Báo cáo hóa học: "Research Article Periodic and Almost Periodic Solutions of Functional Difference Equations with Finite Delay" pdf

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 propertie

fference 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

under certain conditions forF(n, ·) (see below), wheren, j, and τ are integers, and x nwill denote the functionx(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 =Ᏺt,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

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 InSection 2, we review definitions of almost pe-riodic and asymptotically almost pepe-riodic 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) InSection 4, we discuss the existence of almost periodic solutions of (1.1)

2 Preliminaries

We formalize our notation Denote byZ,Z +,Z−, respectively, the set of integers, the set

of nonnegative integers, and the set of nonpositive integers For anya ∈ Z, letZ +

a = { n :

n ≥ a, n ∈ Z} For any integersa < b, let dis[a,b] = { j : a ≤ j ≤ b, j ∈ Z}and dis(a,b]= { j : a < j ≤ b, j ∈ Z}be discrete intervals of integers LetEd denote eitherRd, the

d-dimensional real Euclidean space, orCd, thed-dimensional complex Euclidean space In

the following, we use| · |to denote a norm of a vector inEd

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]

LetX and Y be two Banach spaces with the norm  ·  Xand ·  Y, respectively LetΩ

be a subset ofX.

Definition 2.1 Let f : Z ×Ω→ Y and f (n, ·) be continuous for eachn ∈ Z Then f is said

to be almost periodic inn ∈ Zuniformly forw ∈ Ω if for every ε > 0 and every compact

Σ⊂ Ω corresponds an integer N ε(Σ) > 0 such that among Nε(Σ) consecutive integers there

is one, call itp, such that

f (n + p,w) − f (n,w)

Y < ε ∀ n ∈ Z,w ∈ Σ. (2.1)

Denote byᏭᏼ(Z × Ω : Y) the set of all such functions We may call f ∈Ꮽᏼ(Z × Ω : Y)

a (uniformly) almost periodic sequence inY If Ω is an empty 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, orf :Z +

a →

X by requiring that any N ε(Σ) consecutive integers is inZ +

a For uniformly almost periodic sequences, we have the following results

Theorem 2.2 Let f ∈Ꮽᏼ(Z × Ω : Y) and let 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.

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 → ∞ , and a function ξ : Z ×Ω→ Y such that

f

n + α k,w

uniformly on Z × Σ as k → ∞ , where Σ is any compact set in Ω Moreover, ξ ∈Ꮽᏼ(Z ×Ω :

Y), that is, ξ(n,w) is also almost periodic in n uniformly for w ∈ Ω.

IfΩ is the empty set and Y = X inTheorem 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

uniformly on Z × Σ as k → ∞ , where Σ is any compact set in Ω.

Obviously,{ α k }inTheorem 2.4can be chosen to be a positive integer sequence

Definition 2.5 A sequence { x(n) } n ∈Z+, x(n) ∈ X, or a function x :Z +→ X, is called

asymptotically almost periodic ifx = x1|Z ++x2, wherex1∈Ꮽᏼ(Z,X) and x2:Z +→ X

satisfying x2(n) X →0 asn → ∞ Denote byᏭᏭᏼ(Z +,X) all such sequences

Theorem 2.6 Let x :Z +→ X Then 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 → ∞ , there is a subsequence

{ β k } ⊂ { α k } such that β k → ∞ as k → ∞ and { x(n + β k)} converges uniformly onZ + 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 functionφ : R ×Ω→ X (see also [2] for the case thatX = 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 integerτ > 0, we

defineC to be a Banach space with a norm  · by

C =φ | φ : dis[ − τ,0] −→ E d, φ  =maxφ( j)forj ∈dis[− τ,0]

It is clear thatC is isometric to the spaceEd ×(τ+1)

Letn0∈ Z+and let{ x(n) },n ≥ n0− τ, be a sequence with x(n) ∈ E d For eachn ≥ n0,

we definex n: dis[− τ,0] → E dby the relation

x n(j) = x(n + j), j ∈dis[− τ,0]. (2.5)

Let us return to system (1.1), that is,

x(n + 1) = F

n,x n



whereF : Z × C → E dandx n: dis[− τ,0] → C.

Definition 2.7 Let n0∈ Z+and letφ be a given vector in C A sequence x = { x(n) } n ≥ n0in

Edis said to be a solution of (2.6), passing through (n0,φ), if xn0= φ, that is, x(n0+j) =

φ( j) for j ∈dis[− τ,0], x(n + 1), and x nsatisfy (2.6) forn ≥ n0, wherex n is defined by (2.5) Denote by { x(n,φ) } n ≥ n0 a solution of (2.6) such thatx n0= φ No loss of clarity

arises if we refer to the solution{ x(n,φ) } n ≥ n0asx = { x(n) } n ≥ n0

We make the following assumptions on (2.6) throughout this paper

(H1)F : Z × C → E dandF(n, ·) is continuous onC for each n ∈ Z

(H2) System (2.6) has a bounded solutionu = { u(n) } n ≥0, passing through (0,φ0),φ0∈

C.

For this bounded solution{ u(n) } n ≥0, there is anα > 0 such that | 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 alln ≥0

Definition 2.8 A bounded solutionx= {x(n)} n ≥0of (2.6) is said to be

(i) uniformly stable, abbreviated to read “xisᐁ᏿,” if for any ε > 0 and any integer

n0≥0, there existsδ(ε) > 0 such that xn0− x n0 < δ(ε) implies that xn − x n  < ε

for alln ≥ n0, where{ x(n) } n ≥ n0is any solution of (2.6);

(ii) uniformly asymptotically stable, abbreviated to read “x isᐁᏭ᏿,” if it is uni-formly stable and there exists δ0> 0 such that for any ε > 0, there is a positive

integerN = N(ε) > 0 such that if n0≥0 andxn0− x n0 < δ0, thenxn − x n  < ε

for alln ≥ n0+N, where { x(n) } n ≥ n0is any solution of (2.6);

(iii) globally uniformly asymptotically stable, abbreviated to read “x isᏳᐁᏭ᏿,” if

it is uniformly stable andxn − x n  →0 asn → ∞, whenever{ x(n) } n ≥ n0 is any solution of (2.6)

Remark 2.9 It is easy to see that an equivalent definition forx= {x(n)} n ≥0beingᐁᏭ᏿ is the following:

(ii∗)x= {x(n)} n ≥0isᐁᏭ᏿, if it is uniformly stable and there exists δ0> 0 such that

ifn0≥0 andxn0− x n0 < δ0, thenxn − x n  →0 asn → ∞, where{ x(n) } n ≥ n0 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



under a periodic condition (H3) as follows

(H3) TheF(n, ·) in (3.1) is periodic inn ∈ Z, that is, there exists a positive integerω

such thatF(n + ω,v) = F(n,v) for all n ∈ Zandv ∈ C.

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 ≥0of (3.1) is uniformly stable, then{ u(n) } n ≥0is an asymp-totically almost periodic sequence

Theorem 3.1 Suppose conditions (H1)–(H3) hold If the bounded solution { u(n) } n ≥0of ( 3.1 ) is ᐁ᏿, then { u(n) } n ≥0 is an asymptotically almost periodic sequence inEd , 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 αfor alln ≥0 Let{ n k } k ≥1be any integer sequence such thatn k > 0 and n k → ∞as

k → ∞ For eachn k, there exists a nonnegative integerl k such thatl k ω ≤ n k ≤(lk+ 1)ω Setn k = l k ω + τ k Then 0≤ τ k < ω for all k ≥1 Since{ τ k } k ≥1 is bounded set, we can assume that, taking a subsequence if necessary,τ k = j∗for allk ≥1, where 0≤ j∗ < ω.

Now, setu k(n)= u(n + n k) Notice thatu 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,un+n k



= F

n + n k,uk n

= F

n + j∗,uk n

, (3.2) which implies that{ u k(n)}is a solution of the system

x(n + 1) = F

n + j∗,xn

(3.3)

through (0,u n k) It is readily shown that if{ u(n) } n ≥0isᐁ᏿, then{ u k(n)} n ≥0is alsoᐁ᏿ with the same pair (ε,δ(ε)) as the one for{ u(n) } n ≥0

Since{ u(n + n k)}is bounded for alln ≥ − τ and n k, we can use the diagonal method to get a subsequence{ n k j }of{ n k }such thatu(n + n k j) converges for eachn ≥ − τ as j → ∞ Thus, we can assume that the sequenceu(n + n k) converges for eachn ≥ − τ as k → ∞ Notice thatu k

0(j) = u k(0 +j) = u( j + n k) Then for anyε > 0 there exists a positive integer

N1(ε) such that if k,m≥ N1(ε), then

u k

0− u m

whereδ(ε) is the number for the uniform stability of { u(n) } n ≥0 Notice that{ u m(n)=

u(n + n m)} n ≥0is also a solution of (3.3) and that{ u k(n)} n ≥0is uniformly stable It follows fromDefinition 2.8and (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 ≥ N1(ε). (3.6)

This implies that for any positive integer sequencen k,n k → ∞ ask → ∞, there exists a subsequence{ n k j }of{ n k }for which{ u(n + n k j)}converges uniformly onZ +as j → ∞ Thus,{ u(n) } n ≥0 is an asymptotically almost periodic sequence byTheorem 2.6and the

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 ≥1be 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 is any compact set in C Then { η(n) } n ≥0is a solution of the system

x(n + 1) = G

n,x n

and is ᐁ᏿ Moreover, if { u(n) } n ≥0is ᐁᏭ᏿, then { η(n) } n ≥0is also ᐁᏭ᏿.

Proof Since u k(n)= u(n + n k) is uniformly bounded forn ≥ − τ and k ≥1, we can as-sume that, taking a subsequence if necessary,u(n + n k) also converges for eachn ∈dis [− τ, −1] Defineη( j) =limk →∞ u( j + n k) forj ∈dis[− τ, −1] Thenu(n + n k)→ η(n) for

eachn ∈dis[− τ, ∞), and hence,u k

n → η nask → ∞ for eachn ≥0 Notice thatu k

n ∈ S α

for alln ≥0,k ≥1, andη n ∈ S αforn ≥0 It follows fromTheorem 2.4that there exists

a subsequence{ n k j }of { n k },n k j → ∞as j → ∞, such that F(n + n k j,v)→ G(n,v)

uni-formly onZ × S αasj → ∞andG(n, ·) is continuous onS αuniformly for alln ∈ Z Since

u(n + n k j+ 1)= F(n + n k j,uk n j) and

F

n + n k j,uk 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

−→0 as j −→ ∞,

(3.8)

we haveη(n + 1) = G(n,η n) (n≥0) This shows that{ η(n) } n ≥0is a solution of (3.7)

To prove that{ η(n) } n ≥0isᐁ᏿, we set n k = l k ω + j∗as before, where 0≤ j∗ < ω Then

u k j(n)= u(n + n k j)→ η(n) for each n ∈ Z+asj → ∞ SinceF(n + n k j,v)= F(n + j∗,v)→

G(n,v) as j → ∞, we haveG(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 For anyn0∈ Z+, let{ x(n) } n ≥0be a solution of (3.7) such that η n0− x n0 = μ < δ(ε) Since u k j

n → η nas j → ∞for eachn ≥0, there is a positive integerJ1> 0 such that if j ≥ J1, then

u k j

n0− η n0 < δ(ε) − μ. (3.9)

Thus, for j ≥ J1, we have

u n

0 +j∗+l k j ω − x n0 ≤  u n0 +j∗+l k j ω − η n0 +η n

0− x n0 < 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 ≥ n0. (3.11)

Since { η(n) }is also a solution of (3.7) and u k j

n → η n for each n ≥0 as j → ∞, for an arbitraryν > 0, there is J2> 0 such that if j ≥ J2, then

η n − u n+j∗+l ω< δ(ν), (3.12)

and hence, η n − u n+ j∗+l k j ω  < ν for all n ≥ n0, where (ν,δ(ν)) is a pair for the uniform stability ofu(n + j∗+l k j ω) This shows that if j ≥max(J1,J2), 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 alln ≥ n0, which implies that η n − x n  ≤ ε for all n ≥ n0if η n0− x n0 < δ(ε) because

ν is arbitrary This proves that { η(n) } n ≥0is uniformly stable

To prove that{ η(n) } n ≥0isᐁᏭ᏿, we use definition (ii∗) inRemark 2.9 Let{ x(n) }be

a solution of (3.7) such that η n0− x n0 < δ0, whereδ0is the number for the uniformly asymptotic stability of{ u(n) } Notice thatu(n + j∗+l k j ω) = u k j(n) is a uniformly asymp-totically stable solution of (3.7) withG(n,φ) = F(n + j∗,φ) and with the same δ0as the one for{ u(n) } Set η n0− x n0 = μ < δ0 Again, for sufficient large j, we have the simi-lar relations (3.10) and (3.12) with u n0 +j∗+l k j ω − x n0 < δ0 and u n0 +j∗+l k j ω − η n0 < δ0 Thus,

η n − x n  ≤  η n − u n+ j∗+l k j ω+u n+ j∗+l

as n → ∞ if u n0− x n0 < δ0, because { u k j(n)},{ x(n) }, and{ η(n) } satisfy (3.7) with

Using Theorem 3.1andLemma 3.2, we can show that (3.1) has an almost periodic solution

Theorem 3.3 If the bounded solution { u(n) } n ≥0of ( 3.1 ) is ᐁ᏿, then system ( 3.1 ) has an almost periodic solution, which is also ᐁ᏿.

Proof It follows fromTheorem 3.1that{ u(n) } n ≥0is asymptotically almost periodic Set

u(n) = p(n) + q(n) (n ≥0), where{ p(n) } n ≥0 is almost periodic sequence andq(n) →0

asn → ∞ For positive integer sequence{ n k ω }, there is a subsequence{ n k j ω }of{ n k ω }

such thatp(n + n k j ω) → p ∗(n) uniformly onZasj → ∞and{ p ∗(n)}is almost periodic Thenu(n + n k j ω) → p ∗(n) uniformly for n≥ − τ, and hence, u n+n k j ω → p ∗ n for alln ≥0 as

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 → ∞, we have p ∗(n + 1)= F(n, p ∗ n) for n ≥0, that is, system (3.1) has an almost periodic solution, which is alsoᐁ᏿ byLemma 3.2 

Now, we show that if the bounded solution{ u(n) }is uniformly asymptotically stable, then (3.1) has a periodic solution of periodmω for some positive integer m.

Theorem 3.4 If the bounded solution { u(n) } n ≥0of ( 3.1 ) is ᐁᏭ᏿, then system ( 3.1 ) has a periodic solution of period mω for some positive integer m, which is also ᐁᏭ᏿.

Proof Set u k(n)= u(n + kω), k =1, 2, By the proof ofTheorem 3.1, there is a subse-quence{ u k j(n)}converges to a solution{ η(n) }of (3.3) for eachn ≥ − τ and hence, u k0j →

η0asj → ∞ Thus, there is a positive integerp such that  u k p

0 − u k p+1

0  < δ0(0≤ k p < k p+1), whereδ0 is the one for uniformly asymptotic stability of{ u(n) } n ≥0 Letm = k p+1 − k p

and notice thatu m(n)= u(n + mω) is a solution of (3.1) Sinceu m k p ω(j)= u m(kp 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 < δ0, (3.16)

and hence,

u m

because{ u(n) } n ≥0isᐁᏭ᏿ (see alsoRemark 2.9) On the other hand,{ u(n) } n ≥0is asymp-totically almost periodic byTheorem 3.1, then

where{ p(n) } n ∈Zis almost periodic andq(n) →0 asn → ∞ It follows from (3.17) and (3.18) that

p(n) − p(n + mω)  −→0 asn −→ ∞, (3.19) which implies thatp(n) = p(n + mω) for all n ∈ Zbecause{ p(n) }is almost periodic For integer sequence{ kmω },k =1, 2, , we have u(n + kmω)= p(n) + q(n + kmω).

Thenu(n + kmω) → p(n) uniformly for all n ≥ − τ as k → ∞, and hence, u n+kmω → p n

forn ≥0 ask → ∞ Since u(n + kmω + 1) = F(n,u n+kmω), we have p(n + 1) = F(n, p n) forn ≥0, which implies that (3.1) has a periodic solution{ p(n) } n ≥0of periodmω The

uniformly asymptotic stability of{ p(n) } n ≥0follows fromLemma 3.2 

Finally, we show that if the bounded solution{ u(n) }isᏳᐁᏭ᏿, then (3.1) has a peri-odic solution of periodω.

Theorem 3.5 If the bounded solution { u(n) } n ≥0of ( 3.1 ) is ᏳᐁᏭ᏿, then system ( 3.1 ) has

a periodic solution of period ω.

Proof ByTheorem 3.1,{ u(n) } n ≥0is asymptotically almost periodic Thenu(n) = p(n) + q(n) (n ≥0), where{ p(n) }(n∈ Z) is an almost periodic sequence andq(n) →0 asn → ∞ Notice thatu(n + ω) is also a solution of (3.1) satisfyingu ω ∈ S α Since{ u(n) }isᏳᐁᏭ᏿,

we have u n − u n+ω  →0 asn → ∞, which implies thatp(n) = p(n + ω) for all n ∈ Z By the same technique in the proof ofTheorem 3.4, we can show that{ p(n) }is anω-periodic

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

under the condition (H4) as follows

(H4)F ∈Ꮽᏼ(Z × C :Ed), that is, F(n,v) is almost periodic in n ∈ Z uniformly for

v ∈ C.

ByH(F) we denote the uniform closure of F(n,v), that is, G ∈ H(F) if there is an integer

sequence{ α k }such thatα k → ∞andF(n + α k,v)→ G(n,v) uniformly on Z × Σ as k → ∞, whereΣ is any compact set in C Note that H(F) ⊂Ꮽᏼ(Z × C :Ed) byTheorem 2.3and

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 ≥0of ( 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 2.6that it has the decompositionu(n) = p(n) + q(n) (n ≥0), where{ p(n) } n ∈Z

is almost periodic andq(n) →0 asn → ∞ Notice that{ u(n) }is bounded There is com-pact setS α ∈ C such that u n ∈ S αand p n ∈ S αfor alln ≥0 For anyG ∈ H(F), there is

an integer sequence{ n k },n k > 0, such that n k → ∞ask → ∞andF(n + n k,v)→ G(n,v)

uniformly onZ × S αas k → ∞ Taking a subsequence if necessary, we can also assume thatp(n + n k)→ p ∗(n) uniformly onZand{ p ∗(n)}is also an almost periodic sequence For any j ∈dis[− τ,0], there is positive integer k0such that ifk > k0, thenj + n k ≥0 for anyj ∈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 inC uniformly for n ∈ Z+ask → ∞ Since

u

n + n k+ 1

= F

n + n k,un+n k



=F

n + n k,un+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) tends to zero ask → ∞byTheorem 2.2 and

F(n + n k,p ∗ n)− G(n, p ∗ n)→0 ask → ∞, we have p ∗(n + 1)= G(n, p ∗ n) for all n ∈ Z+, which implies that (4.2) has an almost periodic solution{ p ∗(n)} n ≥0, passing through (0,p ∗0), wherep ∗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 eachG ∈ 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 solution

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 → ∞ , where Σ is any compact subset in C and

G ∈ H(F) If the bounded solution { u(n) } n ≥0is ᐁ᏿, then the solution { η(n) } n ≥0of ( 4.2 ), through (0, ψ), is ᐁ᏿ In addition, if { u(n) } n ≥0is ᐁᏭ᏿, then { η(n) } n ≥0is also ᐁᏭ᏿ Proof Set u k(n)= u(n + n k) It is easy to see thatu k(n) is a solution of

x(n + 1) = F

n + n k,xn

passing though (0,un k) andu k

n ∈ S αfor allk, where S αis compact subset ofC such that

 u n  < α for all n ≥0 Since { u(n) } n ≥0 isᐁ᏿,{ u k(n)}is alsoᐁ᏿ 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 ≥1converges to a vectorη(n) for all n ≥0 ask → ∞ From (4.3) withp n ∗ = η n, we can see that{ η(n) } n ≥0is the unique solution of (4.2), through (0,ψ)

To show that the solution{ η(n) } n ≥0 of (4.2) isᐁ᏿, we need to prove that if for any

ε > 0 and any integer n0≥0, there existsδ ∗(ε) > 0 such that η n0− y n0 < δ ∗(ε) implies that η n − y n  < ε for all n ≥ n0, where{ y(n) } n ≥ n0is a solution of (4.2) passing through (n0,φ) with yn0= φ ∈ C.

For any givenn0∈ Z+, ifk is sufficiently large, say k ≥ k0> 0, we have

u k

n0− η n0 <1

2δ ε

2

whereδ(ε) is the one for uniform stability of { u(n) } n ≥0 Letφ ∈ C such that

φ − η n

0 <1

2δ ε

2

(4.6)

and let{ x(n) } n ≥ n0be the solution of (4.1) such thatx n0 +n k = φ Then { x k(n)= x(n + n k)}

is a solution of (4.4) withx k

n0= φ Since { u k(n)}isᐁ᏿ and x k

n0− u k

n0 < δ(ε/2) for k ≥ k0,

we have

u k

n − x k

n< ε

2 ∀ n ≥ n0,k ≥ k0. (4.7)

It follows from (4.7) that

x k

n  ≤  u k

n+ε

2 < α + ε

2 ∀ n ≥ n0,k ≥ k0. (4.8) Then there exists a number α ∗ > 0 such that x k

n ∈ S α ∗ for alln ≥0 andk ≥ k0, which implies that there is subsequence of{ x k(n)} k ≥ k0for eachn ≥ n0− τ, denoted by { x k(n)}

again, such thatx k(n)→ y(n) for each n ≥ n0− τ, and hence, x k

n → y nfor alln ≥ n0ask →

∞ Clearly,y n0= φ and the set S α ∗is compact set inC Since F(n,v) is almost periodic in n

uniformly forv ∈ C, we can assume that, taking a subsequence if necessary, F(n + n k,v)→

G(n,v) uniformly on Z × S α ∗ ask → ∞ Takingk → ∞in x k(n + 1)= F(n + n n k,xk

n), we havey(n + 1) = G(n, y n), namely,{ y(n) }is the unique solution of (4.2), passing through (n0,φ) with yn0= φ ∈ C On the other hand, for any integer N > 0, there exists k N ≥ k0

such that ifk ≥ k N, then

x k

n − y n< ε

4, u k

n − η n< ε

4 forn0≤ n ≤ n0+N. (4.9) From (4.7) and (4.9), we obtain

η n − y n< ε for n0≤ n ≤ n0+N. (4.10)

SinceN is arbitrary, we have  η n − y n  < ε for all n ≥ n0 if φ − η n0 < [δ(ε/2)]/2 and

φ ∈ C, which implies that the solution { η(n) } n ≥0of (4.2) isᐁ᏿