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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Asymptotically almost automorphic solutions of differential equations with piecewise constant argument

Hui-Sheng Ding
  • Corresponding author
  • College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Shun-Mei Wan
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/math-2017-0051

Abstract

This paper is concerned with the existence and uniqueness of asymptotically almost automorphic solutions to differential equations with piecewise constant argument. To study that, we first introduce several notions about asymptotically almost automorphic type functions and obtain some properties of such functions. Then, on the basis of a systematic study on the associated difference system, the existence and uniqueness theorem is established. Compared with some earlier results, we do not assume directly that the Green’s function is a Bi-almost automorphic type function.

Keywords: Asymptotically almost automorphic; Almost automorphic; Differential equations with piecewise constant argument; DEPCA

MSC 2010: 34C27

1 Introduction

In this paper, we aim to study the existence and uniqueness of asymptotically almost automorphic solution for the following differential equation with piecewise constant argument (DEPCA): y(t)=A(t)y(t)+B(t)y([t])+f(t,y(t),y([t])),tR.(1)

where y(t) is a p-dimensional complex vector (p is a fixed positive integer), A(t) and B(t) are p × p complex matrices, and the coefficients satisfy some conditions recalled in the sequel.

As noted in some earlier works (cf. [2, 3]), differential equations with piecewise constant argument (DEPCA) are of considerable importance in applications to some biomedical dynamics, physical phenomena, discretization problems, etc., and there is a large literature on qualitative properties of solutions to DEPCA, like uniqueness, boundedness, periodicity, almost periodicity, pseudo almost periodicity, stability, etc. However, it seems that there are only few results concerning almost automorphic type solutions for DEPCA (cf. [25, 7, 11]).

Recall that since Bochner [1] introduced the concept of almost automorphy, the automorphic functions have been applied to many areas including ordinary as well as partial differential equations, abstract differential equations, functional-differential equations, integral equations, dynamical systems, etc. We refer the reader to the monographs of N’Guéréata [8, 9] for the basic theory of almost automorphic functions and their applications.

Stimulated by a recent work of Chávez, Castillo, and Pinto [3], in this paper we investigate the existence and uniqueness of asymptotically almost automorphic solution to equation (1). In order to establish our main results, we recall some notions about asymptotically almost automorphic type functions. Also, we make a systematic study on the properties of the introduced asymptotically almost automorphic type functions and the associated difference system for equation (1). Compared with some earlier results (e.g., [2, 3]), we do not assume directly that the Green’s function is a Bi-almost automorphic type function. In fact, under some assumptions, we prove that the Green’s function is Bi-asymptotically almost automorphic (see Lemma 3.4).

Before closing this section, we recall what is understood by a solution of DEPCA (1).

Definition 1.1

([2, 3]). A function y : ℝ → ℂp is called a solution of DEPCA (1) if the following assertions are satisfied:

  1. y is continuous on;

  2. y is differentiable on ℝ\ℤ and for every nZ, y’+(n) and y’_(n) exsit;

  3. there hold y+(n)=A(n)y(n)+B(n)y(n)+f(t,y(n),y(n)) for every n ∈ ℤ and y(t)=A(t)y(t)+B(t)y([t])+f(t,y(t),y([t])),t(n,n+1),nZ.

2 Almost automorphic type functions

In this paper, we denote by ℤ the set of all integers, by ℝ the set of all real numbers, by ℂ the set of all complex numbers, by ℂp the set of all p-dimensional complex vectors, by Mp×p(ℂ) the set of all p × p complex matrices, and by BC(𝕐,𝕏) the Banach space of all bounded and continuous functions from 𝕐 to 𝕏 with norm ||f||=suptY||f(t)|| for every two Banach spaces 𝕏 and 𝕐. Moreover, for convenience, for every c={ci}i=1pCp and C={cij}i,j=1pMp×p(C), we denote their norms by the followings: |c|=max1ip|ci|,|C|=max1i,jp|cij|.

Next, let us recall some basic definitions and results about almost automorphic functions and asymptotically almost automorphic functions. For more details, we refer the reader to [8, 9].

Definition 2.1

A function fBC(ℝ,𝕏) is said to be almost automorphic if given any sequence {n} ⊂ ℝ, there exists a subsequence {sn} ⊂ {n} such that f(t):=limnf(t+sn) is well defined for every t ∈ ℝ, and limnf(tsn)=f(t) for every t ∈ ℝ. We denote the set of all such functions by AA(ℝ,𝕏).

Remark 2.2

The following is a typical example of almost automorphic function: f(t)=sin12+cost+cos2t,tR. We refer the reader to the monographs of N’Guérékata [8, 9] for the basic theory of almost automorphic functions and their applications.

Definition 2.3

A function fBC(ℝ,𝕏) is said to be asymptotically almost automorphic iff it admits a decomposition f=g+h, where gAA(ℝ,𝕏) and hC0(ℝ,𝕏). Here, C0(R,X):={h:RX:hiscontinuousonRandlimnh(t)=0}. We denote the set of all such functions by AAA(ℝ,𝕏).

Lemma 2.4

The following assertions hold:

  1. {g(t):tR}{f(t):tR}¯, where f = g + h, g ∈ AA(ℝ,𝕏) and hC0(ℝ,𝕏).

  2. The decomposition of every asymptotically almost automorphic function is unique.

  3. AA(ℝ,𝕏) and AAA(ℝ,𝕏) are both Banach spaces under the supremum norm ||f||=suptR||f(t)||.

Definition 2.5

A function fBC(ℝ × 𝕐,𝕏) is called almost automorphic if given any compact set K ⊂ 𝕐 and any sequence {n} ⊂ ℝ, there exists a subsequence {sn} ⊂ {n} such that f(t,x):=limnf(t+sn,x) is well defined for every t ∈ ℝ and xK, and limnf(tsn,x)=f(t,x) for every t ∈ ℝ and xK. We denote the set of all such functions by AA(ℝ × 𝕐,𝕏).

Definition 2.6

A function fBC(ℝ × 𝕐,𝕏) is said to be asymptotically almost automorphic iff it admits a decomposition f=g+h,

where gAA(ℝ × 𝕐,𝕏) and hC0(ℝ × 𝕐,𝕏). Here, C0(ℝ × 𝕐,𝕏) denote the set of all continuous functions from to ℝ × 𝕐 to 𝕏 satisfying limnh(t,y)=0 uniformly for y in any compact subset of 𝕐. We denote the set of all such functions by AAA(ℝ × 𝕐,𝕏).

Next, let us recall an interesting notion of ℤ-almost automorphic functions, which is introduced recently by Chávez, Castillo, and Pinto [2, 3].

Let BPC(ℝ,𝕏) be the space of all bounded functions f : ℝ→𝕏 satisfying that f is continuous in ℝ \ ℤ with finite lateral limits in ℤ.

Definition 2.7

A function fBPC(ℝ,𝕏) is said to be ℤ-almost automorphic if given any sequence {n} ⊂ ℤ, there exists a subsequence {sn} ⊂ {n} such that f(t):=limnf(t+sn) is well defined for every t ∈ ℝ, and limnf(tsn)=f(t) for every t ∈ ℝ. We denote the set of all such functions by ZAA(ℝ,𝕏).

Example 2.8

It is not difficult to verify that f(t)=sin12+cos([t]+cos2[t],tR is ℤ-almost automorphic.

Stimulated by the notion of ℤ-almost automorphic functions, in the following, we introduce the notion of ℤ-asymptotically almost automorphic functions and study some basic properties of such functions.

Definition 2.9

A function fBPC(ℝ,𝕏) is said to be ℤ-asymptotically almost automorphic iff it admits a decomposition f=g+h, where gZAA(ℝ,𝕏) and h ∈ ℤC0(ℝ,𝕏). Here, ZC0(R,X)={hBPC(R,X):limnh(t)=0}. We denote the set of all such functions byAAA(ℝ,𝕏).

Lemma 2.10

The following assertions hold:

  1. {g(t):tR}{f(t):tR}¯, where f = g + h, g ∈ ℤAA(ℝ,𝕏) and h ∈ ℤ C0(ℝ,𝕏).

  2. The decomposition of every ℤ-asymptotically almost automorphic function is unique.

  3. Let f = g + h, g ∈ ℤAA(ℝ,𝕏) and h ∈ ℤC0(ℝ,𝕏) If f is uniformly continuous on ℝ, then g is also uniformly continuous on ℝ.

  4. fAAA(ℝ,𝕏) implies f([•]) ∈ ℤAAA(ℝ,𝕏).

  5. Let f be uniformly continuous onand f ∈ ℤAAA(ℝ,𝕏). Then fAAA(ℝ,𝕏).

Proof

The assertion (i) follows from the pointwise limits in the definition of ℤAA(ℝ,𝕏) and h ∈ ℤC0(ℝ,𝕏) Then, (i) yields (ii) since 0 has a unique decomposition.

By the definition of ℤAA(ℝ,𝕏), there exists a sequence {sn} ⊂ ℤ with limnsn=, such that limng(t+sn)=g(t),limng(tsn)=g(t),tR.

For every t1, t2 ∈ ℝ, there holds g(t1)g(t2)g(t1)g(t1+sn)+g(t1+sn)g(t2+sn)+g(t2+sn)g(t2)g(t1)g(t1+sn)+f(t1+sn)f(t2+sn)+g(t2+sn)g(t2)+h(t1+sn)+h(t2+sn).

Combining the above observations and h ∈ ℤC0(ℝ,𝕏), in view of the fact that f is uniformly continuous on ℝ, we conclude that is uniformly continuous on ℝ, which yields that g is also uniformly continuous on ℝ. This completes the proof of (iii).

The assertion (iv) follows from the fact that f([t + k]) = f([t] + k) for every t ∈ ℝ and k ∈ ℤ.

Now, let’s come to the proof of (v). Let f = g + h, g ∈ ℤAA(ℝ,𝕏) and h ∈ ℤC0(ℝ,𝕏) Then, by (iii), g and h are both uniformly continuous on ℝ. It is easy to see that hC0(ℝ,𝕏) By [2, Lemma 2.8], gAA(ℝ,𝕏) Thus, fAAA(ℝ,𝕏).   □

Throughout the rest of this paper, we denote 𝔏𝔦𝔭(ℝ×ℂp×ℂp, ℂp) be the set of all functions f : ℝ×ℂp×ℂp → ℂp satisfying the following property: there exists a constant Lf > 0 such that f(t,x,y)f(t,z,w)Lf(xz+yw),tR,(x,y),(z,w)Cp×Cp.

Lemma 2.11

Let fAAA(ℝ × ℂp × ℂpp) ∩ 𝔏𝔦𝔭(ℝ × ℂp × ℂp; ℂp) and ΨAAA(ℝ, ℂp). Then, f(•,Ψ(•),Ψ([•]))∈ℤAAA(ℝ,ℂp).

Proof

Let f=f1+f2ψ=ψ1+ψ2, where f1AA(ℝ × ℂp × ℂp, ℂp), Ψ1 ∈ AA(ℝ, ℂp) and f2 ∈ C0(ℝ × ℂp × ℂp, ℂp), Ψ2 ∈ C0(ℝ ℂp). Then, we have f(t,ψ(t),ψ([t]))=f1(t,ψ1(t),ψ1([t]))+f(t,ψ(t),ψ([t]))f1(t,ψ1(t),ψ1([t]))=f1(t,ψ1(t),ψ1([t]))+f1(t,ψ(t),ψ([t]))f1(t,ψ1(t),ψ1([t]))+f2(t,ψ(t),ψ([t])).

By a similar proof to that of (iii) in Lemma 2.10, one can show that f1 ∈ 𝔏𝔦𝔭(ℝ × ℂp × ℂp; ℂp) with Lipschitz constant Lf. Combining this with [2, Lemma 2,7], we get f1(•,Ψ(•),Ψ([•])) ∈ ℤAA(ℝ, ℂp). In addition, it follows from f1(t,ψ(t),ψ([t]))f1(t,ψ1(t),ψ1([t]))Lf(ψ2(t)+ψ2([t])),tR, that tf1(t,Ψ(t),Ψ([t]))-f1(t,Ψ1,Ψ1([t]))belongs to ℤC0(ℝ,ℂp). Moreover, it is easy to see that f2(•, Ψ(•), Ψ([•])) ∈ ℤC0(ℝ, ℂp). This completes the proof.   □

Next, we recall some notions about discrete almost automorphic type functions (cf. [2, 3]).

Definition 2.12

A function f: ℤ → 𝕏 is said to be discrete almost automorphic if given any sequence {s¯n} ⊂ ℤ, there exists a subsequence {sn} ⊂ {s¯n} such that f(k):=limnf(k+sn) is well defined for every k ∈ ℤ, and limnf(ksn)=f(k) for every k ∈ ℤ. We denote the set of all such functions by AA(ℤ,𝕏).

Definition 2.13

A function G: ℤ × ℤ → 𝕏 is said to be discrete Bi-almost automorphic if given any sequence {s¯n} ⊂ ℤ, there exists a subsequence {sn} ⊂ {s¯n} such that G(k,m):=limnG(k+sn,m+sn) is well defined for every k, m ∈ ℤ, and limnG(ksn,msn)=G(k,m) for every k, m ∈ ℤ. We denote the set of all such functions by BAA(ℤ × ℤ, 𝕏).

Definition 2.14

A function f: ℤ → 𝕏 is said to be discrete asymptotically almost automorphic iff it admits a decomposition f=g+h, where gAA(ℤ,𝕏) and hC0(Z,X):={h:ZX:limkh(k)=0}. Denote the set of all such functions by AAA(ℤ,𝕏).

In order to establish our main results, we introduce the following notion of Bi-asymptotically almost automorphic functions.

Definition 2.15

A function G : ℤ×ℤ→𝕏 is said to be discrete Bi-asymptotically almost automorphic iff it admits a decomposition G=H+I where HBAA(ℤ × ℤ,𝕏) and IBC0(ℤ × ℤ,𝕏). Here, BC0(Z×Z,X)={I:Z×ZX:limnI(n+k,n+m)=0foreveryk,mZ}. We denote the set of all such functions by BAAA(ℤ × ℤ,𝕏).

Remark 2.16

Note that I ∈ BC0(ℤ × ℤ,𝕏) does not imply that limk,mI(k,m)=0. In fact, letting I(k,m)=(km)sinπk+m, for every k, m ∈ ℤ, we have I(n+k,n+m)=(km)sinπk+m+2n0,n. However, I(2m,m)=msinπ3mπ3asm.

Lemma 2.17

The following assertions hold:

  1. Let G ∈ BAAA(ℤ × ℤ,𝕏). Then, for every k, m ∈ ℤ, G(k + •, m + •) is bounded on ℤ.

  2. Let 𝕏 be a Banach algebra and G1, G2 BAAA(ℤ × ℤ,𝕏). Then, G1G2BAAA(ℤ × ℤ,𝕏).

  3. Let G = H + I, where H ∈ BAA(ℤ × ℤ,𝕏) and I ∈ BC0(ℤ × ℤ,𝕏). Then, for every (k, m) ∈ ℤ × ℤ, there holds {H(k+n,m+n):nZ}{G(k+n,m+n):nZ}¯.

Proof

Fix k, m ∈ ℤ. Let G = H + I, where H ∈ BAA(ℤ × ℤ,𝕏) and IBC0(ℤ × ℤ,𝕏). We claim that H(k + •, m + •) is bounded on ℤ. In fact, if this is not true, there exists a sequence {sn} ⊂ ℤ such that limnH(k+sn,m+sn)=.

This contradicts with H ∈ BAA(ℤ × ℤ,𝕏). Moreover, it follows from the definition of BC0(ℤ × ℤ,𝕏) that I(k + •, m + •) is bounded on ℤ. Thus, G(k + •, m + •) is bounded on ℤ.

The proof of (ii) follows from (i) and the definition of BAA(ℤ × ℤ,𝕏).

It remains to show (iii). Fix (k, m) ∈ ℤ × ℤ. By the definition of BAA(ℤ × ℤ,𝕏), there exists {sl} ⊂ ℤ with limtsl= and a function such that limtH(k+n+sl,m+n+sl)=H(k+n,m+n),nZ,(2) and limtH(k+nsl,m+nsl)=H(k+n,m+n),nZ.(3)

It is easy to see from (3) that {H(k+n,m+n):nZ}{H~(k+n,m+n):nZ}¯.

On the other hand, we have ||H(k+n,m+n)G(k+n+sl,m+n+sl)||||H(k+n,m+n)H(k+n+sl,m+n+sl)||+||I(k+n+sl,m+n+sl)||.

Combing this with (2) and I ∈ BC0(ℤ × ℤ,𝕏), we get limtG(k+n+sl,m+n+sl)=H(k+n,m+n),nZ,

which means that {H(k+n,m+n):nZ}{G(k+n,m+n):nZ}¯.

Thus, {H(k+n,m+n):nZ}{G(k+n,m+n):nZ}¯..   □

Before closing this section, we recall and introduce another two notions (the first one has been mentioned in [3]).

Definition 2.18

A function G : ℝ × ℝ → 𝕏 is said to be Bi-almost automorphic if for every {s¯n} ⊂ ℝ, there exists a subsequence {sn} ⊂ {s¯n} such that G(s,t):=limnG(s+sn,t+sn) is well defined for every (s, t) ∈ ℝ2, and limnG(ssn,tsn)=G(s,t) for every (s, t) ∈ ℝ2. We denote the set of all such functions by BAA(ℝ × ℝ,𝕏).

Definition 2.19

A function G : ℝ × ℝ → 𝕏 is said to be Bi-asymptotically almost automorphic iff it admits a decomposition G=H+I, where HBAA(ℝ × ℝ,𝕏) and IBC0(ℝ × ℝ,𝕏). Here, BC0(R×R,X)={I:R×RX:limrI(s+r,t+r)=0,(s,t)R2}.

We denote the set of all such functions by BAAA(ℝ × ℝ,𝕏).

3 Difference equations

To study equation (1), let us first consider the following linear nonhomogeneous DEPCA: y(t)=A(t)y(t)+B(t)y([t])+f(t).(4)

Let y be an arbitrary solution of (4) on ℝ. Then, by the variation of constants formula, there holds, y(t)=Φ(t,n)+ntΦ(t,u)B(u)duy(n)+ntΦ(t,u)f(u)du,t[n,n+1),nZ,(5) where Φ(t, s) : = Φ(t−1(s) and Φ(t) is a fundamental matrix solution of the system x(t)=A(t)x(t). Since y is continuous on ℝ, taking t → (n + 1) in the equation (5), we obtain the difference system y(n+1)=C(n)y(n)+h(n),nZ,(6) where h(n)=nn+1Φ(n+1,u)f(u)du, and C(n)=Φ(n+1,n)+nn+1Φ(n+1,u)B(u)du. From the above observations, naturally, we need to consider the difference system (6). Firstly, we investigate if C(n) and h(n), n ∈ ℤ, are discrete asymptotically almost automorphic when the coefficients A(t), B(t) and f(t), t ∈ ℝ, are asymptotically almost automorphic.

Lemma 3.1

Let A = A1 + A2AAA(ℝ, Mp×p(ℂ)), BAAA(ℝ, Mp×p(ℂ)), and f ∈ ℤAAA(ℝ, ℂp), where A1AA(ℝ, Mp×p(ℂ)) and A2C0(ℝ, Mp×p(ℂ)). Also, let Φ and Φ1 be the fundamental matrix solutions of systems x’(t) = A(t)x(t) and x’(t) = A1(t)x(t), respectively. Moreover, let Φ(t,s)=Φ(t)Φ1(s),Φ1(t,s)=Φ1(t)Φ11(s)andΦ2=ΦΦ1.Then, the following assertions hold:

  1. For every positive real number l, there exists a constant kl > 0 such that Φ(t,s)kl,Φ1(t,s)kl,(t,s)El:={(t,s)R2:tsl}.

  2. Φ1BAA(ℝ × ℝ, Mp×p(ℂ)), Φ2BC0(ℝ × ℝ, Mp×p(ℂ)) and Φ ∈ BAAA(ℝ × ℝ, Mp×p(ℂ)).

  3. nΦ(n+1,n),nnn+1Φ(n+1,u)B(u)duandnnn+1Φ(n+1,u)f(u)du are all discrete asymptotically almost automorphic. Thus, the two functions C(n) and h(n) in equation (6) are both asymptotically almost automorphic.

  4. The functions Φ(t,[t]),[t]tΦ(t,u)B(u)du,[t]tΦ(t,u)f(u)du,are all ℤ-asymptotically almost automorphic.

Proof

The proof of (i) has been essentially given in [3, Lemma 3.2]. Here, for the reader’s convenience, we give a sketch of the proof. For 0 ≤ tsl, by using Gronwall’s inequality and Φ(t,s)=I+stA(u)Φ(u,s)du, where I is the identity matrix in Mp×p(ℂ), one can show that |Φ(t, s)| ≤ |I|el||A||. For -lt ≤ — s ≤ 0, similarly, by using Φ(t,s)=I+stΦ(t,u)A(u)du,

one can also show that |Φ(t, s)| ≤ |I|el||A||. Analogously, one can prove that |Φ1(t, s)| ≤ |I|el||A|| for every (t, s) ∈ El. In addition, by (i) of Lemma 2.4, {A1(t):tR}{A(t):tR}¯, which implies that ||A1| ≤ ||A||, and thus Φ1(t,s)IelA,(t,s)El.

Next, let us prove (ii). Since A1AA(ℝ, Mp×p(ℂ)), for every {n} ⊂ ℝ, there exist a subsequence {sn} ⊂ {n} and a function Ã1, such that limnA1(t+sn)=A~1(t),limnA~(tsn)=A1(t).

Fix s, t ∈ ℝ with ts. Similar to (i), we have Φ1(t,s)=I+stA1(u)Φ1(u,s)du,Φ~1(t,s)=I+tstA~1(u)Φ~1(u,s)du, where Φ̃1(t) is the fundamental matrix solution of system x’(t) = Ã1(t)x(t) and Φ~(t,s)=Φ~1(t)Φ~11(s). Then, by using ||Ã1|| ≤ ||A1 ≤ ||A||, we get Φ1(t+sn,s+sn)Φ~1(t,s)=stA1(u+sn)Φ1(u+sn,s+sn)dustA~1(u)Φ~1(u,s)dustA1(u+sn)A~1(u)Φ1(u+sn,s+sn)du+||A~1||stΦ1(u+sn,s+sn)Φ~1(u,s)duktsstA1(u+sn)A~1(u)du+||A||stΦ1(u+sn,s+sn)Φ~1(u,s)du,

which yields for n → ∞ Φ1(t+sn,s+sn)Φ1(t,s)e(ts)||A||.ktsstA1(u+sn)A1(u)du0.

Analogously to the above proof, one can show that limnΦ~1(tsn,ssn)=Φ1(t,s) and similarly for the case of t < s. This means that Φ1BAA(ℝ × ℝ, Mp×p(ℂ)). It remains to prove that Φ2BAA(ℝ × ℝ, Mp×p(ℂ)). We only consider the case of st. By a direct calculation, we have Φ2(t+r,s+r)=Φ(t+r,s+r)Φ1(t+r,s+r)=I+s+rt+rA(u)Φ(u,s+r)duIs+rt+rA1(u)Φ1(u,s+r)dus+rt+rA2(u)Φ(u,s+r)du+s+rt+rA1(u)Φ2(u,s+r)dukts.(ts).supu[s+r,t+r]A2(u)+||A||stΦ2(u+r,s+r)du,

which means that Φ(t+r,s+r)kts.(ts).supu[s+r,t+r]A2(u).eA(ts)0,r. Therefore, Φ2BC0(R×R,Mp×p(C)), and thus Φ2BAAA(R×R,Mp×p(C)).)

Now, let’s come to the proof of (iii). Firstly, by (ii), it is easy to see that Φ(.+1,.)AAA(Z,Mp×p(C)). The proof for nnn+1Φ(n+1,u)B(u)du is similar to that of nnn+1Φ(n+1,u)f(u)du. So, we only prove nnn+1Φ(n+1,u)f(u)du belongs to AAA(ℤ, ℂp). Observe that nn+1Φ(n+1,u)f(u)du=nn+1Φ1(n+1,u)f1(u)du+nn+1Φ1(n+1,u)f2(u)du+nn+1Φ2(n+1,u)f(u)du, where f=f1+f2,f1ZAA(R,Cp) and f2ZC0(R,Cp). By [3, Lemma 3.3], nnn+1Φ1(n+1,u)f1(u)du belongs to AA(ℤ, Cℂp). By using Lebesgue dominated convergence theorem and (i), we have limnnn+1Φ1(n+1,u)f2(u)du=limn01Φ1(n+1,u+n)f2(u+n)du=0, and limnnn+1Φ2(n+1,u)f(u)du=limn01Φ2(n+1,n+u)f(u+n)du=0. Thus, we conclude that nnn+1Φ(n+1,u)f(u)du is discrete asymptotically almost automorphic.

It remains to prove (iv). It follows from ΦBAAA(R×R,Mp×p(C)) that the functions Φ(t, [t]) is ℤ-asymptotically almost automorphic. As in the proof of (iii), let f=f1+f2,f1ZAA(R,Cp) and f2ZC0(R,Cp). Then, there holds [t]tΦ(t,u)f(u)du=[t]tΦ1(t,u)f1(u)du+[t]tΦ1(t,u)f2(u)du+[t]tΦ2(t,u)f(u)du. Again by [3, Lemma 3.3], we get t[t]tΦ1(t,u)f1(u)du is ℤ-almost automorphic. Moreover, by (i), f2ZC0(R,Cp), and Φ2BC0(R×R,Mp×p(C)), we conclude [t]tΦ1(t,u)f2(u)duk1.sup[t]utf2(u)0,t, and [t]tΦ2(t,u)f(u)dusup[t]utΦ2(t,u).f0,t. Indeed, by a similar proof to that of Φ2BC0(R×R,Mp×p(C)) in (ii), one can obtain Φ2(t,u)ktu.supuvtA2(v).eA(tu)k1sup[t]vtA2(v).eA,[t]ut, which yields that sup[t]utΦ2(t,u)k1eA.sup[t]vtA2(v)0,t.

In conclusion, we know that the function t[t]tΦ(t,u)f(u)du is ℤ-asymptotically almost automorphic. Analogously, one can also obtain that the function t[t]tΦ(t,u)B(u)du is ℤ-asymptotically almost automorphic.   □

Let us recall the notions of exponential dichotomy and Green’s function for homogenous difference system y(n+1)=C(n)y(n),nZ,(7) where C(n) ∈ Mp×p (ℂ) is invertible and y(n) ∈ ℂp, n ∈ ℤ. We will also study asymptotically almost automorphicity for the Green’s function of difference system (7).

Definition 3.2

(cf. [3]). Let Y (n) be a fundamental matrix of difference system (7). The system (7) is said to have an exponential dichotomy with parameters (α, K, P) if there exist a projection P, which commutes with Y(n), and positive constants K, α > 0 such that G(n,m)Keαnm,n,mZ, where G(n,m):=Y(n)PY1(m),nm,Y(n)(IP)Y1(m),n<m, is called the Green’s function of (7).

Before discussing the asymptotically almost automorphicity of Green’s function, we first establish the following lemma, which is an extension of [6, Lemma 2.9].

Lemma 3.3

Let CAAA(ℤ, Mp×p(ℂ)) such that every C(n) is invertible and the set {C−1(n) : n ∈ ℤ} is bounded. Then, C−1AAA(ℤ, Mp×p(ℂ)).

Proof. Let G(n)=C1(n)+C2(n),nZ, where C1AA(ℤ, Mp×p(ℂ)) and C2C0(ℤ, Mp×p(ℂ)). By (iii) of Lemma 2.17, we have C1(n):nZC(n):nZ¯. Then, for every n ∈ ℤ, there exists a sequence {sk} ⊂ ℤ such that C(sk)C1(n),k. Since C(sk) is invertible, the set {C−1(n) : n ∈ ℤ} is bounded, and C1(n)=C(sk)+C1(n)C(sk)=C(sk).(I+C1(sk)[C1(n)C(sk]), we conclude that C1(n) is invertible. Moreover, there holds C1(sk)C11(n)=C11(n)[C1(n)C(sk)C1(sk), which means that C−1(sk) → C1−1(n) as k → ∞, and thus C11(n):nZC1(n):nZ¯. Then, it follows that {C1−1(n) : n ( ℤ} is also bounded, and thus, by [6, Lemma 2.9], nC1−1(n) belongs to AA(ℤ, Mp×p(ℂ)). In addition, by C2C0(ℤ, Mp×p(ℂ)), we have C1(n)C11(n)=C11(n)[C1(n)C(n)]C1(n)=C11(n)C2(n)C1(n)0,n. Then, we conclude that nC−1 (n) belongs to AAA(ℤ, Mp×p(ℂ)).

Lemma 3.4

Let C ∈ AAA (ℤ, M p×p(ℂ)) such that every C(n) is invertible and the set {C−1(n) : n ∈ ℤ} is bounded. Also, assume that system (7) has an exponential dichotomy with parameters (α, K, P). Then, the Green’s function G ∈ BAAA(ℤ × ℤ, Mp×p(ℂ)).

Proof. For every n, m ∈ℤ, we denote Y(n, m) = Y(n)Y−1(m). Then, Y(n,m)=t=mn1C(m+n11),nm,t=nm1C1(l),n<m. Noting that P commutes with Y(n), it suffices to prove that Y(n, m) is Bi-asymptotically almost automorphic.

Let C(n) = C1(n)+C2(n), n ∈ ℤ, where C1AA(ℤ, Mp×p(ℂ)) and C2C0(ℤ, Mp×p(ℂ)). By Lemma 3.3, C−1(n) is also asymptotically almost automorphic and C−1 (n) is just the almost automorphic component of C−1(n). Denote Y1(n,m)=t=mn1C(m+n11),nm,t=nm1C1(l),n<m. Taking an arbitary sequence {̄sk}⊂ ℝ, there exist a subsequence {sk}, and two functions D, E such that limkC1(n+sk)=D(n),limkD(nsk)=C1(n),nZ, and limkC11(n+sk)=E(n),limkE(nsk)=C11(n),nZ. Denote Y1˜(n,m)={t=mn1D(m+n11),nm,t=nm1E(l),n<m. Fix n, m ∈ ℤ with n > m. We have Y1(n+sk,m)+sk)=l=m+skn+sk1C1(m+n+2sk11)=l=mn1C1(m+n1l+sk)l=mn1D(m+n1l)=Y1~(n,m), as k → 1. Analogously, one can show that Y1~(nsk,msk)Y1(n,m),k.

The proof for the case of n < m is fully similar to the above proof and the proof for the case of m = n is trivial. This shows that Y1BAA(ℤ × ℤ, Mp×p(ℂ)).

Moreover, we claim that [YY1] ∈ BC0(ℤ × ℤ, Mp×p(ℂ)). In fact, for n, m ∈ ℤ with n > m, we have Y(n+k,m+k)Y1(n+k,m+k)=t=m+kn+k1C(m+n+2k1l)t=m+kn+k1C1(m+n+2k1l)=l=mn1C(m+n+k1l)l=mn1C1(m+n1l+k)0,k, since C(n) and C1 (n) are both bounded, and as k → ∞, C(m+n+k1l)C1(m+n1l+k)=C2(m+n1l+k)0,l=m,m+1,...,n1. This shows that YBAAA(ℤZ × ℤ, Mp×p(ℂ)).

4 AAA solutions

In this section, we discuss the existence of asymptotically almost automorphic solutions for difference system (6) and DEPCA (1).

Theorem 4.1

Let CAAA(ℤ, Mp×p(ℂ)) such that every C(n) is invertible and the set {C−1(n) : n ∈ ℤ} is bounded. Also, assume that hAAA(ℤ, ℂp) and system (7) has an exponential dichotomy with parameters (α, K, P). Then, system (6) has a unique asymptotically almost automorphic solution given by y(n)=kZG(n,k+1)h(k),nZ. Moreover y(n)K(1+eα)(1eα)1hforallnZ.

Proof. By [10, Theorem 5.7], system (6) has a unique bounded solution y(n) given by y(n)=kZG(n,k+1)h(k),nZ. It follows from Lemma 3.4 that GBAAA(ℤ × ℤ, Mp×p(ℂ)). Thus, we have the following decomposition: y(n)=kZG(n,k+1)h(k)=kZG1(n,k+1)h1(k)+kZG1(n,k+1)h2(k)+kZG2(n,k+1)h(k), where G = G1+G2, h = h1+h2, G1BAA(ℤ×ℤ, Mp×p(ℂ)), h1AA(ℤ, ℂp), G2BC0(ℤ×ℤ, Mp×p(ℂ)), h2C0(ℤ, ℂp).

It follows from [2, Theorem 3.4] (see also [6, Theorem 3.1]) that kZG1(,k+1)h1(k)AA(Z,Cp). There holds kZG1(n,k+1)h2(k)=k=n1G1(n,k+1)h2(k)+k=n+G1(n,k+1)h2(k)=k=0G1(n,k+n)h2(k+n1)+k=1+G1(n,k+n1). By using the Lebesgue dominated convergence theorem, h2C0(ℤ, ℂp), and G1(n,k+n)Keαk,k,nZ, which is deduced from G1(n,k+n):nZG(n,k+n):nZ¯ (see (iii) of Lemma 2.17), we conclude limnk=0G1(n,k+n)h2(k+n1)=limnk=1+G1(n,k+n)h2(k+n1)=0, which yields that kZG1(.,k+1)h2(k)C0(Z,Cp).

It remains to show that kZG2(.,k+1)h(k)C0(Z,Cp). Noting that kZG2(n,k+1)h(k)=kZG2(n,k+n)h(k+n1), for every limnG2(n,k+n=0) since G2BC0(Z×Z,Mp×p(C)), and G2(n,k+n)=G(n,k+n)G1(n,k+n)2Keαk,k,nZ, again by the Lebesgue dominated convergence theorem, we get limnkZG2(n,k+1)h(k)=0. This completes the proof. □

Theorem 4.2

Let A,BAAA(R,Mp×p(C)),fAAA(R×Cp×Cp,Cp)Lip(R×Cp×Cp,Cp), and I+nn+1Φ(n,u)B(u)du1nZ be bounded, where Φ(t, s) := Φ(t−1(s) and Φ(t) is a fundamental matrix solution of x(t) = A(t)x(t). Also, assume that the system y(n + 1) = C(n)y(n) has an exponential dichotomy with parameters (α, K, P), where C(n)=Φ(n+1,n)+nn+1Φ(n+1,u)B(u)du. Then, there exists a constant L* > 0, such that equation (1) has a unique asymptotically almost automorphic solution provided Lf < L*.

Proof. Taking arbitrary ψ ∈ AAA(ℝ, ℂ p), consider the following equation y(t)=A(t)y(t)+B(t)y([t])+f(t,ψ(t),ψ([t])).(8)

By the arguments in the beginning of Section 3, we know that yψ is a solution of equation (8) if and only if for every n ∈ ℤ and t ∈ [n, n + 1), yΨ(t)=Φ(t,n)+ntΦ(t,u)B(u)duyΨ(n)+ntΦ(t,u)f(u,Ψ([u]))du,(9) and yΨ(n+1)=C(n)yΨ(n)+hΨ(n),(10) where hΨ(n)=nn+1Φ(n+1,u)f(u,Ψ(u),Ψ([u]))du,nZ.

Since fAAA(R×Cp×Cp,Cp)Lip(R×Cp×Cp,Cp),. by Lemma 2.11 f(., ψ(.), ψ([.])) ∈ ℤAAA(ℝ, ℂp). Then, by (iii) of Lemma 3.1, hψAAA(ℤ, ℂp) and CAAA(ℤ, Mp×p(ℂ)). Note that for every nZ,I+nn+1Φ(n,u)B(u)du being invertible implies that C(n)=Φ(n+1,n)+nn+1Φ(n+1,u)B(u)du=Φ(n+1,n)I+nn+1Φ(n,u)B(u)du is also invertible. Moreover, we have C1(n)=I+nn+1Φ(n,u)B(u)du1Φ(n,n+1), which means that {C−1(n)}n∈ℤ is also bounded. Then, it follows from Theorem 4.1 that equation (10) has a unique asymptotically almost automorphic solution given by yΨ(n)=kZG(n,k+1)hΨ(k),nZ.

Then, defining yψ by (9), we get a solution yψ(t) of equation (8). Also, it is easy to see that yψ(t) is the unique solution of equation (8).

Next, let us show that for every ΨAAA(R,Cp),yΨAAA(R,Cp).. Observe that yΨ([.])ZAAA(R,Cp),f(.,Ψ(.),Ψ([.]))(R,Cp), and yΨ(t)=Φ(t,[t])+[t]tΦ(t,u)B(u)duyΨ([t])+[t]tΦ(t,u)f(u,Ψ(u),Ψ([u]))du,tR.(11) It follows from (iv) of Lemma 3.1 that every term on the right-hand side of (11) belongs to ZAAA(, ℂp), and thus yψZAAA(, ℂp). On the other hand, since yψ is continuous on ℝ and yΨ(t)=A(t)yΨ(t)+B(t)yΨ([t])+f(t,Ψ(t),Ψ([t])),t(n,n+1),nZ, where every term on the right-hand side of the above equality is bounded on ℝ, we conclude (cf. [2, Lemma 4.1]) that yψ(t) is uniformly continuous on R. Then, by (v) of Lemma 2.10, yψAAA(ℝ, ℂp).

Now, let us show that equation (1) has a unique asymptotically almost automorphic solution. Define a mapping 𝓜 : AAA(ℝ, ℂp) → AAA(ℝ, ℂp) by (MΨ)(t)=yΨ(t),tR,ΨAAA(R,Cp) From the above proof, 𝓜 is well-defined. For everψ1,ψ2AAA(ℝ,ℂp) and n ∈ ℤ, there holds yΨ1(n)yΨ2(n)=kZG(n,k+1)hΨ1(k)kZG(n,k+1)hΨ2(k)K(1+eα)1eαhΨ1hΨ2K(1+eα)1eα.supnZnn+1Φ(n+1,u).f(u,Ψ1(u),Ψ1(n))f(u,Ψ2(u),Ψ2(n))du2k1LfK(1+eα)1eα.Ψ1Ψ2,

which yields that for every n ∈ ℤ and t ∈ [n, n + 1), there holds (Mψ1)(t)(Mψ2)(t)Φ(t,n)+ntΦ(t,u)B(u)du.yψ1(n)yψ2(n)+ntΦ(t,u).f(u,ψ1(u),ψ1([u]))f(u,ψ2(u),ψ2([u]))duk1(1+||B||).yψ1(n)yψ2(n)+2k1Lf.||ψ1ψ2||2k12LfK(1+eα)(1+||B||)1eα+k1Lf.||ψ1ψ2||.

Thus we have ||Mψ1Mψ2||Lf||ψ1ψ2||L,

where L=1eα2k12K(1+eα)(1+||B||)+2k1(1eα). This means that in the case Lf < L*, 𝓜 has a unique fixed point in AAA(ℝ,ℂp, i.e., equation (1) has a unique asymptotically almost automorphic solution.   □

Remark 4.3

It is not difficult to give some sufficient conditions to ensure that the assumptions of Theorem 4.2 hold. In fact, if ||B|| is sufficiently small, then I+nn+1Φ(n,u)B(u)du1=n=0+nn+1Φ(n,u)B(u)dun is well-defined and bounded for n ∈ ℤ. If, in addition supnZΦ(n+1,n)<1, then supnZ|C(n)|supnZΦ(n+1,n).supnZI+nn+1Φ(n,u)B(u)du<1, which means that the system (7) is exponentially stable (i.e., exponential dichotomy with P = I).

Acknowledgement

The authors are grateful to the anonymous reviewer for his/her careful reading and valuable suggestions.

The work was partially supported by NSFC (11461034), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), the NSF of Jiangxi Province (20143ACB21001), and the Foundation of Jiangxi Provincial Education Department (GJJ150342).

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About the article

Received: 2016-08-19

Accepted: 2017-02-28

Published Online: 2017-05-11


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 595–610, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0051.

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© 2017 Ding and Wan. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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