1. Introduction
The original notion of discrete almost-periodic sequence was proposed by the famous mathematician Walther in 1928 [
1,
2] and the theory as well as applications of discrete almost-periodic sequences was further developed by Halanay [
3] and Corduneanu [
4]. Discrete almost-periodic sequence plays a key role in characterizing the phenomena that are similar to the periodic phenomena. As an important development of the well-known discrete almost-periodic sequence, the notion of a discrete asymptotically almost-periodic sequence, which is the key issue, will be investigated in this work and was introduced in the works of Fan [
5] in 1943 based on the Fréchet concept from [
6,
7]. Since then, the investigation of the existence of solutions with asymptotically almost-periodicity has become one of the important as well as attractive subjects in the theory as well as applications of difference equations due to both the intensive development of the theorem of difference equations itself and the applications in various sciences such as computer science, chemistry, physics, engineering and so forth. For some meaningful and interesting works in this area, we refer readers to Thana [
8] for periodic evolution equations, Agarwal, Cuevas and Dantas [
9] for difference equations of Volterra type, Song [
10] for nonlinear delay difference equations of Volterra type, Cuevas and Pinto [
11] as well as Campo, Pinto and Vidal [
12] for functional difference equations, Matkowski [
13] for functional equations and the references therein.
On the other hand, the tools of fractional calculus are found to be of great utility in studying various scientific processes and systems. This has been mainly due to the ability of fractional order operators to describe long-memory effects of underlying processes. In particular, models described by fractional differential equations have gained significant importance and there has been a great interest in developing the theory and applications of fractional differential equations. For examples and details, we refer the reader to a series of recent research articles [
14,
15,
16,
17,
18,
19] and the references therein. Recently, fractional difference equations have gained more interest by many researchers. Besides their interest in terms of theory, the study of fractional difference equations has great importance in the aspect applications [
20]. The study of modeling with fractional difference equations began with the works of Atici and Seng
l [
21] and then Atici and Eloe [
22,
23], Dassios and Baleanu [
24,
25], Goodrich [
26,
27] studied the initial value problems, boundary value problems and positivity as well as monotonicity properties [
28] of solutions to fractional difference equations, Baleanu and Wu et al. [
29,
30] and Lizama [
31] investigated the stability of fractional difference equations and Wu and Baleanu [
32] studied the logistic map with fractional difference and its chaos. Especially in [
33], Abadias and Lizama consider nonlinear partial difference-differential equations in Banach space
X forms like
where
, the operator
A generates a
-semigroup defined on
X,
denotes the Weyl-like fractional difference operator,
is a function satisfying some Lipschitz type assumptions. By using the operator theoretical method and the Banach fixed point theorem, they prove the existence and uniqueness of solutions with almost automorphy to Equation (
1). Stimulated by the works of Abadias and Lizama [
33], Cao and Zhou [
19] established some criteria for the existence and uniqueness of mild solutions with asymptotical almost-periodicity to Equation (
1) with the nonlinear perturbation
satisfying a Lipschitz assumption or locally Lipschitz assumption.
According to our knowledge, up to now, much less is known about the existence of mild solution with asymptotical almost-periodicity to Equation (
1) when the nonlinear perturbation
loses the Lipschitz assumption with respect to
x. Thus in this work, we will try to fill this gap. Combining a decomposition technique with the Krasnoselskii’s fixed point theorem, we establish some new existence theorems of mild solutions with asymptotic almost-periodicity to Equation (
1). In our results,
does not have to satisfy a Lipschitz assumption or locally Lipschitz assumption with respect to
x (see Remark 2) and
with Lipschitz assumption becomes a special case with our conditions (see Remark 4). Thus our results generalize some related conclusions on this topic. In particular, as an application, we prove the existence and uniqueness of mild solutions with asymptotically almost-periodicity to the fractional difference scalar equation forms like
where
,
is a complex number satisfying
,
is a function to be specified later.
An outline of this article is as follows. We introduce some basic concepts and recall some preliminaries in
Section 2.
Section 3 is concerned with some new existence theorems for mild solutions with asymptotically almost-periodicity to Equation (
1). The last section deal with an example to validate the applications of our theoretical results.
2. Preliminaries
In this section, we introduce some basic concepts and recall some preliminaries.
In this paper, let
,
,
,
,
,
be the sets of all natural numbers, integral numbers, positive integral numbers, real numbers, positive real numbers and complex numbers, respectively. For a Banach space
, let
.
is a set consisting of sequences
.
is a set consisting of sequences
The space
is a Banach space under the norm
is a set consisting of sequences
is also a Banach space under the norm
.
is a set consisting of sequences
Moreover when
, we write
for short.
is a set consisting of sequences
is the space of all
X-valued bounded continuous functions and
is the closed subspace of
consisting of functions vanishing at infinity. Let
be another Banach space,
is a set consisting of functions
The space
is a Banach space under the norm
is a set consisting of functions
Let
be the collection of all bounded linear operators from
X to
Y. Under the uniform operator topology
we denote by
. For
, let
be the resolvent of
A and
be the domain of
A.
Firstly, we recall the definitions and related properties on discrete almost-periodic sequences as well as discrete asymptotically almost-periodic sequences.
Following Bohr, Walther has formulated the notion of discrete almost-periodic sequence.
Definition 1. (Walther [1,2], Corduneanu [34]) Let be a discrete sequence with values in X. If for each , the collectionis relatively dense in , that is for any , there is an integer , such that there exists at least one integer , where Δ is a any collection consisting of N consecutive integers, satisfyingthen is said to be a discrete almost-periodic sequence. The integer , with the property in Definition 1, is said to be an ε-translation number of the sequence .
By we denote the collection of such sequences.
The following notion of a normal process is needed to formulate an important property of the discrete almost-periodic processes.
Definition 2. (Corduneanu [34], Zhang, Liu and Gopalsamy [35]) A discrete sequence is called a normal process, if for any sequence , there is a subsequence , for which converges uniformly with respect to , as . That is to say, for any , there exist an integer and a discrete process such that Lemma 1. (Corduneanu [34]) A discrete process is almost-periodic if and only if it is normal. Definition 3. (Bohr) [36,37] A continuous function is said to be (Bohr) almost-periodic in if for each , the collectionis relatively dense in ; that is for every there exists such that every interval of length contains a number τ with the property that The number τ is called an ε-translation number of and the collection of those functions is denoted by .
Lemma 2. (Corduneanu [34], Zhang [38,39]) (I) For any almost-periodic sequence , there is a function , which is almost-periodic satisfying for . (II) For any almost-periodic function , , is an almost-periodic sequence.
Remark 1. (Campo, Pinto and Vidal [12]) The discretization of a periodic function may not lead to a periodic sequence. For instance, , , is not a periodic sequence, it is an almost-periodic sequence. Lemma 3. (Corduneanu [34], Zhang [38,39]) is bounded if is an almost-periodic sequence. Lemma 4. (Corduneanu [34], Long and Pan [40]) Under the norm , forms a Banach space. Definition 4. (Abadias and Lizama [33]) An operator-valued sequence is summable if The following lemma, which comes from Gohberg and Feldman [41], is the essential property to study almost-periodicity and asymptotic almost-periodicity of difference equations. Lemma 5. (Gohberg and Feldman [41]) Assume is a summable sequence. Then for any discrete sequence which is almost-periodic, the sequence defined byis also an almost-periodic sequence. Definition 5. (Song [10]) Let and Ω be any compact set in Y. If for any , the collectionis relatively dense in , that is for any , there is an integer such that there exists at least one integer , where Δ is any collection consisting of N consecutive integers, satisfyingthen is said to be almost-periodic in uniformly for . The integer , with the property in Definition 4, is said to be the ε-translation number of . By we denote the collection of such functions.
Lemma 6. (Campo, Pinto and Vidal [12], Song [10]) Let Ω be any compact set in Y and assume . Then is continuous on Ω uniformly for , that is for any , there is a constant such thatand is relatively compact in X. Lemma 7. The following statements are equivalent.
(i) ;
(ii) for each and G is continuous in uniformly in , i.e., for , , s.t.where Ω is any compact set in Y. Proof. The proof of follows from Lemma 6. In the following, we prove .
In fact, if
G is continuous in
uniformly in
, then
G is uniformly continuous in
f on
, thus
,
, s.t.
Since
is any compact set in
Y, then there are
so that for each
,
for some
i. It follows from
that for the above
, there is an integer
such that there exists at least one integer
, where
is a any collection consisting of
N consecutive integers, satisfying
Thus for each
, choose some
such that
, then
which yields
. □
Lemma 8. (Song [10]) Assume . Then for any integer sequence , there are a subsequence and a sequence satisfyinguniformly on as . Moreover . Lemma 9. (Song [10]) Assume and Ω is any compact set in Y. Then for any integer sequence , there are a subsequence and a function satisfyinguniformly on as . Moreover . We now give a composition theorem of discrete almost-periodic functions.
Lemma 10. Assume satisfying is continuous in each bounded subset of Y uniformly in , i.e., for and any bounded subset K of Y, s.t. Suppose is a discrete almost-periodic sequence. Then belongs to .
Proof. From
and
, together with Lemmas 8 and 9, it follows that for any integer sequence
, there exist a subsequence
and two functions
,
satisfying
uniformly on
as
and
uniformly on
as
. Moreover
and
. It follows from
and
, together with Lemma 3, that there exists a bounded subset
K of
Y such that
and
for each
. As
is continuous on
K uniformly for
, thus for any
, one can find a constant
satisfying
Moreover for every compact set
, there exists an
such that for all
,
Thus for all
,
which implies
is a normal process, thus
is almost-periodic which follows from Lemma 1.
Let satisfy a Lipschitz assumption in uniformly in , we obtain the following result as an immediate consequence of the previous lemma, Lemma 10. □
Lemma 11. Assume and satisfies a Lipschitz assumption in uniformly for , i.e., Then, the conclusion of Lemma 10 is true.
Lemma 11 admits a new version with local Lipschitz assumption on the function G.
Lemma 12. Assume and satisfies a local Lipschitz condition in uniformly for , i.e., for each and with , ,where is a function. Then, the conclusion of Lemma 10 is true. As an important development of the well-known discrete almost-periodic sequence, the notion of discrete asymptotically almost-periodic sequence, which is based upon the Fréchet concept from [6,7], was introduced in the literature [5] by Fan. Definition 6. (Fan [5], Song [10]) If a sequence with and , then the sequence is said to be asymptotically almost-periodic. The sequences is called the almost-periodic component of and is called the ergodic perturbation of .
By we denote the collection of such sequences.
Definition 7. (Fréchet) [6,7] A continuous function is said to be asymptotically almost-periodic if it can be decomposed as , where Lemma 13. (Zhang [38,39]) (I) For any discrete asymptotically almost-periodic sequence , there is a function , which is asymptotically almost-periodic satisfying for . (II) For any asymptotically almost-periodic function , , is a discrete asymptotically almost-periodic sequence.
Lemma 14. (Zhang [38,39]) The decomposition of an asymptotically almost-periodic sequence with and , is unique. Lemma 15. (Long and Pan [40]) Under the norm , also forms a Banach space. Lemma 16. Assume is the almost-periodic component of the sequence . Then .
Proof. Denote by
, the ergodic perturbation of
. If
is not contained in
, then there are
and
such that
Let us take
. Then,
which is a contradiction with
. □
Definition 8. (Campo, Pinto and Vidal [12], Long and Pan [40]) A function is said to be a discrete asymptotically almost-periodic function in for each ifwith and . By we denote the collection of such functions.
Lemma 17. Assume and Ω is any compact set in Y. Then is bounded on .
Proof. As
, then
with
and
. From Lemma 6, it follows that
is bounded on
. On the other hand, as
, then
is bounded on
. Thus
is bounded on
.
In the following, we give the definitions of the discrete -resolvent family, the fractional sum and fractional difference of .
Let
, defined a sequence
by
where
is the Gamma function. Note that
satisfies the following semigroup property
Define the forward Euler operator
as
Recursively, for each , define and is the identity operator. □
Definition 9. (Abadias and Lizama [33]) Assume and A is a closed linear operator with domain . An operator sequence is said to be a discrete α-resolvent family generated by A if for any and Definition 10. (Abadias and Lizama [33]) For any , let , and be a sequence. The fractional sum of f is given by Definition 11. (Abadias and Lizama [33]) For any , let , and be a sequence. The fractional difference of f is given bywhere , is the largest integer function. In the following, we recall the definitions and properties of Mittag-Lefer functions [42] These functions have the following properties for and .
Lemma 18. (1) [43] , . (2) [44] . (3) [45,46] . Finally, we give a compactness criterion and the so-called Krasnoselskii’s fixed point theorem.
Lemma 19. A set is relatively compact if
(1) uniformly for ,
(2) the set is relatively compact in X for every .
Proof. For any
with
, set
For any
, define
it is clear that
is a metric space. Let
be a sequence in
with
, it follows from the condition (2) that
is relatively compact in
X. Then there exists subsequence
such that
is convergent in
X. Then from the condition (2), it follows that
is relatively compact in
X, thus there exists subsequence
such that
is convergent in
X. Note that
is also convergent in
X. Repeating like this, we can obtain a subsequence
such that
is convergent in
X, that is
is relatively compact in
. Then
is relatively compact in
. Since
is arbitrary, then
is relatively compact.
From the condition (1), it follows that for any
, there exists
such that
As
is relatively compact, then it has a finite
-net
A. Let
For any
, assume that
is a corresponding point of
x and taking
such that
. Assume that
is a corresponding point of
in
, then
This implies that is a finite -net of D, thus D is totally bounded; this together with is a complete space and yields that D is is relatively compact. □
Lemma 20. [47,48]. Let S be a subset of X and Let be maps satisfyingwhere is a bounded closed and convex collection. If is completely continuous and is a contraction, then the equation has a solution on S. 3. Asymptotically Almost-Periodic Mild Solutions
In this section, we will state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of the nonhomogeneous nonlinear difference equations of fractional order given by
where
, the operator
A generates a
-semigroup on
X,
denotes the Weyl-like fractional difference operator,
is a function to be specified later.
In [
33], Abadias and Lizama obtained the following remarkable result.
Lemma 21. (Abadias and Lizama [33]) Assume A generates a -semigroup on X, which is exponentially stable, i.e., Then there exists a discrete α-resolvent family generated by A, which is given bywhere is a function given by Moreover, satisfiesand furthermore is summable with The following definition of mild solutions to Equation (
4), which is given in [
33], is essential for us.
Definition 12. (Abadias and Lizama [33]) Assume is a discrete α-resolvent family generated by A. A sequence is said to be a mild solution of Equation (4) if for each , is summable on and x satisfies The following auxiliary result plays a key role in the proofs of our main results.
Lemma 22. Let be summable. Given sequence , . Let Then and .
Proof. From
, combining with Lemma 3, it follows that
is bounded on
X. In addition, note that
hence
is well defined.
Similarly, is also well defined. □
From Lemma 5, together with
it follows that
.
As
, then for
,
s.t.
Thus
which implies
.
Now, we state and prove our main results. The following assumptions are required:
() The -semigroup generated by A on X is exponentially stable.
(
)
with
and it is bounded on
. Moreover
(
) There is a function
satisfying
and a nondecreasing function
satisfying
() For all , and , the collection is relatively compact in X.
Remark 2. Note that in , does not need to meet the Lipschitz assumption with respect to x. Such classes of functions are more complicated than those with the Lipschitz assumption (see Remark 3).
Remark 3. In Lemma 17, we prove that a discrete asymptotically almost-periodic function is bounded only on for any compact set Ω in X, so the condition boundedness in does not conflict with the condition .
Firstly, we give some important Lemmas.
Lemma 23. Let with Proof. To obtain this result, it suffices to show
If this is not true, then there are
and
s.t.
As
, then
this yields that
s.t.
From
, it follows that for the above
and every compact
, there is an integer
, such that there exists at least one integer
, where
is a any collection consisting of
N consecutive integers, satisfying
Then
which contradicts Equation (
10). □
Remark 4. From Equation (9) it follows that if meets the Lipschitz assumption of Equation (2), then satisfies Equation (7). Then Lipschitz assumptions become a special case of our condition. Note that in [19,33], a Lipschitz assumption (Equation (2)) or a locally Lipschitz assumption (Equation (3)) for of Equation (4) is needed. Thus our condition in () is weaker than those of [19,33] and our results generalize some related conclusions of [19,33]. Let be the sequence in (). Set Lemma 24. .
Proof. From
uniformly for
, it follows that for any
, one can choose a
such that
, for all
. This combining with Lemma 18 and
, implies that
which implies
. □
Firstly, we state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of Equation (
4) when the almost-periodic component
of
F satisfies Equation (
7).
Theorem 1. Let – hold. Put . Then there is a mild solution of Equation (4) whenever Moreover the mild solution is asymptotically almost-periodic.
Proof. We divide into five steps to complete the proof. □
Step 1. Let
be a mapping defined on
given by
There is a unique fixed point of .
Firstly, according to
, there is a discrete
-resolvent family
generated by
A, which is summable (by Lemma 21) and, combining with the discrete function
, is bounded on
, which follows from Lemma 3, and one has
which yields that
exists. Furthermore from
satisfying Equation (
5), combined with Lemmas 3 and 11, it follows that
This, combined with Lemma 2, yields that is well defined.
Then, we show is continuous.
Let
satisfy
as
, then
Therefore, as , , which yields that is continuous.
Finally, we show there is a unique fixed point of .
Let
, similar to the proof of the continuity of
, and we have
which implies
From Equation (
11), it follows that
is a contraction on
. Thus, there is a unique fixed point
of
.
Step 2. Set
. For the above
, define
on
as
There is a constant s.t. maps into itself.
Firstly, according to Equation (7), one has
which yields
According to Equation (
8), one has
Thus is well-defined and maps into itself, which follows from Lemma 22.
From Equations (
8) and (
11), it easily follows that there is a constant
s.t.
Then for any
and
, one has
Thus maps into itself.
Step 3. is a contraction on .
Let
, according to Equation (
7), one has
Thus
is a contraction on
by Equation (
11).
Step 4. is completely continuous on .
Firstly, is continuous on .
In fact,
, let
with
as
. From
(from Lemma 24), one may choose an
big enough s.t.
In addition,
implies that
and
Hence, according to the Lebesgue dominated convergence theorem, there exists an
such that for any
,
Accordingly, is continuous on .
Then we consider the compactness of .
Let
and
for
. First, for all
and
,
in view of
, which follows from Lemma 24, one has
From
it follows that for given
, one can choose an
such that
Thus we get
where
is the convex hull of
K. Using
, the collection
is relatively compact, combing with
, one obtains that
is a relatively compact subset of
X.
Thus is compact by Lemma 19, which further implies is completely continuous on .
Step 5. Show Equation (
4) has at least one mild solution, which is asymptotically almost periodic.
Firstly, from the results of step 4, combining with the results of steps 2 and 3 as well as the Krasnoselskii’s fixed point theorem (see Lemma 20), it follows that
has at least one fixed point
; furthermore,
. Then, consider the following coupled system
Combing the unique fixed point
in step 1 with the fixed point
in Step 4, it follows that
is a solution to System (
13); then
and it is a solution to the equation
that is
is a mild solution to Equation (
4), which is asymptotically almost-periodic
In the following, we state and prove conditions for the existence of mild solutions with asymptotically almost-periodicity of Equation (
4) when the almost-periodic component
satisfies a locally Lipschitz assumption
(
)
with
and it is bounded on
. Moreover, for each
,
where
is a nondecreasing function.
Theorem 2. Assume that , , and hold and if there exist , such that Put . Then there is a mild solution of Equation (4) whenever Moreover the mild solution is asymptotically almost-periodic.
Proof. We also divide into five steps to complete the proof. □
Step 1. Define a mapping
by
has a unique fixed point .
Firstly, according to
, there is a discrete
-resolvent family
generated by
A, which is summable (by Lemma 21) and, combined with the discrete function
, is bounded on
, which follows from Lemma 3, and one has
which yields that
exists. Furthermore from
satisfying Equation (
14), combined with Lemmas 3 and 12, it follows that
Furthermore it follows from Lemma 22 that
. Let
, and one has
Hence , which yields that is well defined.
Then, we show is continuous.
Let
satisfy
as
, then one has
Therefore, as , , hence is continuous.
Finally, we show there is a unique fixed point of .
Let
and, similar to the proof of the continuity of
, one has
which implies
From Equation (
15), it follows that
is a contraction on
. Thus, there is a unique fixed point
of
and
.
Step 2. Set
. For the above
, define
on
as Equation (
12). There exists a constant
s.t.
maps
into itself.
Firstly, as
, then
s.t.
,
. This combined with Equation (
14) implies
which yields
According to Equation (
8), one has
which yields
Thus is well-defined and maps into itself, which follows from Lemma 22.
From Equations (
8) and (
15), it easily follows that there is a constant
s.t.
Then for any
and
, one has
this indicates that
Thus maps into itself.
Step 3. is a contraction on .
Let
and, from Equation (
7), one has
Thus
is a contraction on
by Equation (
15).
Step 4. is completely continuous on .
The proof of this step is the same as step 4 of Theorem 1.
Step 5. Equation (
4) has at least one mild solution which is asymptotically almost-periodic.
The proof of this step is the same as step 5 of Theorem 1.