Space of -Periodic Limit Functions and Its Applications to an Abstract Cauchy Problem

Abstract

We introduce a new space consisting of what we call -periodic limit functions. We investigate some properties of the new function space. In particular, we study inclusion relations among asymptotically periodic type function spaces. Finally, we apply the -periodic limit functions to investigate the existence and uniqueness of asymptotically -periodic mild solutions of an abstract Cauchy problem.

1. Introduction

Let be a Banach space. In this paper, we denote by the interval and by the space consisting of bounded and continuous functions from into , endowed with the uniform convergence norm . Set    and   . A function is said to be asymptotically -periodic if it can be expressed as , where and . The subspace of consisting of the asymptotically -periodic functions will be denoted by .

Let . Since and , one gets The converse is not true. The authors in [1] provided two examples to show that there exists a bounded and continuous function which satisfies (1) but is not asymptotically -periodic. At the same time, (1) leads authors in [2] to propose the following definition.

Definition 1 (see [2, Definition 3.1]). A function is said to be -asymptotically -periodic if there exists such that . The subspace of consisting of the -asymptotically -periodic functions will be denoted by .

The following concept of -asymptotic -periodicity in the Stepanov sense is introduced in [3, Definition 1.3].

Definition 2. A function is called -asymptotically -periodic in the Stepanov sense if as . The subspace of consisting of the -asymptotically -periodic functions in the Stepanov sense will be denoted by .

Let , where consist of those functions from which are uniformly continuous on . We have the following proper inclusions: References [1, Examples 2.1 and 2.2] and [2, Examples 3.1 and 3.2] show that there exists a function but not in . We will give an -asymptotically -periodic function which is not uniformly continuous in Section 3. Reference [3, Example 2.3] shows that there exists a function but not in . Moreover, the function in this example is bounded continuous. That is, but not in . Henríquez et al. in [2] and Pierri in [4] gave some conditions under which an -asymptotically -periodic function is asymptotically -periodic. Later, Henríquez in [3] showed that, if and is uniformly continuous on , then . For some qualitative properties of , we refer the reader to [2, 3]. For their applications we refer the reader to [2–10].

In this paper, we will introduce a new space consisting of what we call -periodic limit functions. Our aim to propose the new function space is twofold. On the one hand, -periodic limit functions generalize asymptotically -periodic functions in a different way from -asymptotically -periodic functions and have some relationship with asymptotically -periodic functions (in the Stepanov sense). On the other hand, the -periodic limit functions contribute to studying the existence and uniqueness of asymptotically -periodic mild solutions of some abstract Cauchy problems.

The paper is organized as follows. In Section 2, we define the space of -periodic limit functions and investigate its properties. In Section 3, we discuss inclusion relations among asymptotically periodic type function spaces. Finally, in Section 4 we study the existence and uniqueness of asymptotically -periodic mild solutions of an abstract Cauchy problem with -periodic limit coefficients.

2. Space of -Periodic Limit Functions

Definition 3. Let and . We call -periodic limit if is well defined for each , where . The collection of such functions will be denoted by .

Remark 4. The function in Definition 3 is measurable but not necessarily continuous.

We list some basic properties of -periodic limit functions in the following proposition, its proof is obvious, and so we omit it.

Proposition 5. If , , and are -periodic limit and is well defined for each , then the following statements are true: (1) is -periodic limit;(2) is -periodic limit for every scalar ;(3) for each ;(4) is bounded on ; moreover ;(5) is -periodic limit for each fixed .

Remark 6. It follows from Proposition 5(1)(2) that is a linear subspace of .

Remark 7. Because of Proposition 5(3), we give name of -periodic limit for functions in Definition 3.

Theorem 8. is a Banach space.

Proof. Let be a sequence of -periodic limit functions such that uniformly in .
By the definition of the -periodic limit function, we have for each and each . So for each the sequence of is a Cauchy sequence in because of the following inequality: Thus the sequence converges to a function pointwise.
To show that we only need to show that pointwise on . But this comes from the following inequality:

Let , where and . For each , one has which shows that .

Next we give an example to show that there exists an -periodic limit function which is not asymptotically -periodic. For concision, we only consider the case . Similarly one can exhibit examples for the general case .

Example 9 (example of a function but not in ). Consider the set . For , define the function by If we define the function by then we have for each , . So is a -periodic limit function.
On the other hand, if we choose , , then we have as , while . This proves that is not uniformly continuous on . So is not asymptotically -periodic.

Fix and denotes the conjugate exponent of .

The Bochner transform , , of a function on , with values in is defined by .

Let be a measurable function. We say that is a Stepanov bounded function, with the exponent , if . The collection of such functions will be denoted by .

The space endowed with the norm is a Banach space.

Define the subspaces of by    and   . A function is called asymptotically -periodic in the Stepanov sense if it can be expressed as , where and . The subspace of consisting of the asymptotically -periodic functions in the Stepanov sense will be denoted by .

Let . Then there exists a measurable function such that for each . By Proposition 5(3)(4), we know that . If we denote , , it is not hard to show that . Thus, .

Therefore, . It is easy to know that the inclusion is proper.

Now we have the following relationship between them: The following proposition is a part of [11, Theorem 2.1]. For the sake of completeness, we include the proof in Appendix.

Proposition 10. Let and be well defined for each . If uniformly on , then .

Corollary 11. Let and be well defined for each . If uniformly on , then .

Proposition 12. If and is uniformly continuous on , then .

Proof. We have that is bounded [3, Proposition 2.2]. Let , where and . Note that Let . Since , we can choose such that when for all . Therefore, as . We denote , . For given, we select such that when for all . Then we have when for all and any . We will prove that is a Cauchy sequence in norm . In fact, if we assume the contrary, there exist , , and sequences such that as and such that Since , . In particular, . So there exists a such that when , . We assume that . For , , we have Thus, when . This implies that Thus, which is a contradiction. Thus is a Cauchy sequence in norm . Hence, there exists a function such that as . Then we have as . Combining with (15), we get for every . From one gets for every . Since , then and . Combining with (26), we get for every . Thus, . Denote . By (24), we have From , we have . Note that . We can show that . If we assume the contrary, there are a constant and a nondecreasing sequence such that as and . Since , there exists a such that when , . We assume that . Therefore, for . This implies that which is a contradiction. Therefore, .

Remark 13. In view of (12) and Proposition 12, it is interesting to know if there exists a function but not in .

Corollary 14. If and is uniformly continuous on , then .

Remark 15. If a function and is also uniformly continuous, then . So the range of is relatively compact. It is interesting to know if there exists a function but the range of is not relatively compact.

3. Inclusion Relations among Asymptotically Periodic Type Function Spaces

In this section, we mainly discuss inclusion relations for these asymptotically periodic type function spaces in (2) and (12).

Let , where and . From one gets which shows .

Example 16 (example of a function but not in ). Let us come back to Example 9. For , we have where . If we choose , , then . So is a -periodic limit function.
On the other hand, we have for , which shows that is not -asymptotically -periodic.

Example 17 (example of a function but not in ). Consider Example 9 again. If satisfies the condition : , then by (32), one has which shows that is -asymptotically -periodic. For example, let , . Clearly and also satisfies the condition . So is -asymptotically -periodic. On the other hand, is not uniformly continuous. Thus is an -asymptotically -periodic function which is not uniformly continuous.

The following proposition plays a role in the example below.

Proposition 18. If , then , where is a nonnegative constant.

Proof. Let , where and . We first prove that there exists a nonnegative constant such that . Note that In the same way, one gets Thus, we obtain Let . By (37), we have In view of , for each . Therefore, there exists a nonnegative constant such that for each . Thus, Since , we have Moreover, one gets Note that By (39), (41), and (42), we have

Example 19 (example of a function but not in ). Let be the function defined by Fix . Let , . Then we have and  . For any , we can choose such that and . Then which shows that is -asymptotically -periodic. From (45) we know that is uniformly continuous. Thus, .
On the other hand, there exist such that , ,  when and , ,  when .
So, when , one gets By (46) the limit of does not exist. By Proposition 18, is not asymptotically -periodic in the Stepanov sense.

Now we can write down the following diagram which summaries asymptotically periodic type function spaces and their inclusion relations:

In view of Examples 16 and 19, and do not contain each other.

4. Existence of Asymptotically -Periodic Solutions of an Abstract Cauchy Problem

In this section we apply the results of Section 2 on to investigate the existence and uniqueness of asymptotically -periodic mild solutions for the following abstract Cauchy problem: where is the infinitesimal generator of an exponentially stable -semigroup ; that is, there exist and such that for all .

The following definition generalizes Definition 3.

Definition 20. A joint continuous function is called -periodic limit in uniformly for in bounded subsets of if for every bounded subset of , is bounded and exists for each and each . Denote by the set of all such functions.

The following is a composition theorem of -periodic limit functions.

Theorem 21. Let be -periodic limit in uniformly for in bounded subsets of and assume that satisfies a Lipschitz condition in uniformly in : for all and , where is a positive constant. Let be -periodic limit. Then the function defined by is -periodic limit.

Proof. Since is a -periodic limit function, we have for each .
By Proposition 5(4), we can choose a bounded subset of such that , for all . Thus is bounded.
On the other hand, we have for each and each .
Let us consider the function defined by . Note that
We deduce from (50) and (51) that for each , which finishes the proof.

Definition 22. A function is called a mild solution of problem (48) if

Now we can establish the main result of this section.

Theorem 23. Let be -periodic limit in uniformly for in bounded subsets of and assume that satisfies a Lipschitz condition in uniformly in : for all and , where is a positive constant. If , then there exists a unique asymptotically -periodic mild solution of problem (48).

Proof. We define the operator on the space by We will show that . Since , it remains to show that . We denote for short. In view of Theorem 21, if , then . By the definition of the -periodic limit function, is well defined for each . Moreover, by Proposition 5(3)(4), there exists a positive constant such that and . Note that Next we will prove that is a Cauchy sequence in for each . Let . For any , , one has Now we estimate the term : We can choose such that when . That is, when uniformly for .
For , we consider Then we have For each , we have Since , for each , one has as . By Lebesgue’s Dominated Convergence Theorem, we obtain Moreover, we have Thus, we can select such that when uniformly for .
Next we estimate the term : uniformly for .
Thus, when . This shows that is a Cauchy sequence. So we can denote for each . We also have that uniformly for .
Now we consider the term . Clearly is well defined for each . For , , one has For each , we have   as and  . By Lebesgue’s Dominated Convergence Theorem, we obtain For given, we select such that when . For any , one has when . Thus, when uniformly for . That is, uniformly for . Now we have    uniformly for . By Corollary 11, we get .
The space is a Banach space. For the sake of completeness, we include the proof of the completeness of the space in Appendix.
Finally, for , one has which shows that is a contraction. To complete the proof of the theorem we only need to invoke the contraction mapping principle.

Finally, we provide an example to illustrate our results.

Example 24. Consider the following partial differential equations: where , , and are appropriate functions. In addition, satisfies where . In what follows we consider and let be the operator given by with domain . It is well known that is the infinitesimal generator of an analytic semigroup on . Moreover, has discrete spectrum with eigenvalues , , and corresponding normalized eigenfunctions given by . Furthermore, is an orthonormal basis of and for . Thus, we have for every . Therefore, if , (75) has a unique asymptotically -periodic mild solution by Theorem 23.

Appendix

Proof of Proposition 10. We first show that . By Proposition 5(3)(4), is bounded on and for each . To show the continuity of the function on we only need to prove that is continuous on .
Now, take any fixed and let . Let . Then, by assumption, we conclude that there exists a positive integer such that for and .
On the other hand, since , then there exists such that for .
Therefore, we have when , which shows that is continuous at . Moreover, is continuous on . Hence .
Next, we will show that . Suppose that and there exists a positive integer such that when uniformly for by assumption again.
Thus, for , , we conclude that uniformly for . Moreover, if we denote , where and , then we obtain for , . That is, , which shows that . Hence .

Proof of the Completeness of the Space . Suppose that ; that is, , where and . We will prove that . In fact, if we assume the contrary, there exist and such that . Then, we have    , which is a contradiction. Thus, . Moreover, . Let be a Cauchy sequence and suppose that , where and . Thus, is Cauchy too and so is . Since and are closed in , there exist and such that and as . Set . Then and as .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research is supported by NSF of China (no. 11071048).