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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 980869, 14 pages

http://dx.doi.org/10.1155/2014/980869

## Weighted Stepanov-Like Pseudoperiodicity and Applications

Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Received 28 June 2013; Revised 28 October 2013; Accepted 28 October 2013; Published 20 January 2014

Academic Editor: Carlos Lizama

Copyright © 2014 Zhinan Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By the weighted ergodic space, we propose a new class of functions called weighted Stepanov-like pseudoperiodic function and explore its properties. Furthermore, the existence and uniqueness of the weighted pseudoperiodic solution to fractional integro-differential equations and nonautonomous differential equations are investigated. Some interesting examples are presented to illustrate the main findings.

#### 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 equation. Many authors have made important contributions to this theory. Recently, in [1], the concept of weighted pseudoperiodicity, which generalizes the notion of periodicity and pseudoperiodicity, is introduced and studied. On the other hand, Stepanov [2, 3] introduced a class of generalized almost periodic functions, for which continuity fails, and only measurability and integrability in the sense of Lebesgue are required. Since then, almost automorphy, pseudo-almost-periodicity, pseudo-almost-automorphy, and so forth are generalized in the Stepanov sense one can see [4–6] for more details on this topics.

Motivated by the above mentioned papers, in this paper, we introduced a new class of functions called weighted Stepanov-like pseudoperiodic function, which generalizes the notation of weighted pseudoperiodic function. We systematically explore the properties of the weighted Stepanov-like pseudoperiodic function in general Banach space including a composition result.

The rapid development of the theory of integrodifferential equations has been strongly promoted by the large number of applications in physics, engineering, biology, and other subjects. This type of equations has received much attention in recent years and the general asymptotic behavior of solutions is at present an active of research [7–12]. To the best of our knowledge, there is no work reported in literature on weighted pseudoperiodicity for fractional integrodifferential equations in . Furthermore, the existence and uniqueness of weighted pseudoperiodic solutions for nonautonomous differential equations are quite new and untreated topic. 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. Next, we propose a new class of functions called weighted Stepanov-like pseudoperiodic function, explore its properties, and establish the composition theorem. Sections 3 and 4 are devoted to the existence and uniqueness of weighted pseudoperiodic solutions to a class of fractional integrodifferential equations and nonautonomous differential equations, respectively. In Section 5, we provide some examples to illustrate our main results.

#### 2. Preliminaries and Basic Results

Let , be two Banach spaces and , , , and stand for the set of natural numbers, integers, real numbers, and complex numbers, respectively. In order to facilitate the discussion below, we further introduce the following notations:(i) (resp., and ): the Banach space of bounded continuous functions from to (resp., from to and from to ) with the supremum norm;(ii) (resp., , ): the set of continuous functions from to (resp., from to and from to );(iii): the Banach space of bounded linear operators from to endowed with the operator topology. In particular, we write when ;(iv): the space of all classes of equivalence (with respect to the equality almost everywhere on ) of measurable functions such that ;(v): standing for the space of all classes of equivalence of measurable functions such that the restriction of to every bounded subinterval of is in .

##### 2.1. Sectorial Operators and Riemann-Liouville Fractional Derivative

*Definition 1 (see [13]). *A closed and densely defined linear operator 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 [13] for more details.

*Definition 2 (see [14]). *Let be given. Let be a closed and linear operator with domain defined on a Banach space . We call is 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 .

Note that if is sectorial of type with , then the generator of a solution operator given by where is a suitable path lying outside the sector (see [15]). Recently, Cuesta [15] has proved that if is a sectorial operator of type for some , , then there exists such that Note that for ; therefore is integrable on .

In the rest of this section, we list some necessary basic definitions in the theory of fractional calculus.

*Definition 3 (see [16]). *The fractional order integral of order with the low limit for a function is defined as
provided the right-hand side is pointwise defined on , where is the Gamma function.

*Definition 4 (see [16]). *Riemann-Liouville derivative of order with the low limit for a function can be written as

##### 2.2. Evolution Family and Exponential Dichotomy

*Definition 5. *A family of bounded linear operators on a Banach space is called a strong continuous evolution family if (i) and for all and ;(ii)the map is continuous for all , and .

*Definition 6. *An evolution family on a Banach space is called hyperbolic (or has an exponential dichotomy) if there exist projections , , uniformly bounded and strong continuous in , and constants and such that (i) for and ;(ii)the restriction of is invertible for (and set );(iii)
for and . Here and below we set .

*Remark 7. *Exponential dichotomy is a classical concept in the study of long-time behaviour of evolution equations. If for , then is exponential stable. One can see [17–21] for more details.

If is hyperbolic, then is called Green's function corresponding to , and

##### 2.3. Weighted Pseudoperiodicity

First, let us recall some definitions of weighted pseudo anti-periodic function and weighted pseudoperiodic function.

*Definition 8. *A function is said to be anti-periodic if there exists a with the property that for all . If there exists a least positive with this property is called the anti-periodic of . The collection of those functions is denoted by .

*Definition 9. *A function is said to be periodic if there exists a with the property that for all . If there exists a least positive with this property is called the periodic of . The collection of those -periodic functions is denoted by .

*Example 10. *An example of a function that is anti-periodic is given by
whose anti-periodic is .

Other examples of nontrivial anti-periodic functions have been constructed in [22].

*Remark 11. *Note that if , then .

*Definition 12. *A function (resp., ) is said to be periodic in uniform in (resp., ) if there exists a with the property that for all , (resp., for all , ). The collection of those -periodic functions is denoted by (resp., ).

Let

*Definition 13 (see [23]). *A function is said to be asymptotically anti-periodic if there exist and such that . Denote by the collection of such functions.

*Definition 14 (see [24]). *A function is said to be asymptotically periodic if there exist and such that . Denote by the collection of such functions.

*Definition 15. *A function is said to be a pseudo anti-periodic if it can be decomposed as , where and with . Denote by the collection of such functions.

*Definition 16 (see [9]). *A function is said to be a pseudoperiodic if it can be decomposed as , where and with . Denote by the collection of such functions.

Let be the set of all functions which are positive and locally integrable over . For a given and each , set Define It is clear that .

*Definition 17. *Let ; is said to be equivalent to (i.e., ) if .

It is trivial to show that “” is a binary equivalence relation on . The equivalence class of a given weight is denoted by . It is clear that .

For , define the weighted ergodic space Particularly, for , define [25] clearly, when , this space coincide with ; that is, , ; this fact suggests that are more interesting when and are not necessarily equivalent. So are general and richer than and give rise to an enlarged space of weighted pseudoperiodic function defined in [1].

*Definition 18. *Let . A function is called weighted pseudo anti-periodic for if it can be decomposed as , where and . Denote by the set of such functions.

*Definition 19. *Let . A function is called weighted pseudoperiodic for if it can be decomposed as , where and . Denote by the set of such functions.

*Remark 20. *If , and coincide with the weighted pseudo anti-periodic function and weighted pseudoperiodic function, respectively, introduced by [1].

Similarly, define

*Definition 21. *Let . A function (resp., ) is called weighted pseudoperiodic in uniform in (resp., ) if it can be decomposed as , where (resp., ) and (resp., ). Denote by (resp., ) the set of such functions.

Next, we show some properties of the space . Similarly results are hold for .

Lemma 22. *Let ; then , , and if and only if, for every ,
**
where .*

*Proof. *The proof is similar to the one in [26].*Sufficiency.* From the statement of the lemma it is clear that, for any , there exists such that, for ,
Then
so
That is, .*Necessity.* Suppose the contrary, that there exists such that
does not converge to as . Then there exists such that, for each ,
Then
which contradicts the fact that , and the proof is complete.

Let .

Lemma 23. *Let ; then if and only if , where , , and .*

*Proof. *By Lemma 22, if and only if, for every ,
where . It is equivalent to for every ,
where . So .

Lemma 24. *Let uniformly on where each ; if , then .*

*Proof. *For ,
Let and then in the above inequality; it follows that .

Let , and define by for . Define [27] It is easy to see that contains many of weights, such as , , , and with et al.

Lemma 25. *Let , , , such that , ; then , .*

*Proof. *Since and , there exist positive constants , , , and such that and ; then,
Let ; then
implies that
then
That is, ; hence
Proceeding in a similar manner, we have . The proof is complete.

Lemma 26. *Let and ; then for .*

*Proof. *Let and ; by Lemma 25, for . Without loss of generality, we assume that ; then
Since implies that , there exists such that for . Then, for ,
Therefore, by ,
which implies that . The proof is complete.

Using similar ideas as in [25, 28], one can easily show the following result.

Lemma 27. *If and , then the decomposition of weighted pseudoperiodic function is unique.*

By Lemma 27, it is obvious that , (resp., ), , and is a Banach space when endowed with the sup norm.

##### 2.4. Weighted Stepanov-Like Pseudoperiodicity

In this subsection, we introduce the new class of functions called weighted -pseudo anti-periodic function and weighted -pseudoperiodic function and investigate the properties of these functions.

Let . The space of all Stepanov bounded functions, with the exponent , consists of all measurable functions such that , where is the Bochner transform of defined by , , . is a Banach space with the norm [29] It is obvious that and for .

For , define the weighted ergodic space in :

*Remark 28. *It is clear that if and only if .

*Definition 29. *Let . A function is said to be weighted Stepanov-like pseudo anti-periodic (or weighted -pseudo anti-periodic) if there is such that the function satisfies a.e. . Denote by the collection of such functions.

*Definition 30. *Let . A function is said to be weighted Stepanov-like pseudoperiodic (or weighted -pseudoperiodic) if there is such that the function satisfies a.e. . Denote by the collection of such functions.

*Remark 31. *Let be defined on by
For each , denote , where is largest integer function; then

Let and ; by a direct calculation, one has
So . In addition, it is obvious that .

Let , ; then , but .

From definition, we have the following diagram that summarizes the different classes of subspaces defined in Figure 1, where , , and so forth. stand for almost periodic and asymptotically almost periodic function, respectively; one can see [6] for more details, where the paper gives the historical development of almost periodicity and almost automorphy and relationship between these functions and their extensions.

Similarly, define

*Definition 32. *Let . A function (resp., ) is called weighted -pseudoperiodic in uniform in (resp., ) if there is (resp., ) such that the function satisfies a.e. and each (resp., a.e. and each ). Denote by (resp., , ) the collection of such functions.

Next, we show some properties of the space . Similarly results hold for .

Lemma 33. * for each , , .*

*Proof. *If , let , where and . Then . By Lemma 23, . Note that for each ; by Lebesgue’s dominated convergence theorem, one has
that is,
which means that , where
By Lemma 23, ; that is,
which means that ; then . The proof is complete.

Theorem 34. *Assume that , with , a.e. , , and there exist constants such that
**
then if .*

The proof is similar to that of Theorem 3.6 in [6] and the details are omitted here.

*Remark 35. *It is not difficult to see that Theorem 34 holds for .

Lemma 36. *Let be a strongly continuous family of bounded and linear operators such that , , where is nonincreasing. If , , , and ; then
*

*Proof. *Let , where and a.e. , then

First, we show that . Consider the integrals
For each , by the principle of uniform boundedness,
Fix and ; we have
In view of , we get
which yields that
This means that is continuous.

By Hölder inequality, one has
since
then is uniform convergent on . Let , ; then
It is obvious that . So, we need to show that
In fact, by Hölder inequality,
then
hence . By Lemma 24, it follows that .

By a.e. , one has
Hence . The proof is complete.

#### 3. Fractional Integrodifferential Equation

This section is devoted to the existence and uniqueness of weighted pseudoperiodic solutions of the following fractional integrodifferential equation: where is a linear densely operator of sectorial type on a complex Banach space and , , and are suitable functions. The fractional derivative is to be understood in the Riemann-Liouville sense.

Before starting our main results, we recall the definition of the mild solution to (62).

*Definition 37 (see [8, 30]). *Assume that generates an integral solution operator . A continuous function satisfying the integral equation
is called a mild solution on to (62).

To study (62), we introduce the following assumptions.(*H*_{1}) is a sectorial operator of type with .(*H*_{2}) is a continuous, nonincreasing function.(*H*_{3}) with , a.e. , , and there exist constants such that
(*H*_{4}) with , a.e. , , and there exist constants such that
(*H*_{5}), and .

Lemma 38. *If , assuming that , , , and hold, then .*

*Proof. *Since , by and Theorem 34, it is clear that . Similarly as the proof of Lemma 36, .

Lemma 39. *Assume that and holds; then
**
lies in .*

*Proof. *By (4), one has
Since and is nonincreasing, and by Lemma 36, .

Theorem 40. *Assume that ( H_{1})–(H_{5}) hold; if
*

*then (62) has a unique mild solution.*

*Proof. *Define the operator by

First, we show that is well defined. In fact, if , by Lemma 38,