Abstract
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.
MSC:65R05, 35B40.
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1 Introduction
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 [14], 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 [17], positivity and contractivity [20], asymptotic equivalence [21], 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 [30]
A closed and densely defined linear operator A is said to be sectorial of type if there exist , , and such that its resolvent exists outside the sector
The sectorial operators are well studied in the literature, we refer to [30] for more details.
Definition 2.2 [31]
Let A be a closed and linear operator with domain defined on a Banach space X. We call A the generator of a solution operator if there exist and a strong continuous function such that and
In this case, is called the solution operator generated by A.
Note that if A is sectorial of type with , then A is the generator of a solution operator given by
where γ is a suitable path lying outside the sector [32]. Recently, Cuesta [32] proved that if A is a sectorial operator of type for some (), , then there exists a constant such that
Note that
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 [19]
The fractional order integral of order with the low limit for a function f is defined as
provided the right-hand side is pointwise defined on , where Γ is the gamma function.
Definition 2.4 [19]
Riemann-Liouville derivative of order with the low limit for a function can be written as
2.2 Compactness criterion and fixed point theorem
First, we recall two useful compactness criteria.
Let be a continuous nondecreasing function such that as . Define
endowed with the norm .
Lemma 2.1 [33]
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 [34])
Let , F is relatively compact in for if and only if
-
(1)
is relatively compact in X.
-
(2)
as uniformly for , where .
Now, we recall the so-called Zima’s fixed point theorem [35] and the Leray-Schauder alternative theorem [36] which will be used in the sequel.
Let denote a Banach space of elements with a binary relation ‘≺’ and a mapping such that
-
(i)
the relation ≺ is transitive;
-
(ii)
and for all ;
-
(iii)
the norm is monotonic, that is, if , then for all .
Theorem 2.1 ([35] Zima’s fixed point theorem)
In the Banach space considered above, let the operators and be given with the following properties:
-
(iv)
B is a bounded linear operator with spectral radius .
-
(v)
B is increasing, that is, if , then for all .
-
(vi)
for all .
Then the equation has a unique solution in Y.
Theorem 2.2 ([36] 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.
2.3 -Asymptotic ω-periodicity in the Stepanov sense
For , define
Definition 2.5 [37]
A function is called asymptotically ω-periodic if there exist , such that . The collection of those functions is denoted by .
Definition 2.6 [5]
A function is said to be -asymptotically periodic if there exists such that . In this case, we say that f is -asymptotically ω-periodic. The collection of those functions is denoted by .
Definition 2.7 [5]
A continuous function is said to be uniformly -asymptotically ω-periodic on bounded sets if for every bounded set K of X, the set is bounded and uniformly in . Denote by the set of such functions.
Definition 2.8 [5]
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 .
We introduce the following composition theorem for an -asymptotically ω-periodic function.
Lemma 2.3 [5]
Assume that is an asymptotically uniformly continuous on bounded sets function. Let , then .
Let . The space of all Stepanov bounded functions, with the exponent p, consists of all measurable functions such that , where is the Bochner transform of f defined by , , . is a Banach space with the norm [38]
It is obvious that and for . We denote by the subspace of consisting of functions f such that as .
Definition 2.9 [14]
A function is called -asymptotically ω-periodic in the Stepanov sense (or --asymptotically ω-periodic) if
Denote by the set of such functions.
It is easy to see that
Definition 2.10 [14]
A function is said to be uniformly -asymptotically ω-periodic on bounded sets in the Stepanov sense if for every bounded set , there exist positive functions and such that for all , and
Denote by the set of such functions.
Definition 2.11 [14]
A function is said to be asymptotically uniformly continuous on bounded sets in the Stepanov sense if for every and every bounded set , there exist and such that
for all and all with .
Lemma 2.4 [14]
Assume that is an asymptotically uniformly continuous on bounded sets in the Stepanov sense function. Let , then .
Lemma 2.5 Let be a strongly continuous family of bounded and linear operators such that , , where is nonincreasing. If , then
Proof For , , one has
that is, Λf is bounded. It is clear that Λf is continuous for each , whence . Moreover, note that
where
By the hypothesis of ϕ, one has
then
On the other hand, since , there exists such that
For , one has
which implies that as . So
The proof is complete. □
3 Semilinear fractional integro-differential equation
Consider the 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 [23]
Assume that A generates a solution operator . A function is called a mild solution of (3.1) if
To study (3.1), we require the following assumptions:
(H1) A is a sectorial operator of type with .
(H2) .
() , .
(H31) f satisfies the Lipschitz condition
(H32) f satisfies the Lipschitz condition
where .
(H33) f satisfies the Lipschitz condition
where .
(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
In this subsection, we study the existence and uniqueness of -asymptotically ω-periodic mild solution of (3.1) when f satisfies the Lipschitz condition.
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 .
Proof Define the operator by
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.
Moreover, let , one has
by the Banach contraction mapping principle, ℱ has a unique fixed point in , which is the unique mild solution to (3.1). □
Theorem 3.2 Assume that (H1), (H2) (or ()), (H32) hold and
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.
For , one has
-
If , in this case
(3.4) -
If , where . In this general case,
where is defined by
then . So we infer that
By (3.4), (3.5), one has
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 .
Proof Define an equivalent norm on as , where and . Define the operator ℱ as in (3.2). Let , one has
consequently,
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 .
Proof Define the operator ℱ as in (3.2), then ℱ is a map from into . Moreover, ℱ is continuous by (3.6). Define the map B on by
It is clear that B is a bounded linear operator from into .
First, we will show that B is a compact operator. For each and each with , define the functions
and
It follows from the Ascoli-Arzelá theorem in the space that the set is relatively compact in , and therefore in .
Since , for each , take such that for ,
For , , one has
then . Since for , one has
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.
Consider the Banach space equipped with both the relation ≺ and the mapping defined by: if
and . It is easy to check that conditions (i), (ii), (iii) are satisfied.
Let , one has
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
In this subsection, we study the existence of -asymptotically ω-periodic mild solution of (3.1) when f does not satisfy the Lipschitz condition.
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 , .
(A3) For each , there exists such that for , implies that
(A4) For all , and , the set is relatively compact in X.
(A5) , where and
C, M are constants given in (2.1).
Then (3.1) has a mild solution .
Proof Define by
Next, we prove that Γ has a fixed point in . We divide the proof into several steps.
-
(i)
For , by (A1), one has
It follows from (A2) that .
-
(ii)
Γ is continuous. In fact, for each , by (A3), there exits , for and , one has
Take into account that , by (A3)
which implies that , so Γ is continuous.
-
(iii)
Γ is completely continuous. Set for the closed ball with center at 0 and radius r in the space Z. Let and for .
Initially, we prove that is a relatively compact subset of X for each , here . Since
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.
Next, we show that is equicontinuous. In fact,
For each , we can choose such that
Moreover, since is a relatively compact set and is strong continuous, we can choose , such that
and
So, for with and for all .
Finally, by (A2), one has
and this convergence is independent of . Hence V satisfies (c1), (c2) of Lemma 2.1, which completes the proof that V is a relatively compact set in .
-
(iv)
If is a solution of the equation for some , then
Hence, one has
and by (A5), we conclude that the set is bounded.
-
(v)
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 .
Let be a sequence in such that it converges to u in the norm . For , let be the constant in (A3), there exists such that for all . For ,
Hence, converges to uniformly in . This implies that and completes the proof. □
Corollary 3.1 Assume that (H1), (H2) (or ()) hold and satisfy the following conditions:
-
(a)
.
-
(b)
f satisfies the Hölder-type condition
where , is a constant.
-
(c)
For all , and , the set is relatively compact in X.
Then (3.1) has a mild solution .
Proof By (b), it is easy to see that (H4), () hold. Let and , then (A1) is satisfied. Take a function h such that , it is not difficult to see that (A2) is satisfied. To verify (A3), note that for each , there exists such that for every , implies that
On the other hand, (A5) can be easily verified using the definition of W. By Theorem 3.5, (3.1) has a mild solution . □
4 Examples
In this section, we provide some examples to illustrate our main results.
Example 4.1 Consider the following fractional differential equation:
where , , . In what follows we consider and let A be the operator given by
with domain
It is well know that A is sectorial of type [30]. Equation (4.1) can be expressed as an abstract system of the form (3.1), where for , , and for , . Moreover, one has
since , we deduce that . From
so (H31) holds with . If is small enough, (4.1) has a unique mild solution by Theorem 3.1.
Example 4.2 Consider the following fractional differential equation:
where , , , . Let , with domain , so A is sectorial of type . Equation (4.2) can be rewritten as the abstract form (3.1), where
Moreover, one has
so and f is asymptotically uniformly continuous on bounded sets by (4.5). By (4.3), we define W by . Let , , , one has
Hence (A1)-(A3) hold.
Next, we prove that the set is relatively compact in by Simon’s theorem. In fact, one has
Hence, for , is bounded uniformly for and , . On the other hand,
therefore,
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.
Author’s contributions
The author has made this manuscript independently. The author read and approved the final version.
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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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Xia, Z. Asymptotically periodic solutions of semilinear fractional integro-differential equations. Adv Differ Equ 2014, 9 (2014). https://doi.org/10.1186/1687-1847-2014-9
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DOI: https://doi.org/10.1186/1687-1847-2014-9