Open Access

Asymptotically almost periodic solution to a class of Volterra difference equations

Advances in Difference Equations20122012:199

https://doi.org/10.1186/1687-1847-2012-199

Received: 1 September 2012

Accepted: 3 November 2012

Published: 16 November 2012

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 periodicVolterra 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 ,
(1.1)

where λ is a fixed positive integer and f i : Z × R R , a i : Z × Z R , g i : Z × R R ( i = 1 , 2 , , λ ) 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 [26] 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 .
(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 [813] 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 + ) the set of (nonnegative) integers, by the set of positive integers, by ( R + ) the set of (nonnegative) real numbers, by Ω a subset of , 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 is called almost periodic if ε, N ( ε ) N such that among any N ( ε ) consecutive integers there exists an integer p with the property that
f ( k + p ) f ( k ) < ε , k Z .

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

Lemma 1.2 [[14], Theorem 1.26]

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

Remark 1.3 Let f , g AP ( Z ) . By Lemma 1.2, it is not difficult to show that ε, N ( ε ) N such that among any N ( ε ) 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

for all k Z .

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

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

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

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

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

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

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

    f E implies that f is bounded.

     
  2. (b)

    f , g E implies that f + g E . Moreover, f g E if X = R .

     
  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 from to . Then f AAP ( Z ) if and only if ε, M ( ε ) , N ( ε ) N such that among any N ( ε ) consecutive integers there exists an integer p with the property that
| f ( k + p ) f ( k ) | < ε

for all k Z with | k | M ( ε ) and | k + p | M ( ε ) .

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

for all k Z with | k | M ( ε ) and | k + p | M ( ε ) .

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

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

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

Thus, g ¯ AP ( Z ) .

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

for all k Z with | k | M ( ε ) .

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

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

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

for all k Z with | k | M ( ε ) and | k + p | M ( ε ) .

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

    for each k Z , { f ( k ) : f F } is bounded;

     
  2. (ii)

    F is equi-asymptotically almost periodic.

     

Proof ‘only if’ part

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

By Remark 1.3, we can get that { f 1 , f 2 , , f N } 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 . Since { f n ( k ) } is bounded for each k Z , we can assume that (if necessary going to a subsequence) { f n ( k ) } is convergent for each k Z . On the other hand, since F is equi-asymptotically almost periodic, for each ε > 0 , there exist M ( ε ) , N ( ε ) N such that among any N ( ε ) consecutive integers there exists an integer p with the property that
sup n Z | f n ( k + p ) f n ( k ) | < ε / 3
(2.3)
for all k Z with | k | M ( ε ) and | k + p | M ( ε ) . For the above ε > 0 , there exists a positive integer K such that for all n , m > K , the following hold:
| f n ( k ) f m ( k ) | < ε / 3 , k [ M ( ε ) , M ( ε ) + N ( ε ) ] Z .
(2.4)
For all k Z with | k | M ( ε ) , taking p [ k + M ( ε ) , k + M ( ε ) + N ( ε ) ] Z , 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 ;
also, for all k Z with | k | < M ( ε ) , by (2.4), we have
| f n ( k ) f m ( k ) | < ε / 3 < ε , n , m > K .
Thus, we get
sup k Z | f n ( k ) f m ( k ) | ε , n , m > K ,

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

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 are two fixed real numbers and
1 p + 1 q = 1 .
In addition, we denote by l p ( Z ) (resp. l q ( Z ) ) the space of all the functions f : Z R satisfying
f p : = ( k Z | f ( k ) | p ) 1 / p < + ( resp.  f q : = ( k Z | f ( k ) | q ) 1 / q < + ) .

For convenience, we first list some assumptions.

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

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

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

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 ;
and
( M x ) ( n ) = i = 1 λ ( A i x ) ( n ) ( B i x ) ( n ) , n Z , x AAP ( Z ) .

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

Step 1. A i and B i both map AAP ( Z ) into AAP ( Z ) , i = 1 , 2 , , λ .

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

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

we know that B i x AAP ( Z ) .

Step 2. For each y AAP ( Z ) with y M , there exists a unique x y AAP ( Z ) such that
x y = i = 1 λ A i x y B i y .
Let
( Y x ) ( n ) = i = 1 λ ( A i x ) ( n ) ( B i y ) ( n ) , n Z , x AAP ( Z ) .
Then, by Step 1, Y maps AAP ( Z ) into AAP ( Z ) . For all x 1 , x 2 AAP ( Z ) and n Z , 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 ,
which yields that
Y x 1 Y x 2 ( i = 1 λ α i L i μ i M p ) x 1 x 2 .

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

Step 3. has a fixed point in AAP ( Z ) .

Let E = { y AAP ( Z ) : y M } and
N y = x y , y E ,

where x y is the unique fixed point of Y (see Step 2).

We claim that N ( E ) E . In fact, if there exists y 0 E such that N y 0 > M , 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 ,

which is a contradiction.

Next, let us show that N : E E is continuous. For all y 1 , y 2 E , 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 ,
which gives that
N y 1 N y 2 i = 1 λ β i B i y 1 B i y 2 ,
(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 .
Letting y k y 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 .
(3.2)
For each i = 1 , 2 , , λ , noting that
| g i ( m , y k ( m ) ) g i ( m , y ( m ) ) | 2 μ i M ( m ) , m Z ,
g i ( m , ) is continuous for each m Z , and y k ( m ) y ( m ) for each m Z , we conclude that
g i ( , y k ( ) ) g i ( , y ( ) ) p 0 .

Combining this with (3.2), we know that N y k N y . So N : E E is continuous.

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

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

By applying Schauder’s fixed point theorem, there exists a fixed point y of N 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 ,

which means that y is a fixed point of . This completes the proof. □

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

Example 3.2 Let λ = 2 , p = 1 , q = ,
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 ) ,
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 .
It is easy to see that (H1) holds with L 1 = 3 10 and L 2 = 1 20 . Also, (H2) holds with μ 1 r ( n ) 1 2 ( 1 + n 2 ) and μ 2 r ( n ) π 2 1 1 + n 2 . In addition, (H3) can be easily verified. By a direct calculation, we can get
α 1 1 , α 2 1 3 ,
and
μ 1 r 1 π + 1 2 , μ 2 r 1 π 2 + π 2 , r > 0 .
Letting M = 1 , we have
i = 1 2 α i L i μ i M 1 3 ( π + 1 ) 20 + π 2 + π 120 < 1 ,
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 .

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

Declarations

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).

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Jiangxi Normal University

References

  1. Agarwal RP Monographs and Textbooks in Pure and Applied Mathematics. In Difference Equations and Inequalities. Theory, Methods, and Applications. 2nd edition. Marcel Dekker, New York; 2000.Google Scholar
  2. Diblík J, Schmeidel E, Ružičková M: Existence of asymptotically periodic solutions of system of Volterra difference equations. J. Differ. Equ. Appl. 2009, 15: 1165–1177. 10.1080/10236190802653653View ArticleGoogle Scholar
  3. Diblík J, Ružičková M, Schmeidel E: Existence of asymptotically periodic solutions of scalar Volterra difference equations. Tatra Mt. Math. Publ. 2009, 43: 51–61.MathSciNetGoogle Scholar
  4. Diblík J, Schmeidel E, Ružičková M: Asymptotically periodic solutions of Volterra system of difference equations. Comput. Math. Appl. 2010, 59: 2854–2867. 10.1016/j.camwa.2010.01.055MathSciNetView ArticleGoogle Scholar
  5. Diblík J, Ružičková M, Schmeidel E, Zbaszyniak M: Weighted asymptotically periodic solutions of linear Volterra difference equations. Abstr. Appl. Anal. 2011., 2011: Article ID 370982. doi:10.1155/2011/370982Google Scholar
  6. Diblík J, Schmeidel E: On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence. Appl. Math. Comput. 2012, 218: 9310–9320. 10.1016/j.amc.2012.03.010MathSciNetView ArticleGoogle Scholar
  7. Ding HS, Chen YY, N’Guérékata GM: C n -almost periodic and almost periodic solutions for some nonlinear integral equations. Electron. J. Qual. Theory Differ. Equ. 2012, 6: 1–13.View ArticleGoogle Scholar
  8. Araya D, Castro R, Lizama C: Almost automorphic solutions of difference equations. Adv. Differ. Equ. 2009., 2009: Article ID 591380Google Scholar
  9. Blot J, Pennequin D: Existence and structure results on almost periodic solutions of difference equations. J. Differ. Equ. Appl. 2001, 7: 383–402. 10.1080/10236190108808277MathSciNetView ArticleGoogle Scholar
  10. Cuevas, C, Henríquez, H, Lizama, C: On the existence of almost automorphic solutions of Volterra difference equations. J. Differ. Equ. Appl. (in press)Google Scholar
  11. Ding HS, Fu JD, N’Guérékata GM: Positive almost periodic type solutions to a class of nonlinear difference equations. Electron. J. Qual. Theory Differ. Equ. 2011, 25: 1–16.View ArticleGoogle Scholar
  12. Hamaya Y: Existence of an almost periodic solution in a difference equation with infinite delay. J. Differ. Equ. Appl. 2003, 9: 227–237. 10.1080/1023619021000035836MathSciNetView ArticleGoogle Scholar
  13. Pennequin D: Existence of almost periodic solutions of discrete time equations. Discrete Contin. Dyn. Syst. 2001, 7: 51–60.MathSciNetView ArticleGoogle Scholar
  14. Corduneanu C: Almost Periodic Functions. 2nd edition. Chelsea, New York; 1989.Google Scholar
  15. Fink AM Lecture Notes in Mathematics. In Almost Periodic Differential Equations. Springer, Berlin; 1974.Google Scholar
  16. Zhang C: Almost Periodic Type Functions and Ergodicity. Kluwer Academic, Dordrecht; 2003.View ArticleGoogle Scholar

Copyright

© Long and Pan; licensee Springer 2012

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