Hindawi Publishing Corporation Advances in Difference Equations Volume 2007, Article ID 68023, 15 pages doi:10.1155/2007/68023 Research Article Periodic and Almost Periodic Solutions of Functional Difference Equations with Finite Delay Yihong Song Received 4 November 2006; Revised 29 January 2007; Accepted 29 January 2007 Recommended by John R. Graef For periodic and almost periodic functional difference equations with finite delay, the ex- istence of periodic and almost periodic solutions is obtained by using stability properties of a bounded solution. Copyright © 2007 Yihong Song. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we study periodic and almost periodic solutions of the following functional difference equations with finite delay: x(n +1) = F  n,x n  , n ≥ 0, (1.1) under certain conditions for F(n, ·) (see below), where n, j,andτ are integers, and x n will denote the function x(n + j), j =−τ,−τ +1, ,0. Equation (1.1) can be regarded as the discrete analogue of the following functional differential equation with bounded delay: dx dt = Ᏺ  t,x t  , t ≥ 0, x t (0) = x(t +0)= φ(t), −σ ≤ t ≤ 0. (1.2) Almost periodic solutions of (1.2) have been discussed in [1]. The aim of this paper is to extend results in [1]to(1.1). Delay difference equations or functional difference equations (no matter with finite or infinite delay), inspired by the development of the study of delay differential equations, have been studied extensively in the past few decades (see, [2–11], to mention a few, and 2AdvancesinDifference Equations references therein). Recently, several papers [12–17] are devoted to study almost periodic solutions of difference equations. To the best of our knowledge, little work has been done on almost periodic solutions of nonlinear functional difference equations with finite de- lay via uniform stability properties of a bounded solution. This motivates us to investigate almost periodic solutions of (1.1). This paper is organized as follows. In Section 2, we review definitions of almost pe- riodic and asymptotically almost periodic sequences and present some related proper- ties for our purposes and some stability definitions of a bounded solution of (1.1). In Section 3, we discuss the existence of periodic solutions of (1.1). In Section 4, we discuss the existence of almost periodic solutions of (1.1). 2. Preliminaries We formalize our notation. Denote by Z, Z + , Z − , respectively, the set of integers, the set of nonnegative integers, and the set of nonpositive integers. For any a ∈ Z,letZ + a ={n : n ≥ a, n ∈ Z}.Foranyintegersa<b,letdis[a,b] ={j : a ≤ j ≤ b, j ∈ Z} and dis(a,b] = { j : a<j≤ b, j ∈ Z} be discrete intervals of integers. Let E d denote either R d ,thed- dimensional real Euclidean space, or C d ,thed-dimensional complex Euclidean space. In the following, we use |·|to denote a norm of a vector in E d . 2.1. Almost periodic sequences. We review definitions of (uniformly) almost periodic and asymptotically almost periodic sequences, which have been discussed by several au- thors (see, e.g., [2, 18]), and present some related properties for our purposes. For almost periodic and asymptotically almost periodic functions, we recommend [19, 18]. Let X and Y betwoBanachspaceswiththenorm · X and · Y , respectively. Let Ω be a subset of X. Definit ion 2.1. Le t f : Z × Ω → Y and f (n,·) be continuous for each n ∈ Z.Then f is said to be almost periodic in n ∈ Z uniformly for w ∈ Ω if for every ε>0andeverycompact Σ ⊂ Ω corresponds an integer N ε (Σ) > 0suchthatamongN ε (Σ) consecutive integers there is one, call it p,suchthat   f (n + p,w) − f (n,w)   Y <ε ∀n ∈ Z, w ∈ Σ. (2.1) Denote by Ꮽᏼ( Z × Ω : Y) the set of all such functions. We may call f ∈ Ꮽᏼ(Z × Ω : Y) a (uniformly) almost periodic sequence in Y.IfΩ is an empt y set and Y = X,then f ∈ Ꮽᏼ(Z : X) is called an almost periodic sequence in X. Almost periodic sequences can be also defined for any sequence { f (n)} n≥a ,or f : Z + a → X by requiring that any N ε (Σ) consecutive integers is in Z + a . For unifor mly almost periodic sequences, we have the following results. Theorem 2.2. Let f ∈ Ꮽᏼ(Z × Ω : Y) and le t Σ be any compact set in Ω. Then f (n,·) is continuous on Σ uniformly for n ∈ Z and the range f (Z × Σ) is relatively compact, which implies that f ( Z × Σ) is a bounded subset in Y. Yihong Song 3 Theorem 2.3. Let f ∈ Ꮽᏼ(Z × Ω : Y). Then for any integer sequence {α  k }, α  k →∞as k → ∞ , there exists a subsequence {α k } of {α  k }, α k →∞as k →∞,andafunctionξ : Z × Ω → Y such that f  n + α k ,w  −→ ξ(n,w) (2.2) uniformly on Z × Σ as k →∞,whereΣ isanycompactsetinΩ.Moreover,ξ ∈ Ꮽᏼ(Z × Ω : Y),thatis,ξ(n,w) is also almost periodic in n uniformly for w ∈ Ω. If Ω is the empty set and Y = X in Theorem 2.3,then{ξ(n)} is an almost periodic sequence. Theorem 2.4. If f ∈ Ꮽᏼ(Z × Ω : Y), then there exists a sequence {α k }, α k →∞as k →∞, such that f  n + α k ,w  −→ f (n,w) (2.3) uniformly on Z × Σ as k →∞,whereΣ is any compact set in Ω. Obviously, {α k } in Theorem 2.4 can be chosen to be a positive integer sequence. Definit ion 2.5. Asequence {x(n)} n∈Z + , x(n) ∈ X, or a function x : Z + → X,iscalled asymptotically almost periodic if x = x 1 | Z + + x 2 ,wherex 1 ∈ Ꮽᏼ(Z,X)andx 2 : Z + → X satisfying x 2 (n) X → 0asn →∞. Denote by ᏭᏭᏼ(Z + ,X) all such sequences. Theorem 2.6. Let x : Z + → X. T hen the following statements are equivalent. (1) x ∈ ᏭᏭᏼ(Z + ,X). (2) For any sequence {α k }⊂Z + , α k > 0, and α k →∞as k →∞,thereisasubsequence {β k }⊂{α k } such that β k →∞ as k →∞ and {x(n + β k )} converges uniformly on Z + as k →∞. Similarly, asymptotically almost periodic sequence can be defined for any sequence {x( n)} n≥a ,orx : Z + a → X. The proof of the above results is omitted here because it is not difficult for readers giving proofs by the similar arguments in [19, 18] for continuous (uniformly) almost periodic function φ : R × Ω → X (see also [2]forthecasethatX = Y = E d ). 2.2. Some assumptions and stability definitions. We now present some definitions and notations that will be used throughout this paper. For a given positive integer τ>0, we define C tobeaBanachspacewithanorm ·by C =  φ | φ : dis[−τ,0] −→ E d ,φ=max    φ( j)    for j ∈ dis[−τ,0]  . (2.4) It is clear that C is isometric to the space E d×(τ+1) . Let n 0 ∈ Z + and let {x(n)}, n ≥ n 0 − τ,beasequencewithx(n) ∈ E d .Foreachn ≥ n 0 , we define x n : dis[−τ,0] → E d by the relation x n ( j) = x(n + j), j ∈ dis[−τ,0]. (2.5) 4AdvancesinDifference Equations Let us return to system (1.1), that is, x(n +1) = F  n,x n  , (2.6) where F : Z × C → E d and x n : dis[−τ,0] → C. Definit ion 2.7. Let n 0 ∈ Z + and let φ be a given vector in C.Asequencex ={x(n)} n≥n 0 in E d is said to be a solution of (2.6), passing through (n 0 ,φ), if x n 0 = φ, that is, x(n 0 + j) = φ( j)forj ∈ dis[−τ,0], x(n +1), and x n satisfy (2.6)forn ≥ n 0 ,wherex n is defined by (2.5). Denote by {x( n, φ)} n≥n 0 asolutionof(2.6)suchthatx n 0 = φ.Nolossofclarity arises i f we refer to the solution {x( n, φ)} n≥n 0 as x ={x(n)} n≥n 0 . We make the following assumptions on (2.6) throughout this paper. (H1) F : Z × C → E d and F(n,·)iscontinuousonC for each n ∈ Z. (H2) System (2.6) has a bounded solution u ={u(n)} n≥0 , passing through (0,φ 0 ), φ 0 ∈ C. For this bounded solution {u(n)} n≥0 , there is an α>0suchthat|u(n)|≤α for all n ≥−τ, which implies that u n ≤α and u n ∈ S α ={φ : φ≤α and φ ∈ C} for all n ≥ 0. Definit ion 2.8. A bounded solution x ={x(n)} n≥0 of (2.6)issaidtobe (i) uniformly stable, abbreviated to read “ x is ᐁ᏿,” i f f o r any ε>0 and any integer n 0 ≥ 0, there exists δ(ε) > 0suchthatx n 0 − x n 0  <δ(ε) implies that x n − x n  <ε for all n ≥ n 0 ,where{x(n)} n≥n 0 is any solution of (2.6); (ii) uniformly asymptotically stable, abbreviated to read “ x is ᐁᏭ᏿,” if it is uni- formly stable and there exists δ 0 > 0 such that for any ε>0, there is a positive integer N = N(ε) > 0suchthatifn 0 ≥ 0andx n 0 − x n 0  <δ 0 ,thenx n − x n  <ε for all n ≥ n 0 + N,where{x( n)} n≥n 0 is any solution of (2.6); (iii) globally uniformly asymptotically stable, abbreviated to read “ x is ᏳᐁᏭ᏿,” if it is uniformly stable and x n − x n →0asn →∞,whenever{x(n)} n≥n 0 is any solution of (2.6). Remark 2.9. It is easy to see that an equivalent definition for x ={x(n)} n≥0 being ᐁᏭ᏿ is the following: (ii ∗ ) x ={x(n)} n≥0 is ᐁᏭ᏿, if it is uniformly stable and there exists δ 0 > 0suchthat if n 0 ≥ 0andx n 0 − x n 0  <δ 0 ,thenx n − x n →0asn →∞,where{x(n)} n≥n 0 is any solution of (2.6). 3. Periodic systems In this section, we discuss the existence of periodic solutions of (2.6), namely, x(n +1) = F  n,x n  , n ≥ 0, (3.1) under a periodic condition (H3) as follows. (H3) The F(n, ·)in(3.1)isperiodicinn ∈ Z, that is, there exists a positive integer ω such that F(n +ω,v) = F(n,v)foralln ∈ Z and v ∈ C. Yihong Song 5 We are now in a position to give our main results in this section. We first show that if the bounded solution {u(n)} n≥0 of (3.1) is uniformly stable, then {u(n)} n≥0 is an asymp- totically almost periodic sequence. Theorem 3.1. Suppose conditions (H1)–(H3) hold. If the bounded solution {u(n)} n≥0 of (3.1)isᐁ᏿, the n {u(n)} n≥0 is an asymptotically almost periodic sequence in E d ,equiva- lently, (3.1) has an asymptotically almost periodic solution. Proof. Since u n ≤α for n ∈ Z + , there is bounded (or compact) set S α ⊂ C such that u n ∈ S α for all n ≥ 0. Let {n k } k≥1 be any integer sequence such that n k > 0andn k →∞as k →∞.Foreachn k , there exists a nonnegative integer l k such that l k ω ≤ n k ≤ (l k +1)ω. Set n k = l k ω + τ k .Then0≤ τ k <ωfor all k ≥ 1. Since {τ k } k≥1 is bounded set, we can assume that, taking a subsequence if necessary, τ k = j ∗ for all k ≥ 1, where 0 ≤ j ∗ <ω. Now, set u k (n) = u(n + n k ). Notice that u n+n k ( j) = u(n + n k + j) = u k (n + j) = u k n ( j)and hence, u n+n k = u k n .Thus, u k (n +1)= u  n + n k +1  = F  n + n k ,u n+n k  = F  n + n k ,u k n  = F  n + j ∗ ,u k n  , (3.2) which implies that {u k (n)} is a solution of the system x(n +1) = F  n + j ∗ ,x n  (3.3) through (0,u n k ). It is readily shown that if {u(n)} n≥0 is ᐁ᏿,then{u k (n)} n≥0 is also ᐁ᏿ with the same pair (ε,δ(ε)) as the one for {u(n)} n≥0 . Since {u(n + n k )} is bounded for al l n ≥−τ and n k , we can use the diagonal method to get a subsequence {n k j } of {n k } such that u(n + n k j )convergesforeachn ≥−τ as j →∞. Thus, we can assume that the sequence u(n + n k )convergesforeachn ≥−τ as k →∞. Notice that u k 0 ( j) = u k (0 + j) = u(j + n k ). Then for any ε>0 there exists a positive integer N 1 (ε)suchthatifk,m ≥ N 1 (ε), then   u k 0 − u m 0   <δ(ε), (3.4) where δ(ε) is the number for the uniform stability of {u(n)} n≥0 . Notice that {u m (n) = u(n + n m )} n≥0 is also a solution of (3.3) and that {u k (n)} n≥0 is uniformly stable. It follows from Definition 2.8 and (3.4)that   u k n − u m n   <ε ∀n ≥ 0, (3.5) and hence,   u k (n) − u m (n)   <ε ∀n ≥ 0, k, m ≥ N 1 (ε). (3.6) This implies that for any positive integer sequence n k , n k →∞as k →∞, there exists a subsequence {n k j } of {n k } for which {u(n + n k j )} converges uniformly on Z + as j →∞. Thus, {u(n)} n≥0 is an asymptotically almost periodic sequence by Theorem 2.6 and the proof is completed.  6AdvancesinDifference Equations Lemma 3.2. Suppose that (H1)–(H3) hold and {u(n)} n≥0 , the bounded solution of (3.1), is ᐁ᏿.Let {n k } k≥1 be an integer sequence such that n k > 0, n k →∞as k →∞, u(n + n k ) → η(n) for each n ∈ Z + and F(n + n k ,v) → G(n,v) uniformly for n ∈ Z + and Σ as k →∞, where Σ isanycompactsetinC. Then {η(n)} n≥0 is a solution of the system x(n +1) = G  n,x n  , n ≥ 0, (3.7) and is ᐁ᏿.Moreover,if {u(n)} n≥0 is ᐁᏭ᏿, then {η(n)} n≥0 is also ᐁᏭ᏿. Proof. Since u k (n) = u(n + n k )isuniformlyboundedforn ≥−τ and k ≥ 1, we can a s- sume that, taking a subsequence if necessary, u(n + n k )alsoconvergesforeachn ∈ dis [ −τ, −1]. Define η( j) = lim k→∞ u( j + n k )for j ∈ dis[−τ,−1]. Then u(n + n k ) → η(n)for each n ∈ dis[−τ,∞), and hence, u k n → η n as k →∞for each n ≥ 0. Notice that u k n ∈ S α for all n ≥ 0, k ≥ 1, and η n ∈ S α for n ≥ 0. It follows from Theorem 2.4 that there exists asubsequence {n k j } of {n k }, n k j →∞as j →∞,suchthatF(n + n k j ,v) → G(n,v)uni- formly on Z × S α as j →∞and G(n, ·)iscontinuousonS α uniformly for all n ∈ Z.Since u(n + n k j +1)= F(n + n k j ,u k j n )and F  n + n k j ,u k j n  − G  n,η n  = F  n + n k j ,u k j n  − G  n,u k j n  + G  n,u k j n  − G  n,η n  −→ 0asj −→ ∞ , (3.8) we have η(n +1) = G(n,η n )(n ≥ 0). This shows that {η(n)} n≥0 is a solution of (3.7). To pro v e th at {η(n)} n≥0 is ᐁ᏿,wesetn k = l k ω + j ∗ as before, where 0 ≤ j ∗ <ω.Then u k j (n) = u(n + n k j ) → η(n)foreachn ∈ Z + as j →∞.SinceF(n + n k j ,v) = F(n + j ∗ ,v) → G(n,v)asj →∞,wehaveG(n, v) = F(n + j ∗ ,v). For any ε>0, let δ(ε) > 0 be the one for uniform stability of {u(n)} n≥0 .Foranyn 0 ∈ Z + ,let{x(n)} n≥0 beasolutionof(3.7)such that η n 0 − x n 0 =μ<δ(ε). Since u k j n → η n as j →∞ for each n ≥ 0, there is a positive integer J 1 > 0suchthatif j ≥ J 1 ,then   u k j n 0 − η n 0   <δ(ε) − μ. (3.9) Thus, for j ≥ J 1 ,wehave   u n 0 +j ∗ +l k j ω − x n 0   ≤   u n 0 +j ∗ +l k j ω − η n 0   +   η n 0 − x n 0   <δ(ε). (3.10) Notice that {u(n + j ∗ + l k j ω)} (n ≥ 0) is a uniformly stable solution of (3.7)withG(n,x n ) = F(n + j ∗ ,x n ). Then,   u n+ j ∗ +l k j ω − x n   <ε ∀n ≥ n 0 . (3.11) Since {η(n)} is also a solution of (3.7)andu k j n → η n for each n ≥ 0asj →∞,foran arbitrary ν > 0, there is J 2 > 0suchthatif j ≥ J 2 ,then   η n 0 − u n 0 +j ∗ +l k j ω   <δ(ν), (3.12) Yihong Song 7 and hence, η n − u n+ j ∗ +l k j ω  < ν for all n ≥ n 0 ,where(ν,δ(ν)) is a pair for the uniform stability of u(n + j ∗ + l k j ω). This shows that if j ≥ max(J 1 ,J 2 ), then   η n − x n   ≤   η n − u n+ j ∗ +l k j ω   +   u n+ j ∗ +l k j ω − x n   <ε+ ν (3.13) for all n ≥ n 0 , which implies that η n − x n ≤ε for all n ≥ n 0 if η n 0 − x n 0  <δ(ε) because ν is arbitrary. This proves that {η(n)} n≥0 is uniformly stable. To pr o ve t ha t {η(n)} n≥0 is ᐁᏭ᏿, we use definition (ii ∗ )inRemark 2.9.Let{x(n)} be asolutionof(3.7)suchthat η n 0 − x n 0  <δ 0 ,whereδ 0 is the number for the uniformly asymptotic stability of {u(n)}. Notice that u(n + j ∗ + l k j ω) = u k j (n)isauniformlyasymp- totically stable solution of (3.7)withG(n,φ) = F(n + j ∗ ,φ) and with the same δ 0 as the one for {u(n)}.Setη n 0 − x n 0 =μ<δ 0 .Again,forsufficient large j, we have the simi- lar relations (3.10)and(3.12)with u n 0 +j ∗ +l k j ω − x n 0  <δ 0 and u n 0 +j ∗ +l k j ω − η n 0  <δ 0 . Thus,   η n − x n   ≤   η n − u n+ j ∗ +l k j ω   +   u n+ j ∗ +l k j ω − x n   −→ 0 (3.14) as n →∞ if u n 0 − x n 0  <δ 0 , because {u k j (n)}, {x(n)},and{η(n)} satisfy (3.7)with G(n,φ) = F(n + j ∗ ,φ). This completes the proof.  Using Theorem 3.1 and Lemma 3.2, we can show that (3.1) has an almost periodic solution. Theorem 3.3. If the bounded solution {u(n)} n≥0 of (3.1)isᐁ᏿,thensystem(3.1) has an almost periodic solution, which is also ᐁ᏿. Proof. It follows from Theorem 3.1 that {u(n)} n≥0 is asymptotically almost periodic. Set u(n) = p(n)+q(n)(n ≥ 0), where {p(n)} n≥0 is almost periodic sequence and q(n) → 0 as n →∞. For positive integer sequence {n k ω}, there is a subsequence {n k j ω} of {n k ω} such that p(n +n k j ω) → p ∗ (n)uniformlyonZ as j →∞and {p ∗ (n)} is almost periodic. Then u( n + n k j ω) → p ∗ (n) uniformly for n ≥−τ, and hence, u n+n k j ω → p ∗ n for all n ≥ 0as j →∞.Since u  n + n k j ω +1  = F  n + n k j ω,u n+n k j ω  = F  n,u n+n k j ω  −→ F  n, p ∗ n  (3.15) as j →∞,wehavep ∗ (n +1)= F(n, p ∗ n )forn ≥ 0, that is, system (3.1) has an almost periodic solution, which is also ᐁ᏿ by Lemma 3.2.  Now, we show that if the bounded solution {u(n)} is uniformly asymptotically stable, then (3.1) has a periodic solution of period mω for some positive integer m. Theorem 3.4. If the bounded solution {u(n)} n≥0 of (3.1)isᐁᏭ᏿,thensystem(3.1)hasa periodic solution of period mω for some positive integer m,whichisalsoᐁᏭ᏿. Proof. Set u k (n) = u(n + kω), k = 1, 2, BytheproofofTheorem 3.1, there is a subse- quence {u k j (n)} converges to a solution {η(n)} of (3.3)foreachn ≥−τ and hence, u k j 0 → η 0 as j →∞. Thus, there is a positive integer p such that u k p 0 − u k p+1 0  <δ 0 (0 ≤ k p <k p+1 ), where δ 0 is the one for uniformly asymptotic stability of {u(n)} n≥0 .Letm = k p+1 − k p 8AdvancesinDifference Equations and notice that u m (n) = u(n + mω) is a solution of (3.1). Since u m k p ω ( j) = u m (k p ω + j) = u(k p+1 ω + j) = u k p+1 ω ( j)for j ∈ dis[−τ,0], we have   u m k p ω − u k p ω   =   u k p+1 ω − u k p ω   =   u k p+1 0 − u k p 0   <δ 0 , (3.16) and hence,   u m n − u n   −→ 0asn −→ ∞ (3.17) because {u(n)} n≥0 is ᐁᏭ᏿ (see also Remark 2.9). On the other hand, {u(n)} n≥0 is asymp- totically almost periodic by Theorem 3.1,then u(n) = p(n)+q(n), n ≥ 0, (3.18) where {p(n)} n∈Z is almost periodic and q(n) → 0asn →∞.Itfollowsfrom(3.17)and (3.18)that   p(n) − p(n + mω)   −→ 0asn −→ ∞ , (3.19) which implies that p(n) = p(n + mω)foralln ∈ Z because {p(n)} is almost periodic. For integer sequence {kmω}, k = 1,2, ,wehaveu(n + kmω) = p(n)+q(n + kmω). Then u(n + kmω) → p(n) uniformly for all n ≥−τ as k →∞, and hence, u n+kmω → p n for n ≥ 0ask →∞.Sinceu(n + kmω +1)= F(n,u n+kmω ), we have p(n +1)= F(n, p n ) for n ≥ 0, which implies that (3.1) has a periodic solution {p(n)} n≥0 of period mω.The uniformly asymptotic stability of {p(n)} n≥0 follows from Lemma 3.2.  Finally, we show that if the bounded solution {u(n)} is ᏳᐁᏭ᏿,then(3.1)hasaperi- odic solution of period ω. Theorem 3.5. If the bounded solution {u(n)} n≥0 of (3.1)isᏳᐁᏭ᏿,thensystem(3.1)has a periodic solution of period ω. Proof. By Theorem 3.1, {u(n)} n≥0 is asymptotically almost periodic. Then u(n) = p(n)+ q(n)(n ≥ 0), where {p(n)} (n ∈ Z) is an almost periodic sequence and q(n) → 0asn →∞. Notice that u(n + ω)isalsoasolutionof(3.1) satisfying u ω ∈ S α .Since{u(n)} is ᏳᐁᏭ᏿, we have u n − u n+ω →0asn →∞, which implies that p(n) = p(n + ω)foralln ∈ Z.By the same technique in the proof of Theorem 3.4, we can show that {p(n)} is an ω-periodic solution of (3.1).  4. Almost periodic systems In this section, we discuss the existence of asymptotically almost periodic solutions of (2.6), that is, x(n +1) = F  n,x n  , n ≥ 0, (4.1) under the condition (H4) as fol lows. (H4) F ∈ Ꮽᏼ(Z × C : E d ), that is, F(n,v) is almost periodic in n ∈ Z uniformly for v ∈ C. Yihong Song 9 By H(F) we denote the uniform closure of F(n,v), that is, G ∈ H(F) if there is an integer sequence {α k } such that α k →∞and F(n + α k ,v) → G(n,v)uniformlyonZ × Σ as k →∞, where Σ is any compact set in C.NotethatH(F) ⊂ Ꮽᏼ(Z × C : E d )byTheorem 2.3 and F ∈ H(F)byTheorem 2.4. We first show that if (4.1) has a bounded asymptotically almost periodic solution, then (4.1) has an almost periodic solution. In fact, we have a more general result in the following. Theorem 4.1. Suppose (H1), (H2), and (H4) hold. If the bounded solution {u(n)} n≥0 of (4.1) is asymptotically almost periodic, then for any G ∈ H(F), the system x(n +1) = G  n,x n  (4.2) has an almost periodic solution for n ≥ 0. Consequently, (4.1) has an almost periodic solu- tion. Proof. Since the solution {u(n)} n≥0 is asymptotically almost periodic, it follows from Theorem 2.6 that it has the decomposition u(n) = p(n)+q(n)(n ≥ 0), where {p(n)} n∈Z is almost periodic and q(n) → 0asn →∞. Notice that {u(n)} is bounded. There is com- pact set S α ∈ C such that u n ∈ S α and p n ∈ S α for all n ≥ 0. For any G ∈ H(F), there is an integer sequence {n k }, n k > 0, such that n k →∞as k →∞and F(n + n k ,v) → G(n,v) uniformly on Z × S α as k →∞. Taking a subsequence if necessary, we can also assume that p(n + n k ) → p ∗ (n)uniformlyonZ and {p ∗ (n)} is also an almost periodic sequence. For any j ∈ dis[−τ,0], there is positive integer k 0 such that if k>k 0 ,then j + n k ≥ 0for any j ∈ dis[−τ,0]. In this case, we see that u(n + n k ) → p ∗ (n) uniformly for all n ≥−τ as k →∞, and hence, u n+n k → p ∗ n in C uniformly for n ∈ Z + as k →∞.Since u  n + n k +1  = F  n + n k ,u n+n k  =  F  n + n k ,u n+n k  − F  n + n k , p ∗ n  +  F  n + n k , p ∗ n  − G  n, p ∗ n  + G  n, p ∗ n  , (4.3) the first term of right-hand side of (4.3)tendstozeroask →∞ by Theorem 2.2 and F(n + n k , p ∗ n ) − G(n, p ∗ n ) → 0ask →∞,wehavep ∗ (n +1)= G(n, p ∗ n )foralln ∈ Z + , which implies that (4.2) has an almost periodic solution {p ∗ (n)} n≥0 , passing through (0, p ∗ 0 ), where p ∗ 0 ( j) = p ∗ ( j)for j ∈ dis[−τ,0].  To deal with almost periodic solutions of (4.1) in terms of uniform stability, we assume that for each G ∈ H(F), system (4.2) has a unique solution for a given initial condition. Lemma 4.2. Suppose (H1), (H2), and (H4) hold. Let {u(n)} n≥0 be the bounded s olution of (4.1). Let {n k } k≥1 be a positive integer sequence such that n k →∞, u n k → ψ, and F(n + n k ,v) → G(n,v) uniformly on Z × Σ as k →∞,whereΣ is any compact subset in C and G ∈ H(F). If the bounded solution {u(n)} n≥0 is ᐁ᏿, then the solution {η(n)} n≥0 of (4.2), through (0,ψ),isᐁ᏿. In addition, if {u(n)} n≥0 is ᐁᏭ᏿, then {η(n)} n≥0 is also ᐁᏭ᏿. Proof. Set u k (n) = u(n + n k ). It is easy to see that u k (n) is a solution of x(n +1) = F  n + n k ,x n  , n ≥ 0, (4.4) 10 Advances in Difference Equations passing though (0,u n k )andu k n ∈ S α for all k,whereS α is compact subset of C such that u n  <αfor all n ≥ 0. Since {u(n)} n≥0 is ᐁ᏿, {u k (n)} is also ᐁ᏿ with the same pair (ε,δ(ε)) as the one for {u(n)} n≥0 . Taking a subsequence if necessary, we can assume that {u k (n)} k≥1 converges to a vector η(n)foralln ≥ 0ask →∞.From(4.3)withp ∗ n = η n ,we can see that {η(n)} n≥0 is the unique solution of (4.2), through (0,ψ). To show that the solution {η(n)} n≥0 of (4.2)isᐁ᏿,weneedtoprovethatifforany ε>0 and any integer n 0 ≥ 0, there exists δ ∗ (ε) > 0suchthatη n 0 − y n 0  <δ ∗ (ε) implies that η n − y n  <εfor all n ≥ n 0 ,where{y(n)} n≥n 0 is a solution of (4.2) passing through (n 0 ,φ)withy n 0 = φ ∈ C. For any given n 0 ∈ Z + ,ifk is sufficiently large, say k ≥ k 0 > 0, we have   u k n 0 − η n 0   < 1 2 δ  ε 2  , (4.5) where δ(ε) is the one for uniform stability of {u(n)} n≥0 .Letφ ∈ C such that   φ − η n 0   < 1 2 δ  ε 2  (4.6) and let {x( n)} n≥n 0 be the solution of (4.1)suchthatx n 0 +n k = φ.Then{x k (n) = x(n + n k )} is a solution of (4.4)withx k n 0 = φ.Since{u k (n)} is ᐁ᏿ and x k n 0 − u k n 0  <δ(ε/2) for k ≥ k 0 , we have   u k n − x k n   < ε 2 ∀n ≥ n 0 , k ≥ k 0 . (4.7) It follows from (4.7)that   x k n   ≤   u k n   + ε 2 <α+ ε 2 ∀n ≥ n 0 , k ≥ k 0 . (4.8) Then there exists a number α ∗ > 0suchthatx k n ∈ S α ∗ for all n ≥ 0andk ≥ k 0 , which implies that there is subsequence of {x k (n)} k≥k 0 for each n ≥ n 0 − τ, denoted by {x k (n)} again, such that x k (n) → y(n)foreachn ≥ n 0 − τ, and hence, x k n → y n for all n ≥ n 0 as k → ∞ .Clearly,y n 0 = φ and the set S α ∗ is compact set in C.SinceF(n,v) is almost periodic in n uniformly for v ∈ C, we can assume that, taking a subsequence if necessary, F(n + n k ,v) → G(n,v)uniformlyonZ × S α ∗ as k →∞.Takingk →∞in x k (n +1)= F(n + n n k ,x k n ), we have y(n +1) = G(n, y n ), namely, {y(n)} is the unique solution of (4.2), passing through (n 0 ,φ)withy n 0 = φ ∈ C. On the other hand, for any integer N>0, there exists k N ≥ k 0 such that if k ≥ k N ,then   x k n − y n   < ε 4 ,   u k n − η n   < ε 4 for n 0 ≤ n ≤ n 0 + N. (4.9) From (4.7)and(4.9), we obtain   η n − y n   <ε for n 0 ≤ n ≤ n 0 + N. (4.10) Since N is arbitrary, we have η n − y n  <εfor all n ≥ n 0 if φ − η n 0  < [δ(ε/2)]/2and φ ∈ C, which implies that the solution {η(n)} n≥0 of (4.2)isᐁ᏿. [...]... Journal of Mathematical Analysis and Applications, vol 313, no 2, pp 678–688, 2006 [15] Y Song, Almost periodic solutions of discrete Volterra equations, ” Journal of Mathematical Analysis and Applications, vol 314, no 1, pp 174–194, 2006 [16] Y Song and H Tian, Periodic and almost periodic solutions of nonlinear Volterra difference equations with unbounded delay,” to appear in Journal of Computational and. .. 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R P Agarwal, Difference Equations and Inequalities, vol 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000 [3] C T H Baker and Y Song, Periodic solutions of discrete Volterra equations, ” Mathematics and Computers in Simulation, vol 64, no 5, pp 521–542, 2004 [4] C Cuevas and M Pinto, “Asymptotic properties of solutions to nonautonomous... “Asymptotic theory for delay difference equations, ” Journal of Difference o Equations and Applications, vol 1, no 2, pp 99–116, 1995 [8] S Elaydi, S Murakami, and E Kamiyama, “Asymptotic equivalence for difference equations with infinite delay,” Journal of Difference Equations and Applications, vol 5, no 1, pp 1–23, 1999 [9] I Gy¨ ri and G Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical... is asymptotically almost periodic sequence by Theorem 2.6 Furthermore, (4.1) has an almost periodic solution, which is ᐁᏭ᏿ by Theorem 4.1 This ends the proof Acknowledgments This work is supported in part by NNSF of China (no 10471102) We thank the referees for helpful comments on our presentation References [1] T Yoshizawa, “Asymptotically almost periodic solutions of an almost periodic system,” Funkcialaj... systems with infinite delay,” Computers & Mathematics with Applications, vol 42, no 3–5, pp 671–685, 2001 [5] S N Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1999 [6] S Elaydi and S Zhang, “Stability and periodicity of difference equations with finite delay,” Funkcialaj Ekvacioj, vol 37, no 3, pp 401–413, 1994 [7] S Elaydi and. .. have y n − ηn < ε ∀n ≥ n 0 + N ε 2 (4.14) if φ − ηn0 < (δ0 /2) and φ ∈ C The proof is completed Before dealing with the asymptotic almost periodicity of {u(n)}, we need the following lemma Lemma 4.3 Suppose that assumptions (H1), (H2), and (H4) hold, the bounded solution {u(n) = u(n,ψ 0 )} of (4.1) is ᐁᏭ᏿ and for each G ∈ H(F), the solution of (4.2) is unique for any given initial data Let S ⊇ Sα be a... following result Theorem 4.4 Suppose that for each G ∈ H(F), the solution of (4.2) is unique for the initial condition If the bounded solution {u(n)}n≥0 of (4.1) is ᐁᏭ᏿, then {u(n)}n≥0 is asymptotically almost periodic Consequently, (4.1) has an almost periodic solution which is ᐁᏭ᏿ Proof Let the bounded solution {u(n)} of (4.1) be ᐁᏭ᏿ with the triple (δ(·),δ0 ,N(·)) Let {nk }k≥1 be any positive integer . in Difference Equations Volume 2007, Article ID 68023, 15 pages doi:10.1155/2007/68023 Research Article Periodic and Almost Periodic Solutions of Functional Difference Equations with Finite Delay Yihong. Graef For periodic and almost periodic functional difference equations with finite delay, the ex- istence of periodic and almost periodic solutions is obtained by using stability properties of a bounded. done on almost periodic solutions of nonlinear functional difference equations with finite de- lay via uniform stability properties of a bounded solution. This motivates us to investigate almost periodic