- Open Access
Asymptotically periodic solutions of semilinear fractional integro-differential equations
© Xia; licensee Springer 2014
Received: 21 September 2013
Accepted: 9 December 2013
Published: 7 January 2014
In this paper, we study the existence of an -asymptotically ω-periodic mild solution of semilinear fractional integro-differential equations in Banach space, where the nonlinear perturbation is -asymptotically ω-periodic or -asymptotically ω-periodic in the Stepanov sense. A fixed point theorem and the nonlinear Leray-Schauder alternative theorem are the main tools in carrying out our proof. Some examples are given to show the efficiency and usefulness of the main findings.
The study of the existence of periodic solutions is one of the most interesting and important topics in the qualitative theory of differential equations, due to its mathematical interest as well as their applications in physics, control theory, mathematical biology, among other areas. Some contributions on the existence of periodic solutions for differential equations have been made. Mostly, the environmental change in the real word is not periodic, but approximately periodic. For this reason, in the past decades many authors studied several extensions of the concept of periodicity, such as asymptotic periodicity, almost periodicity, almost automorphy, pseudo almost periodicity, pseudo almost automorphy, etc. and the same concept in the Stepanov sense, one can see [1–4] for more details.
The notion of -asymptotic ω-periodicity, introduced by Henríquez et al. in [5, 6], is related to and more general than that of asymptotic periodicity. Since then, it has attracted the attention of many researchers [7–13]. Recently, in , the concept of -asymptotic ω-periodicity in the Stepanov sense, which generalizes the notion of -asymptotic ω-periodicity, was introduced and the applications to semilinear first-order abstract differential equations were studied.
Due to their numerous applications in several branches of science, fractional integro-differential equations have received much attention in recent years [15–19]. The properties of solutions of fractional integro-differential equations have been studied from a different point of view, e.g., maximal regularity , positivity and contractivity , asymptotic equivalence , asymptotic periodicity [22–25], almost periodicity [26, 27], almost automorphy [28, 29] and so on. To the best of our knowledge, there is no work reported in literature on -asymptotic ω-periodicity for fractional integro-differential equations if the nonlinear perturbation is -asymptotically ω-periodic in the Stepanov sense. This is one of the key motivations of this study.
The paper is organized as follows. In Section 2, some notations and preliminary results are presented. Section 3 is divided into two parts. In the first one, Section 3.1, we investigate the existence and uniqueness of an -asymptotically ω-periodic mild solution of semilinear fraction integro-differential equations when the nonlinear perturbation f satisfies the Lipschitz condition. In the second part, Section 3.2, when f is a non-Lipschitz case, we explore the properties of solutions for the same equation. In Section 4, we provide some examples to illustrate the main results.
2 Preliminaries and basic results
Let , be two Banach spaces and ℕ, ℝ, , and ℂ stand for the set of natural numbers, real numbers, nonnegative real numbers, and complex numbers, respectively. In order to facilitate the discussion below, we further introduce the following notations:
(resp. ): the Banach space of bounded continuous functions from to X (resp. from to X) with the supremum norm.
(resp. ): the set of continuous functions from to X (resp. from to X).
: the Banach space of bounded linear operators from X to Y endowed with the operator topology. In particular, we write when .
: the space of all classes of equivalence (with respect to the equality almost everywhere on ) of measurable functions such that .
: stand for the space of all classes of equivalence of measurable functions such that the restriction of f to every bounded subinterval of is in .
2.1 Sectorial operators and Riemann-Liouville fractional derivative
Definition 2.1 
The sectorial operators are well studied in the literature, we refer to  for more details.
Definition 2.2 
In this case, is called the solution operator generated by A.
for , therefore is integrable on .
In the rest of this subsection, we list some necessary basic definitions in the theory of fractional calculus.
Definition 2.3 
provided the right-hand side is pointwise defined on , where Γ is the gamma function.
Definition 2.4 
2.2 Compactness criterion and fixed point theorem
First, we recall two useful compactness criteria.
endowed with the norm .
Lemma 2.1 
A set is relatively compact in if it verifies the following conditions:
(c1) For all , the set is relatively compact in .
(c2) uniformly for .
Lemma 2.2 (Simon’s theorem )
is relatively compact in X.
as uniformly for , where .
the relation ≺ is transitive;
and for all ;
the norm is monotonic, that is, if , then for all .
Theorem 2.1 ( Zima’s fixed point theorem)
B is a bounded linear operator with spectral radius .
B is increasing, that is, if , then for all .
for all .
Then the equation has a unique solution in Y.
Theorem 2.2 ( Leray-Schauder alternative theorem)
Let D be a closed convex subset of a Banach space X such that . Let be a completely continuous map. Then the set is unbounded or the map F has a fixed point in D.
Definition 2.5 
A function is called asymptotically ω-periodic if there exist , such that . The collection of those functions is denoted by .
Definition 2.6 
Definition 2.7 
Definition 2.8 
A continuous function is said to be asymptotically uniformly continuous on bounded sets if for every and every bounded set , there exist and such that for all and all with .
Lemma 2.3 
Assume that is an asymptotically uniformly continuous on bounded sets function. Let , then .
It is obvious that and for . We denote by the subspace of consisting of functions f such that as .
Definition 2.9 
Denote by the set of such functions.
Definition 2.10 
Denote by the set of such functions.
Definition 2.11 
for all and all with .
Lemma 2.4 
Assume that is an asymptotically uniformly continuous on bounded sets in the Stepanov sense function. Let , then .
The proof is complete. □
3 Semilinear fractional integro-differential equation
where , is a linear densely defined operator of sectorial type on a complex Banach space X and is an appropriate function.
Before starting our main results, we recall the definition of the mild solution to (3.1).
Definition 3.1 
To study (3.1), we require the following assumptions:
(H1) A is a sectorial operator of type with .
() , .
(H4) f is asymptotically uniformly continuous on bounded sets.
() f is asymptotically uniformly continuous on bounded sets in the Stepanov sense.
3.1 Lipschitz case
If is uniformly Lipschitz continuous at u, i.e., (H31) holds, we reach the following claim.
Theorem 3.1 Assume that (H1), (H2) (or ()), (H31) hold, then (3.1) has a unique mild solution if .
By (2.1), one has , so . By (H31), if (H2) holds, by Lemma 2.3, and if () holds, by Lemma 2.4. Hence ℱ is well defined by Lemma 2.5.
by the Banach contraction mapping principle, ℱ has a unique fixed point in , which is the unique mild solution to (3.1). □
then (3.1) has a unique mild solution .
Proof Define the operator ℱ as in (3.2). If (H2) holds, then . Since (H32) holds, f is asymptotically uniformly continuous on bounded sets in the Stepanov sense, so by Lemma 2.4. If () holds, by Lemma 2.4. Hence ℱ is well defined by Lemma 2.5.
If , in this case(3.4)
If , where . In this general case,
By the Banach contraction mapping principle, ℱ has a unique fixed point in , which is the unique mild solution to (3.1). □
In next results, we relax condition (3.3) to study the existence and uniqueness of mild solution of (3.1).
Theorem 3.3 Assume that (H1), (H2) (or ()), (H32) hold and the integral exists for all . Then (3.1) has a unique mild solution .
Since , ℱ is a contraction and then it has a unique fixed point , which is the unique mild solution to (3.1). □
Theorem 3.4 Assume that (H1), (H2) (or ()), (H33) hold, then (3.1) has a unique mild solution .
It is clear that B is a bounded linear operator from into .
It follows from the Ascoli-Arzelá theorem in the space that the set is relatively compact in , and therefore in .
which implies that is relatively compact, so B is a compact operator. Moreover, it follows from the Gronwall-Bellman lemma that the point spectrum , which implies that the spectral radius of B is equal to zero since B is a compact operator.
and . It is easy to check that conditions (i), (ii), (iii) are satisfied.
hence , and B is increasing with spectral radius . By Theorem 2.1, ℱ has a unique fixed point in , which is the unique mild solution to (3.1). □
3.2 Non-Lipschitz case
The following existence result is based upon the nonlinear Leray-Schauder alternative theorem.
Theorem 3.5 Assume that (H1), (H2), (H4) hold (or (H1), (), () hold) and satisfy the following conditions:
(A1) There exists a continuous nondecreasing function such that for all , .
(A2) For each , .
(A4) For all , and , the set is relatively compact in X.
C, M are constants given in (2.1).
Then (3.1) has a mild solution .
- (i)For , by (A1), one has
- (ii)Γ is continuous. In fact, for each , by (A3), there exits , for and , one has
Γ is completely continuous. Set for the closed ball with center at 0 and radius r in the space Z. Let and for .
where denotes the convex hull of K and . Using the fact that is strong continuous and (A4), we infer that K is a relatively compact set, and is also a relatively compact set.
So, for with and for all .
- (iv)If is a solution of the equation for some , then
If follows from Lemmas 2.3, 2.4 and 2.5 that ; consequently, we consider . Using (i)-(iii), we have that the map is completely continuous. By (iv) and Theorem 2.2, we deduce that Γ has a fixed point .
Hence, converges to uniformly in . This implies that and completes the proof. □
- (b)f satisfies the Hölder-type condition
For all , and , the set is relatively compact in X.
Then (3.1) has a mild solution .
On the other hand, (A5) can be easily verified using the definition of W. By Theorem 3.5, (3.1) has a mild solution . □
In this section, we provide some examples to illustrate our main results.
so (H31) holds with . If is small enough, (4.1) has a unique mild solution by Theorem 3.1.
Hence (A1)-(A3) hold.
uniformly for , , . So (A4) holds by Lemma 2.2. It is not difficult to see that (A5) holds. Whence (4.2) has a mild solution by Theorem 3.5.
The author has made this manuscript independently. The author read and approved the final version.
The author is grateful to the referees for their valuable suggestions. This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ13A010015.
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