Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces

doi:10.1155/2009/380568

Research Article

Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces

Claudio Vidal

Departamento de Matem´atica, Facultad de Ciencias, Universidad del B´ıo B´ıo, Casilla 5-C, Concepci´on, Chile

Correspondence should be addressed to Claudio Vidal,[email protected] Received 20 November 2008; Revised 23 March 2009; Accepted 10 June 2009 Recommended by Donal O’Regan

The existence of periodic, almost periodic, and asymptotically almost periodic of periodic and almost periodic of abstract retarded functional difference equations in phase spaces is obtained by using stability properties of a bounded solution.

Copyrightq2009 Claudio Vidal. 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

In this paper, we study the existence of periodic, almost periodic, and asymptotic almost periodic solutions of the following functional difference equations with infinite delay:

xn1 Fn, xn, n≥n₀≥0, 1.1

assuming that this system possesses a bounded solution with some property of stability. In 1.1F :Nn0× B → C^r, andBdenotes an abstract phase space which we will define later.

The abstract space was introduced by Hale and Kato1to study qualitative theory of functional differential equations with unbounded delay. There exists a lot of literature devoted to this subject; we refer the reader to Corduneanu and Lakshmikantham2, Hino et al.3. The theory of abstract retarded functional difference equations in phase space has attracted the attention of several authors in recent years. We only mention here Murakami 4,5, Elaydi et al.6, Cuevas and Pinto7,8, Cuevas and Vidal9, and Cuevas and Del Campo10.

As usual, we denote byZ,Z, andZ⁻ the set of all integers, the set of all nonnegative integers, and the set of all nonpositive integers, respectively. Let C^r be the r-dimensional complex Euclidean space with norm| · |.Nn0the setNn0 {n∈N:n≥n₀}.

Ifx : Z → C^r is a function, we define for n ∈ Nn0, the function xn : Z⁻ → C^r by x_ns xns,s ∈ Z⁻. Furthermorex_• is the function given for x_• : Nn0 → B, with x_•n x_n.

The abstract phase space B, which is a subfamily of all functions from Z⁻ into C^r denoted by FZ⁻,C^r, is a normed space with norm denoted by · _B and satisfies the following axioms.

AThere is a positive constantJ >0 and nonnegative functionsN·and M·onZ with the property thatx : Z → C^r is a function, such thatx0 ∈ B, then for all n ∈ Z, the following conditions hold:

ix_n∈ B,

iiJ|xn| ≤ xn_B,

iiixn_B≤Nnsup_0≤s≤n|xs|Mnx0_B. BThe spaceB, · _Bis a Banach space.

We need the following property onB.

CThe inclusion map i : BZ⁻,C^r, · _∞ → B, · _Bis continuous, that is, there is a constantK ≥ 0, such thatϕ_B ≤ Kϕ_∞, for allϕ ∈ BZ⁻,C^r, where BZ⁻,C^rrepresents the bounded functions fromZ⁻ intoC^r.

Axiom C says that any element of the Banach space of the bounded functions equipped with the supremum normBZ⁻,C^r, · _∞is onB.

Remark 1.1. Using analogous ideas to the ones of3, it is not difficult to prove that Axiom Cis equivalente to the following.

C’If a uniformly bounded sequence{ϕn}_n inBconverges to a functionϕcompactly onZ⁻i.e., converges on any compact discrete interval inZ⁻in the compact-open topology, thenϕbelong toBandϕn−ϕ_B → 0 asn → ∞.

Remark 1.2. We will denote byxn, τ, ϕ τ ≥ n0, andϕ∈ Bor simply byxn, the solution of 1.1passing through τ, ϕ, that is,xτ, τ, ϕ ϕ, and the functional equation 1.1 is satisfied.

During this paper we will assume that the sequencesMnandNnare bounded.

The paper is organized as follows. In Section 2 we see some important implications of the fading memory spaces. Section 3 is devoted to recall definitions and some important basic results about almost periodic sequences, asymptotically almost periodic sequences, and uniformly asymptotically almost periodic functions. InSection 4we analyze separately the cases whereFis periodic and when it is almost periodic. Thus, inSection 4.1assuming that the system 1.1 is periodic and the existence of a bounded solution particular solution which is uniformly stable and the phase space satisfies only the axiomsA–C, we prove the existence of an almost periodic solution and an asymptotically almost periodic solution. If additionally the particular solution is uniformly asymptotically stable, we prove the existence of a periodic solution. Similarly, inSection 4.2considering that system1.1is almost periodic

and the existence of a bounded solution and whenever the phase space satisfies the axioms A–C, but here it is also necessary that B verifies the fading memory property. If the particular solution is asymptotically almost periodic, then system1.1has an almost periodic solution. While, if the particular solution is uniformly asymptotically stable, we prove the existence of an asymptotically almost periodic solution.

In 11, 12 the problem of existence of almost periodic solutions for functional difference equations is considered in the first case for the discrete Volterra equation and in the second reference for the functional difference equations with finite delay; in both cases the authors assume the existence of a bounded solution with a property of stability that gives information about the existence of an almost periodic solution. In an analogous way in 13the problem of the existence of almost periodic solutions for functional difference equations with infinite delay is considered. These results can be applied to several kinds of discrete equations. However, our approach differs from Hamaya’s because, firstly, in our work we consider both cases, namely, whenFis periodic and when it is almost periodic in the first variable. And secondly, we analyze very carefully the implications of the existence of a bounded solution of1.1with each property: uniformly stable, uniformly asymptotically stable, and globally uniformly stable.

Furthermore, we cite the articles14–16which are devoted to study almost periodic solutions of difference equations, but a little is known about almost periodic solutions, and in particular, for periodic solutions of nonlinear functional difference equations in phase space via uniform stability, uniformly asymptotically stability, and globally uniformly stability properties of a bounded solution.

2. Fading Memory Spaces and Implications

Following the terminology given in3, we introduce the family of operators onB,S·, as

Snϕ θ

⎧⎨

⎩

ϕ0, if −n≤θ≤0,

ϕnθ, ifθ <−n, 2.1

withϕ∈ B. They constitute a family of linear operators onBhaving the semigroup property Snm SnSmforn, m≥0. Immediately, the following result holds from AxiomA:

Sn ≤ Nn

J Mn, for eachn≥0. 2.2

Now, given any function x : Z → C^r such that x0 ∈ B, we have the following decomposition:

xn yn zn, n∈Z, 2.3

where

yn

⎧⎨

⎩

xn, ifn≥0, x0, ifn≥0,

zn

⎧⎨

⎩

0, if n≥0,

xn−x0, if n <0.

2.4

Then, we have the following decomposition ofxnynzn,yn, zn∈ Bforn≥0, where znSn

x₀−x0χ

, 2.5

andχθ 1 for allθ≤0. Note that

z_n0 0, for eachn≥0. 2.6

Let

B0: ϕ∈ B:ϕ0 0

2.7 be a subset ofB, and letS0n Sn|_B0be the restriction ofStoB0. Clearly, the familyS0n, n ∈Nn0, is also a strongly continuous semigroup of bounded linear operators onB0. It is given explicitly by

S₀nϕ θ

0, −n≤θ≤0,

ϕnθ, θ <−n, 2.8

forϕ∈ B0.

Definition 2.1. A phase spaceBthat satisfies axiomsA-BandCorCand such that the semigroupS₀nis strongly stable is called a fading memory space.

Remark 2.2. Remember that a strongly continuous semigroup is strongly stable if for allϕ ∈ B0,S0nϕ → 0 asn → ∞.

Thus, we have the following result.

Lemma 2.3. Letx : Z → C^r, withx₀ ∈ B, whereBis a fading memory space. Ifxn → 0 as n → ∞, thenx_n → 0 asn → ∞.

Proof. Firstly, we note that as before,x_n y_nS₀nx0−x0χ, whereχθ 1, forθ ≤0 and

yθ

⎧⎨

⎩

xθ, θ≥0,

x0, θ <0. 2.9

Then, by definitionS0nx0−x0χ → 0 asn → ∞becausex0−x0χ∈ B0. On the other hand, by hypothesis,xn → 0 as n → ∞, so it follows from AxiomC’thaty_n → 0.

Therefore, we conclude thatxn → 0 asn → ∞.

3. Notations and Preliminary Results

In this section, we review the definitions of uniformly almost periodic, asymptotically almost periodic sequence, which have been discussed by several authors and present some related properties.

For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in3,17, 18for the continuous case. For the discrete case we mention11,12.

Definition 3.1. A sequence x : Z → C^r is called an almost periodic sequence if the - translation set ofx,

E{, x}:{τ ∈Z/|xnτ−xn|< , ∀n∈Z}, 3.1

is a relatively dense set inZfor all >0; that is, for any given >0, there exists an integer ll>0 such that each discrete interval of lengthlcontainsττ∈E{, x}such that

|xnτ−xn|< , ∀n∈Z. 3.2

τ is called the-translation number ofxn. We will denote byAPZ;C^rthe set of all such sequences. We will write thatxis a.p. ifx∈ APZ;C^r.

Definition 3.2. A sequencex:Z → C^ris called an asymptotically almost periodic sequence if

xn pn qn, 3.3

wherepnis an almost periodic sequence, andqn → 0 asn → ∞. We will denote by AAPZ;C^rthe set of all such sequences. We will write thatxis a.a.p. ifx∈ AAPZ;C^r.

In general, we will considerX, · Xa Banach space.

Definition 3.3. A function or sequencex :Z → X is said to be almost periodicabbreviated a.p.inn ∈ Z if for every > 0 there is N N > 0 such that amongN consecutive integers there is one; call itp, such that

x np

−xnX< , ∀n∈Z. 3.4

Denote byAPZ;Xall such sequences, andxis said to be an almost periodica.p.inX.

Definition 3.4. A sequence{xn}_n∈Nn0,or {xn}_n∈Z,xn ∈ X, equivalently, a function x:Nn0 → Xor,x:Z → Xis called asymptotically almost periodic ifxx1x2, where x₁ ∈ APZ;Xandx₂ :Nn0 → X or,x₂ :Z → Xsatisfyingx2n_X → 0 asn → ∞

or,|n| → ∞. Denote byAAPNn0;X orAAPZ;Xall such sequences, andxis said to be an asymptotically almost periodic onNn0 or onZ a.a.p.inX.

Remark 3.5. Almost periodic sequences can be also defined for any sequence{xn}_n∈JJ ⊂Z orx:J → Xby requiring thatNN>0 consecutive integers are inJ.

Definition 3.6. Let f : Z× B → C^r.fn, φ is said to be almost periodic inn uniformly for φ ∈ B, if for any > 0 and every compact Σ ⊂ B, there exists a positive integerl l,Σ such that any interval of lengthli.e., amonglconsecutive integerscontains an integeror equivalently, there is one; call itτ, for which

f

nτ, φ

−f

n, φ< , ∀n∈Z, φ∈Σ. 3.5

τ is called the-translation number offn, φ. We will denote byUAPZ× B;C^rthe set of all such sequences. In brief we will write thatf is u.a.p. if f∈ UAPZ× B;C^r.

Definition 3.7. The hull off, denoted byHf, is defined by

Hf

g n, φ

: lim

k→∞f

nτ_k, φ g

n, φ

uniformly onZ×Σ

, 3.6

for some sequence{τk}, whereΣis any compact set inB.

For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in3,17, 18for the continuous case. For the discrete case we mention11,12. With the objective to make this manuscript self contained we decided to include the majority of the proofs.

Lemma 3.8. aIf{xn}is an a.p. sequence, then there exists an almost periodic functionftsuch thatfn xnforn∈Z.

bIfftis an a.p. function, then{fn}is an a.p. sequence.

Lemma 3.9. aIf{xn}is an a.p. sequence, then{xn}is bounded.

b{xn}is an a.p. sequence if and only if for any sequence{k_i} ⊂Zthere exists a subsequence {ki} ⊂ {k_i}such that xnki converges uniformly on Zas i → ∞. Furthermore, the limits sequence is also an almost periodic sequence.

c{xn}n∈Zis an a.p. sequence if and only if for any sequence of integers{k_i},{l_i}there exist subsequencesk{ki} ⊂ {k_i},l{li} ⊂ {l_i}such that

T_kT_lxn T_klxn, forn∈Z, 3.7

whereT_kxn lim_i→∞xnk_iforn∈Z.

d{xn},n∈Z(or,n∈Z) is an a.a.p. sequence if and only if for any sequence{k_i} ⊂Z (or, Z) such that k_i > 0 andk_i → ∞asi → ∞(or, |k_i| → ∞ as i → ∞), there exists a subsequence{ki} ⊂ {k_i}such thatxnk_iconverges uniformly onZ(orZ) asi → ∞.

Lemma 3.10. Let xnbe an a.a.p. periodic sequence. Then its decomposition,

fn pn qn, 3.8

wherepnis an a.p. sequence whileqn → 0 asn → ∞, is unique.

Lemma 3.11. Let f :Z× B → C^rbe almost periodic innuniformly forφ∈ Band continuous inφ.

Thenfn, φis bounded and uniformly continuous onZ×Σfor any compact setΣinB.

Lemma 3.12. Let fn, φbe the same as in the previous lemma. Then, for any sequence{h_k}, there exist a subsequence{hk}of{h_k}and a functiongn, φcontinuous inφsuch thatfnhk, φ → gn, φuniformly onZ×Σask → ∞, whereΣis any compact set inB. Moreover,gn, φis also almost periodic innuniformly forφ∈ B.

Lemma 3.13. Let fn, φbe the same as in the previous lemma. Then, there exists a sequence{αk}, α_k → ∞ask → ∞such thatfnα_k, φ → fn, φuniformly onZ×Σask → ∞, whereΣ is any compact set inB.

Lemma 3.14. Let f : Z× B → C^r be almost periodic innuniformly forφ ∈ Band continuous in φ∈ B, and letpnbe an almost periodic sequence inBsuch thatpn∈Σfor alln∈Z, whereΣis a compact set inB. Thenfn, pnis almost periodic inn.

Lemma 3.15. Let f : Z× B → C^r be almost periodic innuniformly forφ ∈ Band continuous in φ∈ B, and letpnbe an almost periodic sequence inC^r such thatp_n ∈Σfor alln∈Z, whereΣis a compact set inBandpns pnsfors∈Z⁻. Thenfn, pnis almost periodic inn.

Remark 3.16. Ifx:Nn0 → Xis a.a.p., then the decompositionxx1x2, in the definition of an a.a.p. function, is uniquesee18.

4. Existence of Almost Periodic Solutions

From now on we will assume that the system1.1has a unique solution for a given initial condition onBand without loss of generalityn₀0, thusN_n0 N₀Z.

We will make the following assumptions on1.1.

H1F:Z× B → C^r is continuous in the second variable for any fixedn∈Z.

H2System1.1has a bounded solutiony {yn}_n≥0, passing through0, ϕ,ϕ ∈ B, that is, sup_n≥0|yn|<∞.

For this bounded solution{yn}_n≥0, there is anα >0 such that|yn| ≤αfor alln. So, we will have to assume thatynB≤ αfor alln, andyn ∈Σα {φ ∈ B/φB ≤α}. Next, we will point out the definitions of stability for functional difference equations adapting it from the continuouscase according to Hino et al. in3.

Definition 4.1. A bounded solutionx{xn}_n≥0of1.1is said to be:

istable, if for any > 0 and any integerτ ≥ 0, there isδ : δ, τ > 0 such that xτ −yτB < δ implies thatxn −ynB < for all n ≥ τ, where{yn}_n≥τ is any solution of1.1;

iiuniformly stable, abbreviated as “x ∈ US”, if for any > 0 and any integerτ ≥ 0, there isδ:δ>0δdoes not depend onτsuch thatxτ−yτB< δimplies that xn−y_nB< for alln≥τ, where{yn}_n≥τ is any solution of1.1;

iiiuniformly asymptotically stable, abbreviated as “x ∈ UAS”, if it is uniformly stable and there isδ₀>0 such that for any >0, there is a positive integerNN>0 such that ifτ ≥0 andxτ −y_τB< δ₀, thenxn−y_nB< for alln≥τN, where {yn}_n≥τ is any solution of1.1;

ivglobally uniformly asymptotically stable, abbreviated as “x∈ GUAS”, if it is uniformly stable andxn−ynB → 0 asn → ∞, whenever{yn}_n≥τis any solution of1.1.

Remark 4.2. It is easy to see that an equivalent definition forx{xn}_n≥0, beingUAS, is the following:

iii^∗x {xn}_n≥0isUAS, if it is uniformly stable, and there existsδ0 >0 such that if τ ≥0 andxτ−y_τB< δ₀, thenxn−y_nB → 0 asn → ∞, where{yn}_n≥τis any solution of1.1.

4.1. The Periodic Case

Here, we will assume what follows.

H3The function Fn,·in 1.1 is periodic inn ∈ Z, that is, there exists a positive integerT such thatFnT,· Fn,·for alln∈Z.

Moreover, we will assume what follows.

A The sequences Mn and Nn in Axiom Aiii are bounded by M and N, respectively andM <1.

Lemma 4.3. Suppose that condition (A) holds. If {yn}is a bounded solution of 1.1such that y₀∈ B, theny_nis also bounded inZ.

Proof. Let us say that|yn| ≤Rfor alln∈Z. Then by AxiomAiiiand hypothesisA we have

ynB≤Nsup

0≤s≤n

ysMy0B≤NRMy0B, ∀n∈Z. 4.1

Lemma 4.4. Suppose that condition (A) holds. Let {y^kn}_k≥1be a sequence inC^r such thaty₀^k∈ B for allk≥1. Assume thaty^ks → ηsask → ∞for everys∈Zandη₀∈ B, theny_n^k → ηnin Bask → ∞for eachn∈Z. In particular, ify^ks → ηsask → ∞uniformly ins∈Z, then y^k_n → η_ninBask → ∞uniformly inn∈Z.

Proof. By AxiomAiiiand hypotheses we have that

y_n^k−η_nB≤Nsup

0≤s≤n

y^ks−ηsMy₀^k−η_0B, for anyn≥0. 4.2

In the particular casen0 we obtain

y^k₀ −η₀B≤ N

1−My^k0−η0, 4.3

and soy₀^k−η_0B → 0 ask → ∞. On the other hand, sincenis fixed, it follows that

sup

0≤s≤n

y^ks−ηs−→0 ask−→∞, 4.4

for eachn∈Z. Therefore, we have concluded the proof.

Theorem 4.5. Suppose that condition (A) and (H1)–(H3) hold. If the bounded solution {yn}_n≥0of 1.1isUS, then{yn}is an a.a.p. sequence inC^r, equivalently,1.1has an a.a.p. solution.

Proof. ByLemma 4.3there existsα∈Rsuch thatyn_B≤αfor alln∈Z, and a boundedor compactsetΣα⊂ Bsuch thatyn∈Σαfor alln≥0. Let{nk}_k≥1be any integer sequence such thatn_k > 0 andn_k → ∞ask → ∞. For eachn_k, there exists a nonnegative integerm_k such thatm_kT ≤n_k≤mk1T. Setn_km_kTτ_k. Then 0≤τ_k< T for allk≥1. Since{τk}_k≥1 is a bounded set, we can assume that, taking a subsequence if necessary,τkj_∗for allk≥1, where 0≤j_∗< T. Now, sety^kn ynnk. Thus,

y^kn1 ynn_k1 F

nn_k, y_nnk F

nn_k, y^k_n F

nj_∗, y^k_n

, 4.5

which implies that{y^kn}is a solution of the system, xn1 F

nj_∗, xn

, 4.6

through0, ynk. It is clear that if{yn}_n≥0isUS, then{y^kn}_n≥0 is alsoUSwith the same pair, δas the one for{yn}_n≥0.

Since{ynnk}is bounded for allnandnk, we can use the diagonal method to get a subsequence{nkj}of{nkm_kTj_∗}such thatynnkjconverges for eachn∈Zasj → ∞.

Thus, we can assume that the sequenceynn_kconverges for eachn∈Zask → ∞. Since y^k₀ ynk ∈ B, byLemma 4.4it follows thaty^k_nis also convergent for eachn∈Z. In particular, for any >0 there exists a positive integerN1such that ifk, m≥N1J Jis the constant given in Axiom Aii, then

y^k₀−y^m_0B< δ, 4.7

whereδis the number given by the uniform stability of{yn}_n≥0. Sincey^kn ∈ US, it follows fromDefinition 4.1and4.7that

y^k_n−y_n^m_B< J, ∀n≥0, 4.8

and by Axiom Aiiit follows that

y^kn−y^mn< , ∀n≥0, k, m≥N₁. 4.9

This implies that for any positive integer sequencen_k,n_k → ∞ ask → ∞, there is a subsequence{nkj}of{nk}for which{ynn_kj} converges uniformly onZ asj → ∞.

Thus, the conclusion of the theorem follows fromLemma 3.9d.

Before proving our following result we remark that ifyis a.a.p. then there are unique sequencesp, q : Z → C^r such thatyn pn qn, withpa.p. andqn → 0 asn → 0 asn → ∞. ByLemma 3.9ait follows thatpis bounded and thusp∈BZ⁻,C^r. Hence, by AxiomCwe must have thatpn∈ Bfor alln≥0. In particular,qnyn−pn∈ Bfor alln≥0.

Theorem 4.6. Suppose thatA and (H1)–(H3) hold and the bounded solution{yn}_n≥0of1.1is US, then system1.1has an a.p. solution, which is alsoUS.

Proof. It follows fromTheorem 4.5thatyis an a.a.p. Setyn pn qn n ≥ 0, where {pn}_n≥0is a.p. sequence andqn → 0 asn → ∞. For the positive integer sequence{nkT}, byLemma 3.9b–dand arguments of the previous theorem, we can choice a subsequence {nkjT}of{nkT}such thatynn_kjTconverges uniformly inn∈Zandpnn_kjT → ηn uniformly onZasj → ∞and{ηn}is also a.p. Then,ynnkjT → ηnuniformly in n∈Z, and thus byLemma 4.4y_nnkjT → ηnuniformly inn∈ZonBasj → ∞andηn∈ B.

Since

ηn1←−y

nnkjT1 F

nnkjT, y_nnkjT

F

n, y_nnkjT

−→F n, ηn

4.10

asj → ∞, we haveηn1 Fn, ηnforn ≥ 0, that is, the system1.1has an almost periodic solution, and so we have proved the first statement of the theorem.

In order to prove the second affirmation, notice thatynnkjT∈ USsincey∈ US. For anyn0 ∈Z, let{xn}_n≥0be a solution of1.1such thatx0∈ Bandηn0−xn0B:μ < δ.

Again, byLemma 4.4y_n^kj → η_nasj → ∞for eachn≥0, so there is a positive integerJ₁>0 such that ifj≥J1, then

yn^kj0−η_n0B< δ−μ. 4.11

Thus, forj ≥J1, we have

yn0n_kjT−x_n0B≤ yn0n_kjT−η_n0Bηn0−x_n0B< δ. 4.12

Then,

ynn_kjT−xnB< ∀n≥n0. 4.13

Therefore, there isJ₂>0 such that ifj ≥J₂, then

ηn0−yn0n_kjTB< δν, 4.14

and hence,ηn−y_nnkjTB< νfor alln≥n₀, whereν, δνis a pair for the uniform stability ofynnkjT. This shows that ifj≥max{J1, J₂}, then

ηn−x_nB≤ ηn−y_nnkjTBy_nnkjT−x_nB< ν, 4.15

for alln≥n0, which implies thatηn−xnB≤for alln≥n0ifηn0−xn0B< δbecauseν is arbitrary. This proves thatηnisUS.

In the case when we have an asymptotically stable solution of1.1 we obtain the following result.

Theorem 4.7. Suppose thatA and (H1)–(H3) hold and the bounded solution{yn}_n≥0of1.1is UAS, then the system1.1has a periodic solution of periodmT for some positive integerm, which is alsoUAS.

Proof. Sety^kn ynkT,k 1,2, . . .. By the proof ofTheorem 4.5, there is a subsequence {y^kjn}which converges to a solution{ηn}of4.6for eachn∈Zand hence byLemma 4.4, y^k₀^j → η₀ as j → ∞. Thus, there is a positive integer p such that y^k₀^p −y^k₀^p1_B < δ₀ 0≤k_p< k_p1, whereδ₀is obtained from the uniformly asymptotic stability of{yn}_n≥0. Letm k_p1−kp, and notice thaty^mn ynmTis a solution of1.1. Sincey_k^m_pTj y^mkpTj yk_p1Tj y_kp1Tjforj ∈Z⁻, that is,y^m_kpT y_kp1T, we have

y_k^m_pT−y_kpTBykp1T−y_kpTBy^k₀^p1−y^k₀^p_B< δ₀, 4.16

and hence,

yn^m−ynB−→0 asn−→∞, 4.17

because{yn}_n≥0 isUASsee also Remark 4.2. On the other hand,{yn}_n≥0 is a.a.p. by Theorem 4.5, then

yn pn qn, n≥0, 4.18

where{pn}_n∈Zis a.p. andqn → 0 asn → ∞. It follows from4.17and4.18that pn−pnmT−→0, asn−→∞, 4.19 which implies thatpn pnmTfor alln∈Zbecause{pn}is a.p.

For the integer sequence{kmT},k1,2, . . ., we haveynkmT pn qnkmT.

ThenynkmT → pnuniformly for all n ∈ Zask → ∞, and again byLemma 4.4, y_nkmT → p_nuniformly inn∈Zask → ∞. SinceynkmT1 Fn, y_nkmT, we have pn1 Fn, pnforn ≥ 0, which implies that1.1has a periodic solution{pn}_n≥0 of periodmT.

Now, we will proceed to prove that p ∈ UAS by the use of definition ii^∗ in Remark 4.2. Notice that sincey∈ UAStheny^kjnis aUASsolution of1.1with the same δ₀ as the one for{yn}. Let{xn}be any solution of1.1such thatpn0−x_n0B < δ₀. Set pn0 −x_n0B : μ < δ₀. Again, for sufficient largej, we have the similar relations4.12and 4.14withyn0n_kjT−xn0B< δ₀andyn0n_kjT−ηn0B< δ₀. Thus,

ηn−x_nB≤ ηn−y_nnkjTBy_nnkjT−x_nB−→0, 4.20

asn → ∞if yn0−x_n0B < δ₀, becausey^kj,x, andηnsatisfy1.1. This completes the proof.

Finally, if the particular solution is GUAS, we will prove that system 1.1 has a periodic solution.

Theorem 4.8. Suppose thatA and (H1)–(H3) hold and that the bounded solution{yn}_n≥0 of 1.1isGUAS, then the system1.1has a periodic solution of periodT.

Proof. ByTheorem 4.5,yis a.a.p. Thenyn pn qn n ≥0, where{pn}n ∈Zis an a.p. sequence andqn → 0 asn → ∞. Notice thatynTis also a solution of1.1 satisfyingyT ∈Σα. Since{yn}isGUAS, we have thatyn−y_nTB → 0 asn → ∞, which implies thatpn pnTfor alln∈Z. Using same technique as in the proof ofTheorem 4.7, we can show that{pn}is aT-periodic solution of1.1.

4.2. The Almost Periodic Case Here, we will assume that

H4the function Fn,· in1.1 is almost periodic inn ∈ Z uniformly in the second variable.

By CF we denote the uniform closure of F, that is, CF {G/∃ αk such thatαk → ∞andFn αk,· → Gn,·uniformly on Z × Σ as k → ∞ whereΣis any compact set in B}. Note that CF ⊂ APZ × B,C^r byLemma 3.12 and F∈ CFbyLemma 3.13.

Lemma 4.9. Suppose that Axiom (C) is true, and that{xn}_n∈Zis an a.p. sequence withx0∈ B, then x_nis a.p.

Proof. We know that, given >0, there exists an integerl l> 0 such that each discrete interval of lengthlcontains aτ τ∈E{, x}such that

|xnτ−xn|<

K, ∀n∈Z. 4.21

By AxiomCwe have

x_nτ−x_nB≤Kx_nτ−x_n∞ Ksup

θ≤0|xnτθ−xnθ|

Ksup

θ≤0|xnτθ−xnθ|

< .

4.22

Lemma 4.10. Suppose thatBis a fading memory space and{xn}_n∈Zis a.a.p. withx0∈ B, thenxn

is a.a.p.

Proof. Sincexnis a.a.p. there are unique sequencesynandvnsuch thatyis a.p. and vn → 0 asn → ∞. Then byLemma 4.9it follows thatyn is a.p., and byLemma 2.3it follows thatv_n → 0 asn → ∞. Therefore,x_ny_nv_nis a.a.p.

Theorem 4.11. Suppose that conditionsA, (H1)-(H2), and (H4) hold and that Bis a fading memory space. If the bounded solution{yn}_n≥0of 1.1is an a.a.p. sequence, then the system1.1has an a.p. solution.

Proof. Since the solution{yn}_n≥0is a.a.p., it follows fromLemma 3.10thatynhas a unique decompositionyn pn qn, where{pn}_n∈Zis a.p. andqn → 0 asn → ∞. Notice that{yn}is bounded. ByLemma 4.3there is a compact setΣαinBsuch thatyn, pn∈Σαfor alln≥ 0. ByLemma 3.13, there is an integer sequence{nk},nk > 0, such thatnk → ∞as k → ∞andFnn_k, φ → Fn, φuniformly onZ×Σαask → ∞. Taking a subsequence if necessary, we can also assume thatpnn_k → pn uniformly onZ, and byLemma 3.9b we have that{pn} is also an a.p. sequence. For anys∈Z⁻, there is a positive integerk0such that ifk > k₀, thensn_k≥0. In this case, we see thatynn_k → pn uniformly for allnas k → ∞, and hence byLemma 4.4y_nnk → p_ninBinn∈Zask → ∞. Since

ynnk1 F

nnk, y_nnk

F

nn_k, y_nnk

−F

nn_k,p_n

F

nn_k,p_n

−F n,p_n

F n,p_n

,

4.23

and from the previous considerations the first term of the right-hand side of 4.23 tends to zero as k → ∞ and since Fnnk,pn−Fn,pn → 0 as k → ∞, we have that

pn1 Fn,pnfor alln ∈ Z, which implies that1.1has an a.p. solution {pn}_n≥0 passing through0,p₀, wherep₀j pj forj∈Z⁻.

We are now in a position to prove the following result.

Theorem 4.12. Suppose that the assumptions A, (H1), (H2), and (H4) hold, and that B is a fading memory space. If the bounded solution{yn}_n≥0of 1.1isUAS, then{yn}_n≥0 is a.a.p.

Consequently,1.1has an a.p. solution which isUAS.

Proof. Let the bounded solutionyof1.1beUASwith the tripleδ, δ0, N. Let{nk}_k≥1 be any positive integer such thatnk → ∞ask → ∞. Sety^kn ynnk. As previously y^knis a solution of

xn1 Fnnk, xn, 4.24

and{y^kn}isUASwith the same tripleδ, δ0, N. ByLemma A.2, for the setΣαand any 0< <1 there existsδ₁>0 such that|hn|< δ₁andx^k_n0−x_n0B< δ₁for some n₀≥0 implies thatx_n^k−xnB< /2 for alln≥n₀, where{xn}_n≥n0is a bounded solution of

xn1 Fnn_k, x_n hn, 4.25

passing throughn0, x_n0andx_n ∈ Σα forn ≥ n₀. Sincey^kjis uniformly bounded for all k≥1 andj∈Z, taking a subsequence if necessary, we can assume that{y^kj}is convergent for eachj∈ZandFnnk, φ → Gn, φuniformly onZ×Σα, for some a.p. functionG. In this case, byLemma 4.4there is a positive integerk₁such that ifm, k≥k₁, then

y₀^k−y^m₀B< δ₁. 4.26

On the other hand,y_n^m∈Σαforn∈Zis a solution of4.25withhn hk,mn, that is,

xn1 Fnn_k, x_n h_k,mn, 4.27

whereh_k,mnis defined by the relation hk,mn F

nnm, y^m_n

−F

nnk, y_n^m

, n∈Z. 4.28

To apply Lemma A.2to 4.24 and its associated equation 4.27, we will point out some properties of the sequence{hk,mn}_n≥0. SinceFnn_k, φ → Gn, φuniformly onZ×Σα, for the aboveδ₁>0, there is a positive integerk₂> k₁such that ifk, m≥k₂, then

F

nnm, φ

−F

nnk, φ

< δ1, ∀n∈Z, φ∈Σα, 4.29

which implies that |hk,mn| |Fn n_m, y^m_n − Fn n_k, y^m_n| < δ₁ for all n ∈ Z.

ApplyingLemma A.2to4.24and its associated equation4.27with the above arguments and condition4.26, we conclude that for any positive integer sequence{nk}_k≥1,nk → ∞ ask → ∞, and >0, there is a positive integerk₂>0 such that

y^k_n−y_n^m_B<

J, n≥0 ifk, m > k₂, 4.30

and hence by Axiom Aii|y^kn−y^mn|< for alln≥0 ifk, m > k2. This implies that the bounded solution{yn}_n≥0of1.1is a.a.p. byLemma 3.9d. Furthermore,1.1has an a.p. solution, which isUASbyTheorem 4.11. This ends the proof.

Appendix

The proof of the following lemmas used ideas developed by Hino et al. in 3 for the functional differential equations with infinite delay and by Song12for functional difference equations with finite delay.

Lemma A.1. Suppose thatA, (H1), (H2), and (H4) hold and that Bis a fading memory space. Let ybe the bounded solution of 1.1. Let{nk}_k≥1be a positive integer sequence such thatn_k → ∞, ynk → φ, andFnnk, φ → Gn, φuniformly onZ×Σask → ∞, whereΣis any compact subset inBandG∈ CF. If the bounded solution{yn}_n≥0isUS, then the solution{ηn}_n≥0of

xn1 Gn, xn, A.1

through0, φ, isUS. In addition, if{yn}_n≥0isUAS, then{ηn}_n≥0is alsoUAS.

Proof. Sety^kn ynn_k. It is easy to see thaty^knis a solution of

xn1 Fnnk, xn, n≥0, A.2

passing though0, ynkandy_n^k∈Σαfor allk. Since{yn}_n≥0isUS, then{y^kn}is alsoUS with the same pair, δas the one for{yn}_n≥0. Taking a subsequence if necessary, we can assume that{y^kn}_k≥1converges to a vectorηnfor eachn≥0 ask → ∞. From4.23 with p_n η_n, we can see that{ηn}_n≥0 is the unique solution ofA.1, satisfyingη₀ φ becausey_nk → φ.

To show that the solution{ηn}_n≥0ofA.1isUS, we need to prove that for any >0 and any integer n₀ ≥ 0, there exists δ^∗ > 0 such thatηn0 −y_n0B < δ^∗ implies that ηn−y_nB< for alln≥n₀, where{yn}_n≥n0is a solution ofA.1withy_n0 χ∈ B.

We know fromLemma 4.4thaty_n^k → ηn ask → ∞for eachn; thus, for any given n₀∈Z, ifkis sufficiently large; sayk≥k₀>0, we have

y_n^k₀−η_n0B< 1 2δ

2

, A.3

whereδcomes from the uniform stability of{yn}_n≥0. Letχ∈ Bbe such that

χ−ηn0B< 1 2δ

2

, A.4

and let{xn}_n≥n0be the solution of1.1such thatxn0nk φ. Then{x^kn xnnk}is a solution ofA.2withx^k_n0 φ. Since{y^kn}isUSandx^k_n0−y_n^k_0B< δ/2fork ≥k₀, we have

y_n^k−x^k_nB<

2 ∀n≥n₀, k≥k₀. A.5

It follows fromA.5that

x^k_nB≤ y^k_nB

2 < α

2 n≥n0, k≥k0. A.6

Then there exists a numberα^∗ >0 such thatx_n^k∈S_α∗ for alln≥0 andk ≥k₀, which implies that there is a subsequence of{x^kn}_k≥0for eachn≥n₀−τ, denoted by{x^kn}again, such thatx^kn → ynfor eachn ≥n0−τ, and hence byLemma 4.4x_n^k → yn for alln≥ n0 as k → ∞. Clearly,y_n0 χ, and the setS_α∗is compact setB. SinceFn, φis almost periodic in nuniformly forφ∈ B, we can assume that, taking a subsequence if necessary,Fnn_k, φ → Gn, φuniformly onZ×Sα^∗ ask → ∞. Takingk → ∞inx^kn1 Fnnnk, xⁿ_k, we haveyn1 Gn, yn, namely,{yn}is the unique solution ofA.1, passing through n0, χwithy_n0χ∈ B. On the other hand, for any integerN >0, there existsk_N ≥k₀such that ifk≥kN, then

x^k_n−y_nB<

4, y^k_n−η_nB<

4 forn₀≤n≤n₀N. A.7 FromA.5andA.7, we obtain

ηn−ynB< forn0≤n≤n0N. A.8

SinceNis arbitrary, we haveηn−y_nB< for alln≥n₀ifχ−η_n0B< δ/2/2 andφ∈ B, which implies that the solution{ηn}_n≥0ofA.1isUS.

Now, we consider the case where{yn}_n≥0isUAS. Then the solution{y^kn}ofA.2 is alsoUASwith the same pairδ0, , Nas the one for{yn}_n≥0. Letδ^∗, be the pair for uniform stability of{ηn}.

For any givenn0∈Z, ifkis sufficiently large; sayk≥k0>0, we have

y^k_n0−η_n0B< 1

2δ₀, A.9

whereδ0 is the one for uniformly asymptotic stability of{yn}_n≥0. Letφ ∈ Bsuch thatφ− η_n0B < δ0/2, and let{xn}_n≥n0, for each fixed k ≥ k₀, be the solution of1.1such that x_n0nk χ. Thenx^kis a solution ofA.2withx^k_n0χ. Since{y^kn}isUASandx^k_n0−y_n^k_0B<

δ0/2for each fixedk≥k₀, we have

y^k_n−x^k_nB<

2 ∀n≥n₀N 2

, k≥k₀. A.10

By the same argument as above, there is a subsequence ofnk, which we will continue calling n_k, such that{x^kn} converges to the solution{yn} of A.1 through n0, χand Fn nk, φ → Gn, φuniformly onZ×Sα^∗ ask → ∞, whereSα^∗ is a compact set inBwith

|x^kn| ≤ α^∗ for allk ≥ k0 and n ∈ Z. Then{yn}is the unique solution ofA.1, passing throughn0, χwithy_n0 χ ∈ B. On the other hand, byLemma 4.4for any integerN > 0 there existsk_N≥k₀such that ifk≥k_N, then

x_n^k−ynB<

4, y_n^k−ηnB<

4 forn0N 2

≤n≤n0N 2

N, A.11

and henceyn−η_nB < forn₀N/2 ≤ n≤n₀N/2 N. SinceNis arbitrary, we have

yn−η_nB< , ∀n≥n₀N 2

, A.12

ifφ−ηn0B<δ0/2andφ∈ B; thus,ηn∈ UASand the proof is complete.

Now, we need to prove the following important lemma.

Lemma A.2. Suppose that the assumptionsA, (H1), (H2), and (H4) hold, that Bis a fading memory space, that the bounded solutionyof 1.1isUAS, and that for eachG∈ CF, the solution ofA.1 is unique for any given initial data. LetS⊃Σαbe a given compact set inB. Then for any >0, there existsδ δ>0 such that ifn₀≥0,yn0−xn0B < δ, and{hn}is a sequence with|hn| ≤δ forn≥n0, one hasyn−xnB< for alln≥n0, where{xn}is any bounded solution of the system

xn1 Fn, xn hn, n≥n0, A.13

passing throughn0, x_n0and such thatx_n∈Sfor alln≥n₀.

Proof. Suppose that the bounded solution {yn}_n≥0 of 1.1 is UAS with the triple δ•, δ0, N•. The proof will be by contradiction, we assume thatLemma A.2is not true.

Then for some compact setS_∗Σα, there exist, 0< < δ₀, sequences{nk} ⊂Z,{rk} ⊂Z, mapping sequencesh_k:nk,∞ → C^r,ϕ^k:−∞, nk → C^r, and

ynk−x^k_nkB< 1

k, |hkn| ≤ 1

k for n≥nk,

yn−x^k_nB≤ forn_k≤n≤n_kr_k−1, ynkrk−x^k_nkrkB≤,

A.14

for sufficiently largek, where{x^kn}is a solution of

xn1 Fn, xn hkn, n≥nk, A.15

passing throughnk, ϕ^ksuch thatx^k_n ∈ S_∗for alln ≥ n_k andk ≥ 1. SinceS_∗is a bounded subset ofB, it follows that{x^knkrkn}_k≥1and{x^knkn}_k≥1are uniformly bounded for all nkandn≥ −∞. We first consider the case where{rk}_k≥1contains an unbounded subsequence.

SetN N > 1. Taking a subsequence if necessary, we may assume from Lemmas 3.12