is relatively compact in X.

Then (1.1)has a solution u ∈PSAPω(Z,X).

ProofBy (c),it is easy to see that (A4)holds.Let C0=||q||d,M(·)=1 and W(ξ)=C0+C1ξα.Then(A1)is satisfied.Take a function h such thatC2<∞.It is not difcfiult to see that(A2)is satisfeid.To verify(A3),note that for each ε>0,there existssuch that for every u,v ∈C0h(Z,X),||u−v||h ≤δ implies t hat

for all n ∈Z.Moreover,(A5)can be easily verified using the definition of W.By Theorem 3.5,(1.1)has a solution u ∈PSAPω(Z,X).

AcknowledgementThe author would like to thank the anonymous referees for their careful reading of the manuscript and numerous suggestions for its improvement.