Research Article Almost Periodic Solutions of Prey-Predator Discrete Models with Delay

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1 Hindawi Publishing Corporation Advances in Difference Equations Volume 009, Article ID , 19 pages doi: /009/ Research Article Almost Periodic Solutions of Prey-Predator Discrete Models with Delay Tomomi Itokazu and Yoshihiro Hamaya Department of Information Science, Okayama University of Science, 1-1 Ridai-cho, Okayama , Japan Correspondence should be addressed to Yoshihiro Hamaya, Received 10 February 009; Revised 18 May 009; Accepted 9 July 009 Recommended by Elena Braverman The purpose of this article is to investigate the existence of almost periodic solutions of a system of almost periodic Lotka-Volterra difference equations which are a prey-predator system x 1 n 1 x 1 n expb 1 n a 1 n x 1 n c n n K n s x s, x n 1 x n exp b n a n x n c 1 n n K 1 n s x 1 s and a competitive system x i n 1 x i n expb i n a ii x i n l n j 1,j / i K ij n s x j s, by using certain stability properties, which are referred to as K, ρ -weakly uniformly asymptotic stable in hull and K, ρ -totally stable. Copyright q 009 T. Itokazu and Y. Hamaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and Preliminary For ordinary differential equations and functional differential equations, the existence of almost periodic solutions of almost periodic systems has been studied by many authors. One of the most popular methods is to assume the certain stability properties 1 4. Recently, Song and Tian 5 have shown the existence of periodic and almost periodic solutions for nonlinear Volterra difference equations by means of K, ρ -stability conditions. Their results are to extend results in Hamaya to discrete Volterra equations. To the best of our knowledge, there are no relevant results on almost periodic solutions for discrete Lotka-Volterra models by means of our approach, except for Xia and Cheng s special paper 6. However, they treated only nondelay case in 6. We emphasize that our results extend 3, 6, 7 as a discret delay case. In this paper, we will discuss the existence of almost periodic solutions for discrete Lotka-Volterra difference equations with time delay. In what follows, we denote by R m real Euclidean m-space, Z is the set of integers, Z is the set of nonnegative integers, and will denote the Euclidean norm in R m. For any interval

2 Advances in Difference Equations I Z :,, we denote by BS I the set of all bounded functions mapping I into R m, and set φ I sup φ s : s I. Now, for any function x :,a R m and n<a,define a function x n : Z, 0 R m by x n s x n s for s Z. Let BS be a real linear space of functions mapping Z into R m with sup-norm: BS φ φ : Z R m with φ sup φ s <. 1.1 s Z We introduce an almost periodic function f n, x : Z D R m, where D is an open set in R m. Definition 1.1. It holds that f n, x is said to be almost periodic in n uniformly for x D, if for any ɛ>0 and any compact set K in D, there exists a positive integer L ɛ, K such that any interval of length L ɛ, K contains an integer τ for which f n τ, x f n, x ɛ 1. for all n Z and all x K. Such a number τ in the above inquality is called an ɛ-translation number of f n, x. In order to formulate a property of almost periodic functions, which is equivalent to the above difinition, we discuss the concept of the normality of almost periodic functions. Namely, let f n, x be almost periodic in n uniformly for x D. Then, for any sequence h k Z, there exist a subsequence h k of h and function g n, x such that k f n h k,x g n, x 1.3 uniformly on Z K as k, where K is a compact set in D. There are many properties of the discrete almost periodic functions 5, 8, which are corresponding properties of the continuous almost periodic functions f t, x C R D, R m 4. We will denote by T f the function space consisting of all translates of f, that is, f τ T f, where f τ n, x f n τ, x, τ Z. 1.4 Let H f denote the uniform closure of T f in the sense of 1.4. H f is called the hull of f. In particular, we denote by Ω f the set of all limit functions g H f such that for some sequence n k, n k as k and f n n k,x g n, x uniformly on Z S for any compact subset S in R m.by 1.3, iff : Z D R m is almost periodic in n uniformly for x D, so is a function in Ω f. The following concept of asymptotic almost periodicity was introduced by Frechet in the case of continuous function cf. 4. Definition 1.. It holds that u n is said to be asymptotically almost periodic if it is a sum of an almost periodic function p n and a function q n defined on I a, Z 0, which tends to zero as,thatis, u n p n q n. 1.5

3 Advances in Difference Equations 3 However, u n is asymptotically almost periodic if and only if for any sequence n k such that n k as k, there exists a subsequence n k for which u n n k converges uniformly on n; a n <.. Prey-Predator Model We will consider the existence of a strictly positive component-wise almost periodic solution of a system of Volterra difference equations: x 1 n 1 x 1 n exp b 1 n a 1 n x 1 n c n K n s x s, x n 1 x n exp b n a n x n c 1 n K 1 n s x 1 s, F which describes a model of the dynamics of a prey-predator discrete system in mathematical ecology. In F, setting a i n, b i n, and c i n R-valued bounded almost periodic in Z: a i inf n Z a i n, B i supb i n, n Z A i supa i n, n Z c i inf n Z c i n C i supc i n n Z b i inf n Z b i n, i 1,,.1 and K i : Z 0, R i 1, denote delay kernels such that K i s 0, K i s 1, sk i s <, i 1,.. Under the above assumptions, it follows that for any n 0,φ i Z BS, i 1,, there is a unique solution u n u 1 n,u n of F through n 0,φ i, i 1,, if it remains bounded. We set α 1 exp B 1 1, α exp b C 1 α 1 1, a 1 a β 1 min exp b 1 A 1 α 1 C α b 1 C α, b 1 C α, A 1 A 1 ( ) B A α c 1 β 1 B c 1 β 1 B c 1 β 1 β min exp, A A.3 cf. 6, Application 4.1.

4 4 Advances in Difference Equations We now make the following assumptions: i a i > 0, b i > 0 i 1,, c 1 > 0, c 0, ii b 1 >C α, B <c 1 β 1, iii there exists a positive constant m such that a i >C i m i 1,..4 Then, we have 0 <β i <α i for each i 1,. We can show the following lemmas. Lemma.1. If x n x 1 n,x n is a solution of F through n 0,φ i, i 1, such that β i φ i s α i i 1, for all s 0, then one has β i x i n α i i 1, for all n n 0. Proof. First, we claim that lim sup x 1 n α 1..5 To do this, we first assume that there exists an l 0 n 0 such that x 1 l 0 1 x 1 l 0. Then, it follows from the first equation of F that b 1 l 0 a 1 l 0 x 1 l 0 c l 0 l 0 K l 0 s x s 0..6 Hence, It follows that x 1 l 0 b 1 l 0 c l 0 l0 K l 0 s x s a 1 l 0 x 1 l 0 1 x 1 l 0 exp b 1 l 0 a 1 l 0 x 1 l 0 c l 0 b 1 l 0 a 1 l 0 B 1 a 1. l 0 x 1 l 0 expb 1 a 1 x 1 l 0 expb 1 1 a 1 : α 1, K l 0 s x s.7.8 where we use the fact that Now we claim that max x R x exp ( p qx ) exp ( p 1 ) q, for p, q > 0..9 x 1 n α 1, for n l 0..5

5 Advances in Difference Equations 5 By way of contradiction, we assume that there exists a p 0 >l 0 such that x 1 p 0 >α 1. Then, p 0 l 0. Let p 0 l 0 be the smallest integer such that x 1 p 0 >α 1. Then, x 1 p 0 1 x 1 p 0. The above argument shows that x 1 p 0 α 1, which is a contradiction. This proves our assertion. We now assume that x 1 n 1 <x 1 n for all n n 0. Then lim x 1 n exists, which is denoted by x 1. We claim that x 1 exp B 1 1 /a 1. Suppose to the contrary that x 1 > exp B 1 1 /a 1. Taking limits in the first equation in System F, wesetthat ( 0 lim b 1 n a 1 n x 1 n c n ) K n s x s lim b 1 n a 1 n x 1 n B 1 a 1 x 1 < 0,.10 which is a contradiction. It follows that.5 holds, and then we have x i n α 1 for all n n 0 from.5. Next, we prove that lim sup x n α..11 We first assume that there exists an k 0 n 0 such that k 0 l 0 and x k 0 1 x k 0. Then b k 0 a k 0 x k 0 c 1 k 0 k 0 K 1 k 0 s x 1 s 0..1 Hence, x k 0 b k 0 c 1 k 0 k0 K 1 k 0 s x 1 s b C 1 α a k 0 a It follows that x k 0 1 x k 0 exp b k 0 a k 0 x k 0 c 1 k 0 x k 0 exp b a x k 0 C 1 α 1 exp b C 1 α 1 1 : α, a k 0 K 1 k 0 s x 1 s.14 where we also use the two facts which are used to prove.5. Now we claim that x n α n k Suppose to the contrary that there exists a q 0 >k 0 such that x q 0 >α. Then q 0 k 0. Let q 0 k 0 be the smallest integer such that x q 0 >α. Then x q 0 1 <x q 0. Then the above argument shows that x q 0 α, which is a contradiction. This prove our claim from.11 and.11.

6 6 Advances in Difference Equations Now, we assume that x n 1 <x n for all n n 0. Then lim x n exists, which is denoted by x. We claim that x exp b C 1 α 1 1 /a. Suppose to the contrary that x > exp b C 1 α 1 1 /a. Taking limits in the first equation in System F, wesetthat ( 0 lim b n a n x n c 1 n b a x C 1 α 1 < 0, ) K 1 n s x 1 s.15 which is a contradiction. It follows that.11 holds. We show that lim inf x 1 n β According to the above assertion, there exists a k n 0 such that x 1 n α 1 ɛ and x n α ɛ, for all n k. We assume that there exists an l 0 k such that x 1 l 0 1 x 1 l 0.Note that for n l 0, x 1 n 1 x 1 n exp b 1 n a 1 n x 1 n c n x 1 n expb 1 C α A 1 x 1 n. K n s x s.17 In particular, with n l 0, we have b 1 A 1 x 1 l 0 C α 0,.18 which implies that x 1 l 0 b 1 C α A Then, x 1 l 0 1 b 1 C α A 1 exp b 1 C α A 1 α 1 ɛ : x 1ɛ..0 We assert that x 1 n x 1ɛ, n l By way of contradiction, we assume that there exists a p 0 l 0 such that x 1 p 0 <x 1ɛ. Then p 0 l 0. Let p 0 be the smallest integer such that x 1 p 0 <x 1ɛ.Then x 1 p 0 x 1 p 0 1. The above argument yields x 1 p 0 x 1ɛ, which is a contradiction. This proves our claim. We now assume that x 1 n 1 <x 1 n for all n n 0. Then lim x 1 n exists, which is denoted

7 Advances in Difference Equations 7 by x 1. We claim that x 1 b 1 C α /A 1. Suppose to the contrary that x 1 < b 1 C α /A 1. Taking the limits in the first equation in System F, wesetthat ( 0 lim b 1 n a 1 n x 1 n c n b 1 A 1 x 1 C α > 0, ) K n s x s.1 which is a contradiction. It follows that.16 holds, and then β 1 x 1 n for all n n 0 from.16 and.16. Finally, by using the inequality B <c 1 β 1, similar arguments lead to lim inf x β, and then x n β for all n k 0. This proof is complete. Lemma.. Let K be the closed bounded set in R such that K x 1,x R ; β i x i α i for each i 1,.. Then K is invariant for System F, that is, one can see that for any n 0 Z and any ϕ i such that ϕ i s K, s 0 i 1,, every solution of F through n 0,ϕ i remains in K for all n n 0 and i 1,. Proof. From Lemma.1,itissufficient to prove that this K / φ. To do this, by assumption of almost periodic functions, there exists a sequence n k,n k as k, such that b i n n k b i n,a i n n k a i n, and c i n n k c i n as k uniformly on Z and i 1,. Let x n be a solution of System F through n 0,ϕ that remains in K for all n n 0, whose existence was ensured by Lemma.1. Clearly, the sequence x n n k is uniformly bounded on bounded subset of Z. Therefore, we may assume that the sequence x n n k converges to a function y n y 1 n,y n as k uniformly on each bounded subset of Z taking a subset of x n n k if necessary. We may assume that n k n 0 for all k. For n 0, we have x 1 n n k 1 x 1 n n k exp b 1 n n k a 1 n n k x 1 n n k c n n k x n n k 1 x n n k exp b n n k a n n k x n n k c 1 n n k n n k n n k K n n k s x s, K 1 n n k s x 1 s. F nk Since x n n k K and y n K for all n Z, there exists r>0 such that x n n k r and y n r for all n Z. Then, by assumption of delay kernel K i,forthisr and any ɛ>0, there exists an integer S S ɛ, r > 0 such that n S K i n n k s x i s ɛ, n S K i n s y i s ɛ..3

8 8 Advances in Difference Equations Then, we have K i n n k s x i s n S ɛ K i n s y i s K i n n k s x i s s n S n S Ki n s y i s Ki n n k s x i s K i n s y i s s n S K i n n k s x i s K i n s y i s..4 Since x i n n k s converges to y i n s on discrete interval s n S, n as k, there exists an integer k 0 ɛ >k, for some k > 0, such that s n S K i n n k s x i s K i n s y i s ɛ i 1,.5 when k k 0 ɛ. Thus, we have K i n n k s x i s K i n s y i s.6 as k. Letting k in F nk, we have y 1 n 1 y 1 n exp b 1 n a 1 n y 1 n c n K n s y s, y n 1 y n exp b n a n y n c 1 n K 1 n s y 1 s,.7 for all n n 0. Then, y n y 1 n,y n is a solution of System F on Z. It is clear that y n K for all n Z. Thus,K / φ.

9 Advances in Difference Equations 9 We denote by Ω F the set of all limit functions G such that for some sequence n k such that n k as k, b i n n k b i n,a i n n k a i n, and c i n n k c i n uniformly on Z as k. Here, the equation for G is x 1 n 1 x 1 n exp b 1 n a 1 n x 1 n c n K n s x s, x n 1 x n exp b n a n x n c 1 n K 1 n s x 1 s. G Moreover, we denote by v, G Ω u, F when for the same sequence n k,u n n k v n uniformly on any compact subset in Z as k. Then a system G is called a limiting equation of F when G Ω F and v n is a solution of G when v, G Ω u, F. Lemma.3. If a compact set K in R of Lemma. is invariant for System F, thenk is invariant also for every limiting equation of System F. Proof. Let G be a limiting equation of system F. Since G Ω F, there exists a sequence n k such that n k as k and that b i n n k b i n,a i n n k a i n, and c i n n k c i n uniformly on Z as k.letn 0 0,φ BS such that φ s K for all s 0, and let y n be a solution of system G through n 0,φ. Letx k n be the solution of System F through n 0 n k,φ. Then x k n 0 n k s φ s K for all s 0andx k n is defined on n n 0 n k. Since K is invariant for System F, x k n K for all n n 0 n k.ifweset z k n x k n n k,k 1,,..., then z k n is defined on n n 0 and is a solution of x 1 n 1 x 1 n exp b 1 n n k a 1 n n k x 1 n c n n k n n k K n n k s x s, n n k x n 1 x n exp b n n k a n n k x n c 1 n n k K 1 n n k s x 1 s,.8 such that z k n 0 s xn k 0 n k s φ s K for all s 0. Since x k n K for all n n 0 n k, z k n K for all n n 0. Since the sequence z k n is uniformly bounded on n 0, and z k n 0 φ, z k n can be assumed to converge to the solution y n of G through n 0,φ uniformly on any compact set n 0,, because y n is the unique solution through n 0,φ and the same argument as in the proof of Lemma.. Therefore, y n K for all n n 0 since z k n K for all n n 0 and K is compact. This shows that K is invariant for limiting G.

10 10 Advances in Difference Equations Let K be the compact set in R m such that u n K for all n Z, where u n φ 0 n for n 0. For any θ, ψ BS, we set ρ ( θ, ψ ) ( ) ρ j θ, ψ [ ( ( ))], j.9 1 ρ j θ, ψ j 1 where ( ) ρ j θ, ψ sup θ s ψ s. j s 0.30 Clearly, ρ θ n,θ 0asif and only if θ n s θ s uniformly on any compact subset of, 0 as. In what follows, we need the following definitions of stability. Definition.4. The bounded solution u n of System F is said to be as follows: i K, ρ -totally stable in short, K, ρ -TS if for any ɛ > 0, there exists a δ ɛ > 0 such that if n 0 0, ρ x n0,u n0 <δ ɛ and h h 1,h BS n 0, which satisfies h n0, <δ ɛ, then ρ x n,u n <ɛfor all n n 0, where x n is a solution of x 1 n 1 x 1 n exp b 1 n a 1 n x 1 n c n K n s x s h 1 n, x n 1 x n exp b n a n x n c 1 n K 1 n s x 1 s h n, F h through n 0,φ such that x n0 s φ s K for all s 0. In the case where h n 0, this gives the definition of the K, ρ -US of u n ; ii K, ρ -attracting in Ω F in short, K, ρ -A in Ω F if there exists a δ 0 > 0 such that if n 0 0andany v, G Ω u, F, ρ x n0,v n0 <δ 0, then ρ x n,v n 0asn, where x n is a solution of limiting equation of.5 ; G through n 0,ψ such that x n0 s ψ s K for all s 0; iii K, ρ -weakly uniformly asymptotically stable in Ω F in short, K, ρ -WUAS in Ω F if it is K, ρ -US in Ω F, that is, if for any ε>0 there exists a δ ɛ > 0 such that if n 0 0andany v, G Ω u, F, ρ x n0,v n0 <δ ɛ, then ρ x n,v n <ɛfor all n n 0, where x n is a solution of G through n 0,ψ such that x n0 s ψ s K for all s 0, and K, ρ -A in Ω F.

11 Advances in Difference Equations 11 Proposition.5. Under the assumption (i), (i), and (iii), if the solution u n of System F is K, ρ - WUAS in Ω F, then the solution u n of System F is K, ρ -TS. Proof. Suppose that u n is not K, ρ -TS. Then there exist a small ɛ>0, sequences ɛ k, 0 < ɛ k <ɛand ɛ k 0ask, sequences s k, n k, h k, x k such that s k as k, 0 <s k 1 <n k, h k : Z R is bounded function satisfying h k n <ɛ k for n s k and such that ) ) ρ (u sk,x k sk <ɛ k, ρ (u nk,x k nk ɛ, ( ) ρ u n,xn k <ɛ s k,n k,.31 where x k n is a solution of x 1 n 1 x 1 n exp b 1 n a 1 n x 1 n c n K n s x s h k1 n, x n 1 x n exp b n a n x n c 1 n K 1 n s x 1 s h k n, F h k such that x k s k s K for all s 0. We can assume that ɛ<δ 0 where δ 0 is the number for K, ρ -A in Ω F of Definition.4. Moreover, by.31, we can chose sequence τ k such that s k <τ k <n k, δ ɛ/ ρ ( ) u τk,xτ k δ ɛ/ k,.3 ρ ( u n,x k n) ɛ for n τk,n k,.33 where δ is the number for K, ρ -US in Ω F. We may assume that u n τ k v n as k on each bounded subset of Z for a function v, and for the sequence τ k, τ k as k, taking a subsequence if necessary, there exists a v, G Ω u, F. Moreover, we may assume that x k n τ k z n as k uniformly on any bounded subset of Z for function z, since the sequence x k n τ k is uniformly bounded on Z. Because, if we set y k n x k n τ k, then y k n is defined on n n 0 τ k and y k n is a solution of x 1 n 1 x 1 n exp b 1 n τ k a 1 n τ k x 1 n c n τ k K n n k s x s h k1 n τ k,

12 1 Advances in Difference Equations x n 1 x n exp b n τ k a n τ k x n c 1 n τ k K 1 n n k s x 1 s h k n τ k,.34 such that y k 0 s xk τ k s K for all s 0. Then we may show that taking a subsequence if necessary, y k n converges to a solution z n of G such that z 0 s K for s 0, by the same argument for Σ-calculations with condition of K i as in the proof of Lemma.. Then, the same argument as in the proof of Lemma. shows that z K. Now, suppose that n k τ k as k. Letting k in.33, we have δ ɛ/ / ρ v n,z n ɛ on n 0. Since ɛ < δ 0 and u n is K, ρ -A in Ω F, we have δ ɛ/ / ρ v n,z n 0as, which is a contradiction. Thus n k τ k as k. Taking a subsequence again if necessary, we can assum that n k τ k r< as k. Letting k in.3, we have ρ v 0,z 0 δ ɛ/ / <δ ɛ/ and hence ρ v n,z n <ɛ/for all n 0, because u is K, ρ -US in Ω F. On the other hand, from.31, we have ρ v n,z n ɛ, which is a contradiction. This shows that u n is K, ρ -TS. Now we will see that the existence of a strictly positive almost periodic solution of System F can be obtained under conditions i, ii, and iii. Theorem.6. one assumes conditions i, ii, and iii. Then System F has a unique almost periodic solution p n in compact set K. Proof. For System F, we first introduce the change of variables: u i n expv i n, x i n exp y i n, i 1,..35 Then, System F can be written as y 1 n 1 y 1 n b 1 n a 1 n exp y 1 n c n K n s exp y s, y n 1 y n b n a n exp y n c 1 n K 1 n s exp y 1 s, F We now consider Liapunov functional: V ( v n,y n ) vi n y i n K i s i 1 n 1 l n s c i s l expvi l exp y i l,.36

13 Advances in Difference Equations 13 where y n and v n are solutions of F which remains in K. Calculating the differences, we have ΔV ( v n,y n ) v 1 n 1 v 1 n y1 n 1 y 1 n K 1 s ( c 1 s n expv1 n exp y 1 n c1 n expv1 n s exp y 1 n s ) v n 1 v n y n 1 y n K s ( c s n expv n exp y n c n expv n s exp y n s ) b 1 n a 1 n expv 1 n c n K n s expv s b 1 n a 1 n exp y 1 n c n K n s exp y s K 1 s ( c 1 s n expv1 n exp y 1 n c 1 n expv1 n s exp y 1 n s ) b n a n expv n c 1 n K 1 n s expv 1 s b n a n exp y n c 1 n K 1 n s exp y 1 s K s ( c s n expv n exp y n c n expv n s exp y n s ) b1 n a 1 n expv 1 n c n expv n s b1 n a 1 n exp y 1 n c n exp y n s c 1 s n expv1 n exp y 1 n c1 n expv1 n s exp y 1 n s b n a n expv n c 1 n expv 1 n s b n a n exp y n c 1 n exp y 1 n s c s n expv n exp y n c n expv n s exp y n s a 1 n expv1 n exp y 1 n c1 s n expv1 n exp y 1 n a n expv n exp y n c s n expv n exp y n..37 From the mean value theorem, we have expvi n exp y i n expθi n vi n y i n, i 1,,.38

14 14 Advances in Difference Equations where θ i n lies between v i n and y i n i 1,. Then, by iii, we have ΔV ( v n,y n ) md v i n y i n, i 1.39 where set D maxexpβ 1, expβ, and let solutions x i n of System F be such that x i n β i for n n 0 i 1,.Thus i 1 v i n y i n 0as, and hence ρ v n,y n 0as. Moreover, we can show that v n is K, ρ -US in Ω of F, by the same argument as in 5. By using similar Liapunov functional to.36, we can show that v n is K, ρ -A in Ω of F. Therefore, v n is K, ρ -WUAS in Ω F. Thus,fromProposition.5, v n is K, ρ - TS, because K is invariant. By the equivalence between F and F, solution u n of System F is K, ρ -TS. Therefore, it follows from Theorem 4.4 in 5 and 9 that System F has an almost periodic solution p n such that β i p i n α i, i 1,, for all n Z. 3. Competitive System We will consider the l-species almost periodic competitive Lotka-Volterra system: x i n 1 x i n exp b i n a ii x i n a ij n K ij n s x j s, j 1,j / i i 1,,...,l, H where b i n,a ij n are positive almost periodic sequences on Z; a ij n are strictly positive, and, moreover, a ij inf a ij n, A ij supa ij n, b i inf b i n, B i supb i n, n Z n Z n Z n Z K ij : Z 0, R ( ) 3.1 i, j 1,,...,l, which can be seen as the discretization of the differential equation in 3.Weset β i min α i exp B i 1 a ii, exp b i A ii α i ( l A j 1,j / i ijα j b i ) l A j 1,j / i ijα j, A ii b i l j 1,j / i α j A ii. 3. Now, we make the following assumptions: iv K ij s 0, and K ij s 1, sk ij s < i 1,,...,l ; v b i > j 1,j / i A ijα j for i 1,,...,l;

15 Advances in Difference Equations 15 vi there exists a positive constant m such that a ii > A ij m i 1,,...,l. 3.3 j 1,j / i Then, we have 0 <β i <α i for each i 1,,...,l. Under the assumptions iv and v,it follows that for any n 0,φ Z BS, there is a unique solution u n u 1 n,u n,...,u l n of H through n 0,φ, if it remains bounded. Then, we can show the similar lemmas to Lemma.1. Lemma 3.1. If x n x 1 n,x n,...,x l n is a solution of H through n 0,φ such that β i φ s α i i 1,,...,l for all s 0, then one has β i x i n α i i 1,,...,l for all n n 0. Proof. First, we claim that lim sup x i n B i, i 1,,...,l. 3.4 Clearly, x i n > 0forn n 0. To prove this, we first assume that there exists an l 0 n 0 such that x i l 0 1 x i l 0. Then, it follows from the first equation of H that b i l 0 a ii l 0 x i l 0 a ij l 0 l 0 j 1,j / i K ij l 0 s x j s Hence b i l 0 l x i l 0 j 1,j / i a ij l 0 l 0 K ij l 0 s x j s a ij l 0 b i l 0 a ii l 0 B i a ii. 3.6 It follows that x i l 0 1 x i l 0 exp b i l 0 a ii l 0 x i l 0 j 1,j / i a ij l 0 x i l 0 expb i a ii x i l 0 expb i 1 a ii : α i. l 0 K ij l 0 s x j s 3.7 Now we claim that x i n B i, for n l By way of contradiction, we assume that there exists a p 0 >l 0 such that x i p 0 >α i. Then, p 0 l 0. Let p 0 l 0 be the smallest integer such that x i p 0 >α i. Then, x i p 0 1 x i p 0.The

16 16 Advances in Difference Equations above argument shows that x i p 0 α i, which is a contradiction. This proves our assertion. We now assume that x i n 1 <x i n for all n n 0. Then lim x i n exists, which is denoted by x i. We claim that x i exp B i 1 /a ii. Suppose to the contrary that x i > exp B i 1 /a ii. Taking limits in the first equation in System H,wesetthat 0 lim b i n a ii n x i n j 1,j / i a ij n lim b i n a ii n x i n B i a ii x i < 0, which is a contradiction. It follows that 3.4 holds. We first show that K ij n s x j s 3.9 lim inf x i n β i According to above assertion, there exists a k n 0 such that x i n α i ε, for all n k.we assume that there exists an l 0 k such that x i l 0 1 x i l 0. Note that for n l 0, x i n 1 x i n exp b i n a ii n x i n a ij n j 1,j / i x i n exp b i A ij α j A ii x i n. j 1,j / i K ij n s x j s 3.11 In particular, with n l 0, we have b i A ii x i l 0 A ij α j 0, j 1,j / i 3.1 which implies that x i l 0 b i l j 1,j / i A ijα j A ii Then, x i l 0 1 b i l j 1,j / i A ijα j A ii exp b i j 1,j / i A ij α j A ii α i ɛ : x iɛ We assert that x i n x iɛ, n l

17 Advances in Difference Equations 17 By way of contradiction, we assume that there exists a p 0 l 0 such that x i p 0 <x iε. Then p 0 l 0. Let p 0 be the smallest integer such that x i p 0 <x iε.then x i p 0 x i p 0 1. The above argument yields x i p 0 x iε, which is a contradiction. This proves our claim. We now assume that x i n 1 <x i n for all n n 0. Then lim x i n exists, which is denoted by x i. We claim that x i b i l j 1,j / i A ijα j /A ii. Suppose to the contrary that x i < b i l j 1,j / i A ijα j /A ii. Taking limits in the first equation in System F, wesetthat 0 lim b i n a ii n x i n b i A ii x i A ij α j > 0, j 1,j / i a ij n K ij n s x j s j 1,j / i 3.16 which is a contradiction. It follows that 3.10 holds. This proof is complete. By the same arguments of Lemmas.,.3 and Proposition.5, we obtain Lemmas 3., 3.3 and Proposition 3.4. So, we will omit to these proofs. Lemma 3.. Let K be the closed bounded set in R l such that K x 1,x,...,x l R l ; β i x i α i for each i 1,,...,l Then K is invariant for System H, that is, one can see that for any n 0 Z and any ϕ such that ϕ s K, s 0, every solution of H through n 0,ϕ remains in K for all n n 0 and i 1,,...,l. Lemma 3.3. If a compact set K in R l of Lemma 3. is invariant for System H, thenk is invariant also for every limiting equation of System H. Proposition 3.4. Under the assumption (iv), (v), and (vi), if the solution u n of System H is K, ρ -WUAS in Ω H, then the solution u n of System H is K, ρ -TS. By making changes of the variables x i n expy i n and defining the Liapunov functional V by V ( v n,y n ) vi n y i n K ij s i 1 n 1 l n s c i s l expvi l exp y i l, 3.18 where y n and v n are solutions of changing equation Ĥ for H which remains in K, the arguments similar to the Theorem.6 lead to the following results. Theorem 3.5. one assumes conditions (iv), (v), and (vi). Then System H has a unique almost periodic solution p n in compact set K. From Theorem 3.5, one obtains the following result, which was proved by Gopalsamy in [10] when System H is continuous case.

18 18 Advances in Difference Equations Corollary 3.6. Under the assumption (iv), (v), and (vi), suppose that b i n and a ij n are positive ω- periodic sequences ω Z for all i, j 1,,...,l. Then System (H) has a unique ω-periodic solution in K. 4. Examples For simplicity, we consider the following prey-predator system with finite delay: x 1 n 1 x 1 n exp 1.5 sin n 1 sin n ( ) 1 s 1 0.5x 1 n x n s, sinn x n 1 x n exp 0.5x n 1 ( ) 1 s 1 x 1 n s E 1 Then, we have b 1 1, B 1, a 1 A 1 0.5, c 1 C 1 1 ( ) 1 s 1 4, K 1 s, b 0.016, B 0.04, a A 0.5, c 0, C 1 ( ) 1 s 1 8, K s 4.1 for System F. Thus, α , α.818, β It is easy to verify that System E 1 satisfies all the assumptions in our Theorem.6. Thus, System E 1 has an almost periodic solution. We next consider the following competitive system with finite delay: x 1 n 1 x 1 n exp sin n x 1 n 1 ( ) 1 s 1 x n s, 16 x n 1 x n exp cos n x n 1 ( ) 1 s 1 x 1 n s. 4 E Then, we have b 1 1.5, B 1.5, a 11 A 11 1, a 1 A 1 1 ( ) 1 s 1 8, K 1 s, b 1, B 3, a A 1, a 1 A 1 1 ( ) 1 s , K 1 s

19 Advances in Difference Equations 19 for System E.Thus, α , α It is easy to verify that System E satisfies all the assumptions in Theorem 3.5. Thus, System E has an almost periodic solution. Acknowledgment The authors would like to express their gratitude to the referees for their many helpful comments. References 1 S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 005. Y. Hamaya, Periodic solutions of nonlinear integro-differential equations, The Tohoku Mathematical Journal, vol. 41, no. 1, pp , Y. Murakami, Almost periodic solutions of a system of integrodifferential equations, The Tohoku Mathematical Journal, vol. 39, no. 1, pp , T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol. 14 of Applied Mathematical Sciences, Springer, New York, NY, USA, Y. Song and H. Tian, Periodic and almost periodic solutions of nonlinear Volterra difference equations with unbounded delay, Journal of Computational and Applied Mathematics, vol. 05, no., pp , Y. Xia and S. S. Cheng, Quasi-uniformly asymptotic stability and existence of almost periodic solutions of difference equations with applications in population dynamic systems, Journal of Difference Equations and Applications, vol. 14, no. 1, pp , Y. Hamaya, Total stability property in limiting equations of integrodifferential equations, Funkcialaj Ekvacioj, vol. 33, no., pp , C. Corduneanu, Almost periodic discrete processes, Libertas Mathematica, vol., pp , Y. Hamaya, Existence of an almost periodic solution in a difference equation with infinite delay, Journal of Difference Equations and Applications, vol. 9, no., pp. 7 37, K. Gopalsamy, Global asymptotic stability in a periodic integro-differential system, The Tohoku Mathematical Journal, vol. 37, no. 3, pp , 1985.