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Abstract and Applied Analysis

Volume 2014, Article ID 263690, 11 pages

http://dx.doi.org/10.1155/2014/263690
Research Article

Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations

1Department of Applied Mathematics, Guangdong University of Foreign Studies, Guangzhou 510420, China

2Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China

Received 10 April 2014; Accepted 4 May 2014; Published 22 May 2014

Academic Editor: Hui-Sheng Ding

Copyright © 2014 R. N. Wang and Y. Zhou. 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

This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the form , , , , where (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach space . Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on , we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.

1. Introduction and Motivation

Antiperiodic problems have recently proved to be valuable tools in the modelling of many phenomena in physical processes. For the background on this class of problems we refer the reader to [13] and the references therein. For this reason, much attention is attracted by questions of existence of antiperiodic solutions to the various antiperiodic problems represented by linear and nonlinear abstract evolution equations since the work of Okochi [4] in 1988 (see also [5, 6]). The literature related to such problems is quite extensive; see, for instance, Haraux [7] for nonlinear first order evolution equations in Hilbert spaces and Aftabizadeh et al. [8] and Aizicovici and Pavel [9] for second order evolution equations in Hilbert and Banach spaces. In particular, using the maximal monotone property of the derivative operator with antiperiodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings, Liu [10] recently studied the antiperiodic problem for nonlinear evolution equation with nonmonotone perturbation of the form in a real reflexive Banach space , where is monotone and is not. For more details about development and applications along this line, see, for example, [1, 1113] and the references therein. Let us note that equations in the research mentioned above are all autonomous.

Motivated by these works, in this paper we will carry out our investigation to the semilinear nonautonomous parabolic evolution equation having the form subject to antiperiodic condition in the Banach space . Here, stands for the set of real numbers, (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on and has domains , and is a given function to be specified later.

Note also that the problem (2)-(3) has been considered by Wang [14] under different situations, in which the author proved the existence of antiperiodic mild (strict) solutions in the case of (2) being a mild (strict) dissipative differential equation by using the modular degree theorem. The line, which we will go along in this study, is that we establish some results concerning the existence of antiperiodic mild solutions to the problem (2)-(3) under new criteria (without the assumption of (2) being dissipative). Then, we also consider the antiperiodic mild solutions to nonautonomous semilinear parabolic evolution equation of neutral type Finally, as a sample of possible applications, we give a result on the existence of antiperiodic mild solutions to a partial differential equation with homogeneous Dirichlet boundary condition, whose operator in the linear part generates an evolution family of exponential stability. As can be seen, our results extend and unify many existing results in this area. Banach’s contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem, as well as the theory of evolution families such as exponential dichotomy techniques, are employed in our approach. Note also that the conditions we used in this paper are quite different from those in [14].

We organize this paper as follows. We present some definitions and preliminary facts in Section 2. In Section 3, starting with introducing the assumptions that are needed in the proofs of our main results, we then establish some existence theorems of antiperiodic mild solutions to the problem (2)-(3). In Section 4, we extend the result obtained in Section 3 to the neutral problem (4). An example is given to illustrate the theorem in Section 5.

2. Preliminaries

Throughout this paper, is assumed to be a Banach space with norm and stands for the Banach space of all bounded linear operators from to equipped with its natural topology. Denote by the Banach space of all bounded, continuous functions from to equipped with the sup norm by the Banach space of all Bocher integrable functions from to equipped with the norm and by the set of all locally Bocher integrable functions from to .

A function is said to be -antiperiodic if

By , we denote the set of all -antiperiodic functions from to . It is easy to see that , equipped with the sup norm, is a Banach space. For every , write which is convex closed subset of .

The following lemma provides some useful information on compactness criterion, which can be regarded as an extension of the known Arzéla-Ascoli theorem.

Lemma 1 (see [15, Lemma 3.1]). A set is relatively compact in if is equicontinuous and the set is relatively compact in for every .

Definition 2. A family of bounded linear operators on a Banach space is called a strongly continuous evolution family if(1) and for all and ,(2)the map is continuous for all , and .

Throughout the paper, we assume that is a family of closed and densely defined operators on , which satisfies the conditions of Acquistapace and Terreni ( ) and ( ).( ) are linear operators on and there are constants , , and such that and for all and , ( )There are constants and with such that for all and Here .

Conditions ( ) and ( ), which are initiated by Acquistapace and Terreni [16, 17] for , are well understood and widely used in the literature.

Remark 3. It should be mentioned that when has a constant domain , can be replaced with the following: there exist constants , such that for all (see, e.g., [18, 19]).

By an obvious rescaling from [16, Theorem 2.3] and [20, Theorem 2.1] (see also [17, 21]), it follows that conditions ( ) and ( ) ensure that there exists a unique evolution family on such that(I) , for , and for , ;(II) for and with .

In this case we say that generates the evolution family .

Definition 4. An evolution family is called hyperbolic (or has exponential dichotomy) if there are projections , , that are uniformly bounded and strongly continuous in , and constants , such that(a) for all ;(b)the restriction is invertible for all (and we set );(c) and for all .Here and below . Specially, if for , then is said to be exponentially stable.

Exponential dichotomy is a classical concept in the study of the long-term behavior of evolution equations; see [2224] and references therein.

3. The Existence of Antiperiodic Mild Solutions

In this section we establish some existence theorems of antiperiodic mild solutions to the problem (2)-(3).

To prove our main results, we introduce the following assumptions. For sake of brevity, put for some .( )The evolution family , generated by , is hyperbolic. Moreover, for all , and for all .( )The function satisfies the following conditions.(i) is measurable for each and for all , .(ii)There exists a constant with such that for a.e. and all .( )(i)The function is a Carathéodory function; that is, for every , is measurable and for a.e. , is continuous, and for all , .(ii)There exists a function such that for a.e. and all , and with .( ) is compact for .

Remark 5. From it is clear that for all .

Definition 6. A mild solution to (2) is a function satisfying the integral equation for all and .

Before stating the existence theorem, we first prove the following lemma.

Lemma 7. Let the assumption be satisfied. Given , suppose that for a.e. . Define Then is well defined for each and belongs to .

Proof. From (c) and our assumptions on it follows that for each , which implies that is well defined for and is bounded.

To prove belongs to , we first verify the continuity of . For fixed and , we obtain upon changing of variables that Thus, we have Noticing that , we have which yields . An analogue argument shows .

Since is strongly continuous, we obtain for all , . This, together with (c), yields that for all , . Now, using the Lebesgue dominated convergence theorem we obtain . Similarly, we can show that .

Next, it remains to prove that is -antiperiodic. Noting that for a.e. , we have for any . Therefore, we can conclude that belongs to . This completes the proof.

Now we are ready to state the first main result.

Theorem 8. Let ( ) and ( ) hold. Then the problem (2)-(3) has a unique -antiperiodic mild solution.

Proof. Set, for , . It easily follows from that the function satisfies the conditions of Lemma 7. From this, we obtain that the mapping , defined by is well defined and maps into itself.

To prove the theorem, we first show that has a unique fixed point in . Let . Then, by we have Consequently, which, together with our assumption , implies that is a strict contraction on . Thus, using the Banach contraction principle we conclude that has a unique fixed point in .

To the end of the proof, we will prove that is a mild solution of (2) if and only if it is a fixed point of .

We first suppose that is a mild solution of (2); that is, satisfies the integral equation for all and . From this and (c), it immediately follows that So, one has This proves that is a fixed point of .

Conversely, if is a fixed point of , then satisfies the integral equations For any , , we obtain upon multiplying both sides of (32) by that which implies that is a mild solution to problem (2).

Now, according to the discussed above we deduce that the problem (2)-(3) has a unique -antiperiodic mild solution. The proof is completed.

Now we are in a position to prove our second existence result of antiperiodic mild solutions for the problem (2)-(3).

Theorem 9. Let ( ), ( ), and ( ) hold with for . Then the problem (2)-(3) has at least one -antiperiodic mild solution.

Proof. Let us define the mapping by We first notice, thanks to assumptions ( ) and ( ) (i) and Lemma 7, that is well defined and maps into itself.

Next, by applying Schauder’s fixed point theorem we show that has at least one fixed point in . From ( ) (ii) it is easy to see that there exists some such that Using this, a direct calculation yields that, for every and all , which implies that for every .

In the sequel, we show that is completely continuous on . The proof will be divided into two steps.

Step 1. is continuous on .

Take . Then it follows from (i) that This, together with the Lebesgue dominated convergence theorem and the continuity of with respect to second variable, shows that which implies the continuity of .

Step 2. is a compact operator on .

For each , set From it follows that, for each and , the set is relatively compact in . Thus, for each and , the set is also relatively compact in . Then, for every and , as in , we conclude, in view of the total boundedness, that for each , the set is relatively compact in .

Next, we will show that is equicontinuous. Taking with , we have where , are positive constants yet to be determined.

Given . We first note that there exist , small enough such that For , one can take a big enough which is independent of and such that For such fixed , , it is easy to find that there exists a big enough such that , which, together with (ii) and (c), yields that Therefore, from the continuity of for in the uniform operator topology it follows that whenever , where is small enough.

Thus, from the arguments above one can deduce that there exist such that whenever and , which implies that the set is equicontinuous. Consequently, is compact operator on due to Lemma 1.

Now, applying Schauder’s fixed point theorem, we deduce that has at least one fixed point . Moreover, following from the same idea as the last part of the proof in Theorem 8, we obtain that is a -antiperiodic mild solution of the problem (2)-(3). This completes the proof.

Remark 10. Theorems 8 and 9 generalize corresponding results for antiperiodic problems due to [12]. Note in particular that Theorems 8 and 9 cover results in [12].

Remark 11. (i) It can be easily shown that if is antiperiodic with period , then it is periodic with period . Hence, from the arguments of Theorems 8 and 9 we can also obtain the existence results of -periodic solutions of the problem (2)-(3).

(ii) It is clear that if is periodic with period , may or may not be antiperiodic with period .

Additional information is contained in the following. We consider the following nonautonomous semilinear parabolic evolution equation with periodic condition

From the arguments of Theorems 8 and 9 it is easy to see that if(1)the hypotheses in Theorem 8 are satisfied except that the antiperiodic on is replaced by the following then there exists a unique -periodic mild solution for the problem (48);(2)the hypotheses in Theorem 9 are satisfied except that the antiperiodic conditions on are replaced by (49), then there exists at least a -periodic mild solution for the problem (48).

The following remark indicates one motivation of the present paper.

Remark 12. As in [25], under certain conditions, the existence result is valid for the case of antiperiodic solutions, while there is no such a result in the periodic case. It is also noted that in dealing with the existence of certain problems, there is an essential difference between the periodic solutions and antiperiodic solutions (see also [26] for more details).

4. Neutral Problems

In this section, it is assumed that has a constant domain and verifies the conditions of Acquistapace and Terreni ( ) and ( ) with .

If the hypothesis is satisfied, then it follows readily that is periodic. Also, from Remark 3 it is easy to see that there exist constants , such that for all . Therefore, we deduce that for all .

Let denote the Banach space endowed with the graph norm for . By , we denote the set of all -antiperiodic functions from to . It is clear that , equipped with the sup norm, is a Banach space.

In this section, we extend the result obtained in Section 3 to the antiperiodic problem of neutral type (4).

Definition 13. A mild solution to (4) is a function satisfying the integral equation for all and .

Remark 14. It will be seen later that the last two terms on right side in (52), being integrals in sense of Bocher (see [27]), are reasonable.

To prove the existence of antiperiodic mild solutions to the problem (4), let us introduce the following assumptions:( )the function is continuous and for all , . Moreover,(i)there exists a constant such that for all , ;(ii)there exists a nondecreasing function such that for all , , and ( )Consider .( ) , where .

Theorem 15. Under the hypotheses ( ), ( ), ( ) (i), and ( ), the problem (4) has a unique -antiperiodic mild solution.

Proof. From ( ) (i) note that for each . Define, for , Then by (c) and (51) a direct calculation gives This proves that is well defined for and is bounded. Moreover, noticing ( ) (i) and using a similar argument with that in Lemma 7 one can show easily that maps into itself.

Let us assume that the mapping is defined the same as in Theorem 8. Then from the proof of Theorem 8 with it follows that is well defined and maps into itself.

Now, consider the mapping . We see, from the arguments above, that maps into itself. Also, for , , as in view of ( ) and ( ) (i), we conclude that is a strict contraction on due to . This allows us to obtain, in view of the contraction mapping principle, that has a unique fixed point . Moreover, an application of the same idea as the last part of the proof in Theorem 8 justifies that is a mild solution of (4) if and only if it is a fixed point of . The proof is then completed.

The following fixed point theorem plays a key role in the proof of our subsequent result; see, for example, [28].

Lemma 16. Krasnoselskii’s fixed point theorem: let be a Banach space and be a nonempty closed convex subset of , and let , be maps of into such that for every pair . If is a strict contraction and is completely continuous, then the equation has a solution on .

Theorem 17. Under the hypotheses ( ), ( ), and ( )–( ) with for , the problem (4) has at least one -antiperiodic mild solution.

Proof. From our hypotheses on , and ( ), it is easy to see that there exists a such that

Assume that the mapping is defined as in Theorem 9. Let us define the mapping by Note, thanks to the proofs of Theorem 9 with assumptions ( ) and ( ) (i) and Theorem 15 with assumption ( ) (i), that and are well defined and map into itself. Moreover, for every pair and , a direct calculation yields in view of (59), which implies that for every pair .

To obtain the fixed points of , we will use Krasnoselskii’s fixed point theorem. In what follows, we show that and satisfy the conditions of Lemma 16. For , from (i) we infer that for all , which together with yields that is a contraction on . On the other hand, by a similar proof with that in Theorem 9 the mapping is completely continuous on . Hence, applying Lemma 16 we deduce that has at least one fixed point , which is a -antiperiodic mild solution to (4) due to the same idea as the last part of the proof in Theorem 8. This completes the proof of theorem.

Remark 18. Let us note that in Theorems 8 and 15, exponential dichotomy on evolution equations is involved. However, as can be seen from the proofs of Theorems 9 and 17, such condition is not enough to obtain our desired results and therefore is replaced by the special one: is exponentially stable.

5. Application

In this section, we give an example to illustrate our abstract results, which do not aim at generality but indicate how our theorems can be applied to concrete problem.

Consider the antiperiodic problem for partial differential equation in the form supplemented with homogeneous Dirichlet boundary condition , where , are given functions which will be specified later.

Here, our objective is to show the existence of -antiperiodic solutions for the antiperiodic problem (63).

Take and define the operator by , , where It is well-known that has a discrete spectrum and its eigenvalues are , with the corresponding normalized eigenvectors .

Assume that is a Hölder continuous function such that Take , where

Define

It follows from [19, Lemma 6.1 in Chapter 7] that there are constants and such that satisfy for all and . Moreover, we note that for , , , from which we see that for, , Accordingly, satisfy the conditions ( ) and ( ). Thus, the family generates an evolution family : A direct calculation gives

Also, is compact due to the boundedness of for (cf. [19, Theorem 6.1 in Chapter 5]). Moreover, it is easy to verify that for all .

On the other hand, observe that is a Carathéodory function, for all , , and which implies that hypothesis is satisfied with

Therefore, the antiperiodic problem (63) can be transformed into the abstract problem (2)-(3) and assumptions ( ), ( ), and ( ) hold with for . Hence, (63) has at least one -antiperiodic mild solution due to Theorem 9.

Conflict of Interests

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

Acknowledgments

This research was supported by the Natural Science Foundations of China (no. 11271309, 11101202), the Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001), and Hunan Provincial Natural Science Foundation of China (12JJ2001). The authors also want to express their thanks to the anonymous referees for their suggestions and comments that improved the quality of the paper.

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