Weighted Stepanov-Like Pseudoperiodicity and Applications

Xia, Zhinan

doi:https://doi.org/10.1155/2014/980869

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 980869 | https://doi.org/10.1155/2014/980869

Weighted Stepanov-Like Pseudoperiodicity and Applications

Zhinan Xia¹

Academic Editor: Carlos Lizama

Received28 Jun 2013

Revised28 Oct 2013

Accepted28 Oct 2013

Published20 Jan 2014

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.(H₁) is a sectorial operator of type with .(H₂) is a continuous, nonincreasing function.(H₃) with , a.e. , , and there exist constants such that (H₄) with , a.e. , , and there exist constants such that (H₅), 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 (H₁)–(H₅) 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, , so from Theorem 34. By Lemma 39, it is not difficult to see that is well defined.
For any , By the Banach contraction mapping principle, has a unique fixed point in , which is the unique mild solution to (62).

Remark 41. It is not difficult to see that similar result of Theorem 40 holds for mild solution. In this case, instead of in , , then (62) has a unique mild solution.

4. Nonautonomous Differential Equation

This section is devoted to the existence and uniqueness of weighted pseudoperiodic solutions of the following nonautonomous differential equation in Banach space : where satisfies “Acquistapace-Terreni” conditions, evolution family generated by has exponential dichotomy, and is a suitable function.

In this section, we consider the following assumptions.(A₁) There exists constants , , and with such that for .(A₂) The evolution family generated by has an exponential dichotomy with constants and , dichotomy projections , , and Green's function .(A₃) There exists such that .(A₄) with , a.e. , , and there exist constants such that (A₅) , and .

Remark 42. is usually called “Acquistapace-Terreni” conditions, which was first introduced in [31] and widely used to study nonautonomous differential equations in [19–21, 31, 32]. If holds, there exists a unique evolution family on , which governs the homogeneous version of (71) [32].

Definition 43 (see [21]). A mild solution of (71) is a continuous function satisfying for all , .

Lemma 44. Assume that and hold; then lies in .

Proof. Let , where and a.e. ; then where
Next, we show that . Consider the integrals for each , .
By the Hölder inequality and the exponential dichotomy of , it follows that where . Since the series is convergent, by the Weierstrass test, is uniformly convergent on . Furthermore, It is obvious that . So, we need to show that In fact, by Hölder inequality, one has where is a constant, depending on , , and . Then hence . By Lemma 24, it follows that . It is not difficult to see that a.e. , whence .

Theorem 45. Assume that (A₁)–(A₅) hold; if , then (71) has a unique mild solution such that

Proof. First, we show that (71) admits a unique bounded solution given by (84), which is similar to the proof of [21, Theorem 3.3]. For , it is clear that since holds; then by the definition of exponential dichotomy of , it is not difficult to see that (84) is well defined for each .
For all , , then so is a mild solution of (71). To prove the uniqueness, let be another mild solution of (71), then by the exponential dichotomy of , Similarly, So
Next, define the operator by If , it is not difficult to see that since holds. By Lemma 44, , so is well defined.
For any , By the Banach contraction mapping principle, has a unique fixed point in , which is the unique mild solution to (71).

Remark 46. It is not difficult to see that similar result of Theorem 45 holds for mild solution. In this case, instead of in ; then (71) has a unique mild solution.

5. Example

In this section, we provide some examples to illustrate our main results.

Example 1. Consider the fractional partial differential equation where , and is a real valued continuous nonincreasing function such that for and and are positive constants.

Take , and define the operator by It is well know that is the infinitesimal generator of an analytic semigroup on . Thus is of sectorial type .

Let , , ; then (93) can be rewritten as the abstract form (62). Take and ; by Theorem 40, one has the following.

Theorem 47. Under assumptions and , then (93) has a unique mild solution whenever

Example 2. Consider the heat equation with Dirichlet conditions where is weighted pseudoperiodic function, is a continuous function, for , and there exists such that .
Take being equipped with its natural topology and define Let for all . It is well known that is the infinitesimal generator of an analytic semigroup on with for . Moreover, for each .
Define a family of linear operators by Since , one can see that the system has an associated evolution family on , which can be explicitly expressed by Moreover, It is not difficult to verify that satisfy (A₁)–(A₃) with . One can see [21, 33] for more details.

By Theorem 45, we claim that

Theorem 48. Under the assumptions and , (96) admits a unique mild solution if .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ13A010015.

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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.

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