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Advances in Difference Equations

, 2012:199 | Cite as

Asymptotically almost periodic solution to a class of Volterra difference equations

  • Wei LongEmail author
  • Wen-Hai Pan
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

This paper is concerned with an asymptotically almost periodic solution to a class of Volterra-type difference equations. We establish a compactness criterion for the sets of asymptotically almost periodic sequences. Then, by using the compactness criterion and Schauder’s fixed point theorem, we present an existence theorem for an asymptotically almost periodic solution to the addressed Volterra-type difference equation. Our existence theorem extends and complements a recent result due to (Ding et al. in Electron. J. Qual. Theory Differ. Equ. 6:1-13, 2012).

MSC:39A24, 34K14.

Keywords

asymptotically almost periodic Volterra difference equation 

1 Introduction and preliminaries

In this paper, we consider the following nonlinear Volterra-type difference equation:
x ( n ) = i = 1 λ [ f i ( n , x ( n ) ) m Z a i ( n , m ) g i ( m , x ( m ) ) ] , n Z , Open image in new window
(1.1)

where λ is a fixed positive integer and f i : Z × R R Open image in new window, a i : Z × Z R Open image in new window, g i : Z × R R Open image in new window ( i = 1 , 2 , , λ Open image in new window) satisfy some conditions recalled in Section 3.

For the background of discrete Volterra equations, we refer the reader to the well-known monograph [1] by Agarwal. The first motivation for this paper is some recent work on asymptotical periodicity for Volterra-type difference equations in [2, 3, 4, 5, 6] by Diblík et al. In fact, asymptotical behavior for Volterra-type difference equations, including periodicity, asymptotical periodicity, etc., has been of great interest for many mathematicians. However, to the best of our knowledge, there is seldom literature available about asymptotically almost periodicity for Equation (1.1). Thus, in this paper, we will investigate this problem. In addition, it is needed to note that compared with asymptotically periodic sequences, in general, it is more difficult to obtain the compactness for a set of asymptotically almost periodic sequences.

On the other hand, in a recent work [7], by using the classical Schauder fixed point theorem, Ding et al. established an interesting existence theorem for the following functional integral equation:
y ( t ) = e ( t , y ( α ( t ) ) ) + g ( t , y ( β ( t ) ) ) [ h ( t ) + R k ( t , s ) f ( s , y ( γ ( s ) ) ) d s ] , t R . Open image in new window
(1.2)

In fact, the existence of almost periodic type solutions has been an interesting and important topic in the study of qualitative theory of difference equations. We refer the reader to [8, 9, 10, 11, 12, 13] and references therein for some recent developments on this topic. Equation (1.1) can be seen as a discrete analogue (but more general) of Equation (1.2). That is another main motivation for this work.

Throughout the rest of this paper, we denote by ℤ ( Z + Open image in new window) the set of (nonnegative) integers, by ℕ the set of positive integers, by ℝ ( R + Open image in new window) the set of (nonnegative) real numbers, by Ω a subset of ℝ, and by X a Banach space.

First, let us recall some notations and basic results of almost periodic type sequences (for more details, see [11, 14, 15]).

Definition 1.1 [14]

A function f : Z X Open image in new window is called almost periodic if ∀ε, N ( ε ) N Open image in new window such that among any N ( ε ) Open image in new window consecutive integers there exists an integer p with the property that
f ( k + p ) f ( k ) < ε , k Z . Open image in new window

Denote by AP ( Z , X ) Open image in new window the set of all such functions. Moreover, we denote AP ( Z , R ) Open image in new window by AP ( Z ) Open image in new window for convenience.

Lemma 1.2 [[14], Theorem 1.26]

A necessary and sufficient condition for the sequence f : Z R Open image in new window to be almost periodic is that for any integer sequence { n k } Open image in new window, one can extract a subsequence { n k } Open image in new window such that { f ( n + n k ) } Open image in new window converges uniformly with respect to n Z Open image in new window.

Remark 1.3 Let f , g AP ( Z ) Open image in new window. By Lemma 1.2, it is not difficult to show that ∀ε, N ( ε ) N Open image in new window such that among any N ( ε ) Open image in new window consecutive integers there exists a common integer p with the property that
| f ( k + p ) f ( k ) | < ε and | g ( k + p ) g ( k ) | < ε Open image in new window

for all k Z Open image in new window.

Next, we denote by C 0 ( Z , X ) Open image in new window the space of all the functions f : Z X Open image in new window such that lim | n | f ( n ) = 0 Open image in new window.

Definition 1.4 A function f : Z X Open image in new window is called asymptotically almost periodic if it admits a decomposition f = g + h Open image in new window, where g AP ( Z , X ) Open image in new window and h C 0 ( Z , X ) Open image in new window. Denote by AAP ( Z , X ) Open image in new window the set of all such functions. Moreover, we denote AAP ( Z , R ) Open image in new window by AAP ( Z ) Open image in new window for convenience.

Definition 1.5 Let Ω R Open image in new window and f be a function from Z × Ω Open image in new window to ℝ such that f ( n , ) Open image in new window is continuous for each n Z Open image in new window. Then f is called almost periodic in n Z Open image in new window uniformly for ω Ω Open image in new window if for every ε > 0 Open image in new window and every compact Σ Ω Open image in new window, there corresponds an integer N ε ( Σ ) > 0 Open image in new window such that among N ε ( Σ ) Open image in new window consecutive integers there exists an integer p with the property that
| f ( k + p , ω ) f ( k , ω ) | < ε Open image in new window

for all k Z Open image in new window and ω Σ Open image in new window. Denote by AP ( Z × Ω ) Open image in new window the set of all such functions.

Similarly, for each subset Ω R Open image in new window, we denote by C 0 ( Z × Ω ) Open image in new window the space of all the functions f : Z × Ω R Open image in new window such that f ( n , ) Open image in new window is continuous for each n Z Open image in new window, and lim | n | f ( n , x ) = 0 Open image in new window uniformly for x in any compact subset of Ω.

Definition 1.6 A function f : Z × Ω R Open image in new window is called asymptotically almost periodic in n uniformly for x Ω Open image in new window if it admits a decomposition f = g + h Open image in new window, where g AP ( Z × Ω ) Open image in new window and h C 0 ( Z × Ω ) Open image in new window. Denote by AAP ( Z × Ω ) Open image in new window the set of all such functions.

Lemma 1.7 Let E { AP ( Z , X ) , AAP ( Z , X ) } Open image in new window. Then the following hold true:
  1. (a)

    f E Open image in new window implies that f is bounded.

     
  2. (b)

    f , g E Open image in new window implies that f + g E Open image in new window. Moreover, f g E Open image in new window if X = R Open image in new window.

     
  3. (c)

    E is a Banach space equipped with the supremum norm.

     

Proof The proof is similar to that of the continuous case (cf. [14, 15]). So, we omit the details. □

2 A compactness criterion

The following theorem is a well-known result for the continuous case (see, e.g., [[16], p.24, Theorem 2.5]). Here, we give a discrete version.

Theorem 2.1 Let f be a function fromto ℝ. Then f AAP ( Z ) Open image in new window if and only ifε, M ( ε ) , N ( ε ) N Open image in new window such that among any N ( ε ) Open image in new window consecutive integers there exists an integer p with the property that
| f ( k + p ) f ( k ) | < ε Open image in new window

for all k Z Open image in new window with | k | M ( ε ) Open image in new window and | k + p | M ( ε ) Open image in new window.

Proof We first show the ‘only if’ part. Let f AAP ( Z ) Open image in new window. Then there exist g AP ( Z ) Open image in new window and h C 0 ( Z , R ) Open image in new window such that f = g + h Open image in new window. By g AP ( Z ) Open image in new window, for each ε > 0 Open image in new window, N ( ε ) N Open image in new window such that among any N ( ε ) Open image in new window consecutive integers there exists an integer p with the property that
| g ( k + p ) g ( k ) | < ε 3 , k Z . Open image in new window
In addition, since h C 0 ( Z , R ) Open image in new window, for the above ε > 0 Open image in new window, there exists M ( ε ) N Open image in new window such that | h ( k ) | < ε 3 Open image in new window for all k Z Open image in new window with | k | M ( ε ) Open image in new window. Thus, we have
| f ( k + p ) f ( k ) | | g ( k + p ) g ( k ) | + | h ( k + p ) | + | h ( k ) | < ε Open image in new window

for all k Z Open image in new window with | k | M ( ε ) Open image in new window and | k + p | M ( ε ) Open image in new window.

Next, let us prove the ‘if’ part. First, let us show that f is bounded. Letting ε = 1 Open image in new window, there exists M ( 1 ) , N ( 1 ) N Open image in new window such that among any N ( 1 ) Open image in new window consecutive integers there exists an integer p with the property that
| f ( k + p ) f ( k ) | < 1 Open image in new window
for all k Z Open image in new window with | k | M ( 1 ) Open image in new window and | k + p | M ( 1 ) Open image in new window. Then, for each k Z Open image in new window with | k | M ( 1 ) Open image in new window, there exists p k [ M ( 1 ) k , M ( 1 ) + N ( 1 ) k ] Z Open image in new window such that
| f ( k + p k ) f ( k ) | < 1 . Open image in new window
Noting that k + p k [ M ( 1 ) , M ( 1 ) + N ( 1 ) ] Open image in new window, we get
| f ( k ) | | f ( k + p k ) | + 1 max k [ M ( 1 ) , M ( 1 ) + N ( 1 ) ] | f ( k ) | + 1 Open image in new window
for all k Z Open image in new window with | k | M ( 1 ) Open image in new window. Thus,
sup k Z | f ( k ) | max k [ M ( 1 ) , M ( 1 ) + N ( 1 ) ] | f ( k ) | + 1 < + . Open image in new window

Now, let us show that f AAP ( Z ) Open image in new window. We divide the remaining proof into three steps.

Step 1. Since f is bounded, we can choose a sequence { s n } N Open image in new window such that lim n + s n = + Open image in new window and lim n + f ( k + s n ) Open image in new window exists for each k Z Open image in new window. Let
g ¯ ( k ) = lim n + f ( k + s n ) , k Z . Open image in new window
For each ε > 0 Open image in new window, among any N ( ε ) Open image in new window consecutive integers there exists an integer p with the property that
| f ( k + p ) f ( k ) | < ε Open image in new window
for all k Z Open image in new window with | k | M ( ε ) Open image in new window and | k + p | M ( ε ) Open image in new window. Then, for each fixed k Z Open image in new window, we have
| f ( k + s n + p ) f ( k + s n ) | < ε Open image in new window
for sufficiently large n, which yields that
| g ¯ ( k + p ) g ¯ ( k ) | ε . Open image in new window

Thus, g ¯ AP ( Z ) Open image in new window.

Step 2. Now fix ε > 0 Open image in new window. Then, for each n N Open image in new window, there exists t n [ s n N ( ε ) , s n ] Z Open image in new window such that
| f ( k + t n ) f ( k ) | < ε Open image in new window
(2.1)
for all k Z Open image in new window with | k | M ( ε ) Open image in new window and | k + t n | M ( ε ) Open image in new window. Let r n = s n t n Open image in new window. Then r n { 0 , 1 , 2 , , N ( ε ) } Open image in new window, which means that there exist a subsequence { r n } { r n } Open image in new window and r ( ε ) { 0 , 1 , 2 , , N ( ε ) } Open image in new window such that
r n r ( ε ) . Open image in new window
Thus, for all k Z Open image in new window with | k | M ( ε ) Open image in new window, we have
| f ( k ) g ¯ ( k r ( ε ) ) | = | f ( k ) f ( k + t n ) | + | f ( k r ( ε ) + s n ) g ¯ ( k r ( ε ) ) | . Open image in new window
Combining this with (2.1), lim n + t n = + Open image in new window, and
g ¯ ( k ) = lim n + f ( k + s n ) , k Z , Open image in new window
we conclude
| f ( k ) g ¯ ( k r ( ε ) ) | ε Open image in new window

for all k Z Open image in new window with | k | M ( ε ) Open image in new window.

Step 3. By Step 2, we know that for each ε > 0 Open image in new window, there exists r ( ε ) { 0 , 1 , 2 , , N ( ε ) } Open image in new window such that
| f ( k ) g ¯ ( k r ( ε ) ) | ε Open image in new window
for all k Z Open image in new window with | k | M ( ε ) Open image in new window. Taking ε = 1 , 1 / 2 , Open image in new window , we get a sequence { r ( 1 / m ) } Open image in new window. On the other hand, it follows from Step 1 that g ¯ AP ( Z ) Open image in new window. Thus, going to a subsequence, if necessary, we may assume that g ¯ ( r ( 1 / m ) ) Open image in new window is uniformly convergent on ℤ. Let
g ( k ) = lim m + g ¯ ( k r ( 1 / m ) ) , k Z . Open image in new window
Then g AP ( Z ) Open image in new window. In addition, noting that
| f ( k ) g ( k ) | | f ( k ) g ¯ ( k r ( 1 / m ) ) | + | g ¯ ( k r ( 1 / m ) ) g ( k ) | 1 m + | g ¯ ( k r ( 1 / m ) ) g ( k ) | Open image in new window

for all k Z Open image in new window with | k | M ( 1 / m ) Open image in new window, we know that f g C 0 ( Z ) Open image in new window. This completes the proof. □

Definition 2.2 F AAP ( Z ) Open image in new window is said to be equi-asymptotically almost periodic if for each ε > 0 Open image in new window, there exist M ( ε ) , N ( ε ) N Open image in new window such that among any N ( ε ) Open image in new window consecutive integers there exists an integer p with the property that
sup f F | f ( k + p ) f ( k ) | < ε Open image in new window

for all k Z Open image in new window with | k | M ( ε ) Open image in new window and | k + p | M ( ε ) Open image in new window.

Theorem 2.3 Let F AAP ( Z ) Open image in new window. Then F is precompact in AAP ( Z ) Open image in new window if and only if the following two conditions hold:
  1. (i)

    for each k Z Open image in new window, { f ( k ) : f F } Open image in new window is bounded;

     
  2. (ii)

    F is equi-asymptotically almost periodic.

     

Proof ‘only if’ part

Let F AAP ( Z ) Open image in new window be precompact. Then F is bounded in AAP ( Z ) Open image in new window. So, (i) obviously holds. In addition, ε > 0 Open image in new window, there exists N N Open image in new window and f 1 , f 2 , , f N F Open image in new window such that
F i = 1 N B ( f i , ε ) . Open image in new window
(2.2)

By Remark 1.3, we can get that { f 1 , f 2 , , f N } Open image in new window is equi-asymptotically almost periodic. Combing this with (2.2), we can show that F is equi-asymptotically almost periodic, i.e., (ii) holds.

‘if part’

Let { f n } F Open image in new window. Since { f n ( k ) } Open image in new window is bounded for each k Z Open image in new window, we can assume that (if necessary going to a subsequence) { f n ( k ) } Open image in new window is convergent for each k Z Open image in new window. On the other hand, since F is equi-asymptotically almost periodic, for each ε > 0 Open image in new window, there exist M ( ε ) , N ( ε ) N Open image in new window such that among any N ( ε ) Open image in new window consecutive integers there exists an integer p with the property that
sup n Z | f n ( k + p ) f n ( k ) | < ε / 3 Open image in new window
(2.3)
for all k Z Open image in new window with | k | M ( ε ) Open image in new window and | k + p | M ( ε ) Open image in new window. For the above ε > 0 Open image in new window, there exists a positive integer K such that for all n , m > K Open image in new window, the following hold:
| f n ( k ) f m ( k ) | < ε / 3 , k [ M ( ε ) , M ( ε ) + N ( ε ) ] Z . Open image in new window
(2.4)
For all k Z Open image in new window with | k | M ( ε ) Open image in new window, taking p [ k + M ( ε ) , k + M ( ε ) + N ( ε ) ] Z Open image in new window, by (2.3) and (2.4), we get
| f n ( k ) f m ( k ) | | f n ( k ) f n ( k + p ) | + | f n ( k + p ) f m ( k + p ) | + | f m ( k + p ) f m ( k ) | < ε / 3 + ε / 3 + ε / 3 = ε , n , m > K ; Open image in new window
also, for all k Z Open image in new window with | k | < M ( ε ) Open image in new window, by (2.4), we have
| f n ( k ) f m ( k ) | < ε / 3 < ε , n , m > K . Open image in new window
Thus, we get
sup k Z | f n ( k ) f m ( k ) | ε , n , m > K , Open image in new window

which means that { f n ( k ) } Open image in new window is uniformly convergent on ℤ, i.e., { f n } Open image in new window is convergent in AAP ( Z ) Open image in new window. So, F is precompact in AAP ( Z ) Open image in new window. □

3 Application to Volterra difference equations

In this section, we discuss the existence of an asymptotically almost periodic solution to Volterra difference equation (1.1). Throughout the rest of this paper, p , q 1 Open image in new window are two fixed real numbers and
1 p + 1 q = 1 . Open image in new window
In addition, we denote by l p ( Z ) Open image in new window (resp. l q ( Z ) Open image in new window) the space of all the functions f : Z R Open image in new window satisfying
f p : = ( k Z | f ( k ) | p ) 1 / p < + ( resp.  f q : = ( k Z | f ( k ) | q ) 1 / q < + ) . Open image in new window

For convenience, we first list some assumptions.

(H1) For each i { 1 , 2 , , λ } Open image in new window, f i ( , x ) AAP ( Z ) Open image in new window for any fixed x R Open image in new window, and there exists a constant L i 0 Open image in new window such that
| f i ( k , x ) f i ( k , y ) | L i | x y | , k Z , x , y R . Open image in new window
(H2) For each i { 1 , 2 , , λ } Open image in new window, g i ( k , ) Open image in new window is continuous for each k Z Open image in new window, and for each r > 0 Open image in new window, there exists a sequence { μ i r } l p ( Z ) Open image in new window such that
| g i ( k , x ) | μ i r ( k ) , | x | r , k Z . Open image in new window

(H3) For each i { 1 , 2 , , λ } Open image in new window, a ˜ i AAP ( Z , l q ( Z ) ) Open image in new window, where [ a ˜ i ( k ) ] ( l ) = a i ( k , l ) Open image in new window, k , l Z Open image in new window.

(H4) There exists a constant M > 0 Open image in new window such that
i = 1 λ α i L i μ i M p < 1 , Open image in new window
where α i = sup n Z a ˜ i ( n ) q Open image in new window; and
i = 1 λ [ sup n Z , | x | K | f i ( n , x ) | α i μ i M p ] < K , K > M , Open image in new window

Theorem 3.1 Assume that (H1)-(H4) hold. Then Equation (1.1) has an asymptotically almost periodic solution.

Proof We denote
( A i x ) ( n ) = f i ( n , x ( n ) ) , n Z , x AAP ( Z ) , i = 1 , 2 , , λ ; ( B i x ) ( n ) = m Z a i ( n , m ) g i ( m , x ( m ) ) , n Z , x AAP ( Z ) , i = 1 , 2 , , λ ; Open image in new window
and
( M x ) ( n ) = i = 1 λ ( A i x ) ( n ) ( B i x ) ( n ) , n Z , x AAP ( Z ) . Open image in new window

It suffices to prove that ℳ has a fixed point in AAP ( Z ) Open image in new window. We give the proof in three steps.

Step 1. A i Open image in new window and B i Open image in new window both map AAP ( Z ) Open image in new window into AAP ( Z ) Open image in new window, i = 1 , 2 , , λ Open image in new window.

Since f i Open image in new window is Lipschitz, by Remark 1.3, we can first show that for each compact subset K R Open image in new window and each i { 1 , 2 , , λ } Open image in new window, { f i ( , x ) : x K } Open image in new window is equi-asymptotically almost periodic. Then it is easy to show that A i x AAP ( Z ) Open image in new window for each x AAP ( Z ) Open image in new window.

Since a ˜ i AAP ( Z , l q ( Z ) ) Open image in new window, there exist b i AP ( Z , l q ( Z ) ) Open image in new window and c i C 0 ( Z , l q ( Z ) ) Open image in new window such that a ˜ i = b i + c i Open image in new window. For each x AAP ( Z ) Open image in new window, noting that for n , p Z Open image in new window,
and
| m Z [ c i ( n ) ] ( m ) g i ( m , x ( m ) ) | c i ( n ) q μ i x p , Open image in new window

we know that B i x AAP ( Z ) Open image in new window.

Step 2. For each y AAP ( Z ) Open image in new window with y M Open image in new window, there exists a unique x y AAP ( Z ) Open image in new window such that
x y = i = 1 λ A i x y B i y . Open image in new window
Let
( Y x ) ( n ) = i = 1 λ ( A i x ) ( n ) ( B i y ) ( n ) , n Z , x AAP ( Z ) . Open image in new window
Then, by Step 1, Y Open image in new window maps AAP ( Z ) Open image in new window into AAP ( Z ) Open image in new window. For all x 1 , x 2 AAP ( Z ) Open image in new window and n Z Open image in new window, we have
| ( Y x 1 ) ( n ) ( Y x 2 ) ( n ) | i = 1 λ | ( A i x 1 ) ( n ) ( A i x 2 ) ( n ) | | ( B i y ) ( n ) | = i = 1 λ | f i ( n , x 1 ( n ) ) f i ( n , x 2 ( n ) ) | | ( B i y ) ( n ) | i = 1 λ L i | x 1 ( n ) x 2 ( n ) | | m Z a i ( n , m ) g i ( m , y ( m ) ) | i = 1 λ L i x 1 x 2 | m Z a i ( n , m ) g i ( m , y ( m ) ) | i = 1 λ L i x 1 x 2 m Z | [ a ˜ i ( n ) ] ( m ) | μ i M ( m ) i = 1 λ L i x 1 x 2 a ˜ i ( n ) q μ i M p ( i = 1 λ α i L i μ i M p ) x 1 x 2 , Open image in new window
which yields that
Y x 1 Y x 2 ( i = 1 λ α i L i μ i M p ) x 1 x 2 . Open image in new window

Noting that i = 1 λ α i L i μ i M p < 1 Open image in new window, Y Open image in new window has a unique fixed point x y Open image in new window in AAP ( Z ) Open image in new window.

Step 3. ℳ has a fixed point in AAP ( Z ) Open image in new window.

Let E = { y AAP ( Z ) : y M } Open image in new window and
N y = x y , y E , Open image in new window

where x y Open image in new window is the unique fixed point of Y Open image in new window (see Step 2).

We claim that N ( E ) E Open image in new window. In fact, if there exists y 0 E Open image in new window such that N y 0 > M Open image in new window, then by (H4), we have
N y 0 = x y 0 = sup n Z | i = 1 λ ( A i x y 0 ) ( n ) ( B i y 0 ) ( n ) | sup n Z ( i = 1 λ | f i ( n , x y 0 ( n ) ) | | m Z a i ( n , m ) g i ( m , y 0 ( m ) ) | ) sup n Z ( i = 1 λ | f i ( n , x y 0 ( n ) ) | α i μ i M p ) i = 1 λ [ sup n Z , | x | N y 0 | f i ( n , x ) | α i μ i M p ] < N y 0 , Open image in new window

which is a contradiction.

Next, let us show that N : E E Open image in new window is continuous. For all y 1 , y 2 E Open image in new window, we have
N y 1 N y 2 = x y 1 x y 2 = i = 1 λ A i x y 1 B i y 1 i = 1 λ A i x y 2 B i y 2 i = 1 λ A i x y 1 B i y 1 A i x y 2 B i y 1 + A i x y 2 B i y 1 A i x y 2 B i y 2 ( i = 1 λ α i L i μ i M p ) x y 1 x y 2 + i = 1 λ ( M L i + sup n Z | f i ( n , 0 ) | ) B i y 1 B i y 2 , Open image in new window
which gives that
N y 1 N y 2 i = 1 λ β i B i y 1 B i y 2 , Open image in new window
(3.1)
where
β i : = M L i + sup n Z | f i ( n , 0 ) | 1 ( i = 1 λ α i L i μ i M p ) , i = 1 , 2 , , λ . Open image in new window
Letting y k y Open image in new window in E, by (3.1), we have
N y k N y i = 1 λ β i B i y k B i y i = 1 λ β i sup n Z ( m Z | a i ( n , m ) | | g i ( m , y k ( m ) ) g i ( m , y ( m ) ) | ) i = 1 λ β i sup n Z ( a ˜ i ( n ) q g i ( , y k ( ) ) g i ( , y ( ) ) p ) i = 1 λ α i β i g i ( , y k ( ) ) g i ( , y ( ) ) p . Open image in new window
(3.2)
For each i = 1 , 2 , , λ Open image in new window, noting that
| g i ( m , y k ( m ) ) g i ( m , y ( m ) ) | 2 μ i M ( m ) , m Z , Open image in new window
g i ( m , ) Open image in new window is continuous for each m Z Open image in new window, and y k ( m ) y ( m ) Open image in new window for each m Z Open image in new window, we conclude that
g i ( , y k ( ) ) g i ( , y ( ) ) p 0 . Open image in new window

Combining this with (3.2), we know that N y k N y Open image in new window. So N : E E Open image in new window is continuous.

Now, let us show that N ( E ) Open image in new window is precompact in AAP ( Z ) Open image in new window. In order to show that, we first prove each B i ( E ) Open image in new window is precompact in AAP ( Z ) Open image in new window. By a direct calculation, we can get
| ( B i y ) ( n ) | α i μ i M p , i = 1 , 2 , , λ , Open image in new window
for all y E Open image in new window and n Z Open image in new window. In addition, for all n 1 , n 2 Z Open image in new window and y E Open image in new window, we have
| ( B i y ) ( n 1 ) ( B i y ) ( n 2 ) | m Z | a i ( n 1 , m ) a i ( n 2 , m ) | | g i ( m , y ( m ) ) | a ˜ i ( n 1 ) a ˜ i ( n 2 ) q μ i M p , Open image in new window

which yields that each B i ( E ) Open image in new window is equi-asymptotically almost periodic since Open image in new window . Then, by Theorem 2.3, each B i ( E ) Open image in new window is precompact in AAP ( Z ) Open image in new window. Let { y k } E Open image in new window. Then { B i y k } Open image in new window, if necessary going to a subsequence, is convergent in AAP ( Z ) Open image in new window for each i { 1 , 2 , , λ } Open image in new window. By (3.1), we conclude that { N y k } Open image in new window is convergent in AAP ( Z ) Open image in new window. So, N ( E ) Open image in new window is precompact in AAP ( Z ) Open image in new window.

By applying Schauder’s fixed point theorem, there exists a fixed point y Open image in new window of N Open image in new window in E. Then we have
y = N y = x y = i = 1 λ A i x y B i y = i = 1 λ A i y B i y = M y , Open image in new window

which means that y Open image in new window is a fixed point of ℳ. This completes the proof. □

Finally, we give a simple example to illustrate our result.

Example 3.2 Let λ = 2 Open image in new window, p = 1 Open image in new window, q = Open image in new window,
f 1 ( n , x ) = x 10 ( sin n + sin π n + 1 | n | + 1 ) , g 1 ( n , x ) = sin ( x e n 2 ) 2 ( 1 + n 2 ) , a 1 ( n , m ) = cos n + cos 2 n + 1 n 2 + 1 3 ( 1 + m 2 ) , Open image in new window
and
f 2 ( n , x ) = cos n sin x 20 , g 2 ( n , x ) = arctan ( n x ) 1 + n 2 , a 2 ( n , m ) = 1 3 e m 2 sin n . Open image in new window
It is easy to see that (H1) holds with L 1 = 3 10 Open image in new window and L 2 = 1 20 Open image in new window. Also, (H2) holds with μ 1 r ( n ) 1 2 ( 1 + n 2 ) Open image in new window and μ 2 r ( n ) π 2 1 1 + n 2 Open image in new window. In addition, (H3) can be easily verified. By a direct calculation, we can get
α 1 1 , α 2 1 3 , Open image in new window
and
μ 1 r 1 π + 1 2 , μ 2 r 1 π 2 + π 2 , r > 0 . Open image in new window
Letting M = 1 Open image in new window, we have
i = 1 2 α i L i μ i M 1 3 ( π + 1 ) 20 + π 2 + π 120 < 1 , Open image in new window
and
i = 1 2 [ sup n Z , | x | K | f i ( n , x ) | α i μ i M 1 ] 3 ( π + 1 ) 20 K + π 2 + π 120 < K , K > 1 . Open image in new window

Thus, (H4) holds with M = 1 Open image in new window. Then, by using Theorem 3.1, Equation (1.1) has an asymptotically almost periodic solution.

Notes

Acknowledgements

The work was supported by the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province, the Foundation of Jiangxi Provincial Education Department (GJJ12205), and the Research Project of Jiangxi Normal University (2012-114).

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© Long and Pan; licensee Springer 2012

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Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceJiangxi Normal UniversityNanchangPeople’s Republic of China

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