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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 475956, 24 pages
http://dx.doi.org/10.1155/2012/475956
Research Article

Periodic Solutions of Second-Order Differential Inclusions Systems with -Laplacian

1School of Mathematical Sciences, University of Jinan, Jinan, Shangdong 250022, China
2School of Business, Inner Mongolia University of Science & Technology, Inner Mongolia, Baotou 014010, China

Received 19 June 2012; Accepted 26 September 2012

Academic Editor: Toka Diagana

Copyright © 2012 Liang Zhang and Peng Zhang. 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

The existence of periodic solutions for nonautonomous second-order differential inclusion systems with -Laplacian is considered. We get some existence results of periodic solutions for system, a.e. , , by using nonsmooth critical point theory. Our results generalize and improve some theorems in the literature.

1. Introduction

Consider the second-order system with -Laplacian where , denotes the Clarke subdifferential, and satisfies the following assumption:

(A) and , where which satisfies .

Moreover, we suppose that satisfies the following assumption:

(A’) is measurable in for every and locally Lipschitz in for a.e. , and there exist positive constants , , and such that

for a.e. and all .

If and , system (1.1) reduces to the ordinary -Laplacian system:

Especially, when , then system (1.3) reduces to

The corresponding functional on given by is continuously differentiable and weakly lower semicontinuous on (see [1]), where is a Hilbert space with a norm defined by for .

Since Mawhin and Willem studied the periodic solutions of Hamilton system and obtained a series of results (see [2]). Considerable attention has been paid to the existence of periodic and subharmonic solutions for system (1.3) and (1.4) by the use of critical point theory in variational methods. Many solvability conditions are given, such as Ambrosetti-Rabinowitz conditions, coercivity condition, the convexity condition, the boundedness condition, the subadditive condition, the sublinear condition, and the periodicity condition, see [37] and the references therein.

The classical critical point theory was developed in the sixties and seventies for functionals. The needs of specific applications (such as nonsmooth mechanics and nonsmooth gradient systems) and the impressive progress in nonsmooth analysis and multivalued analysis led to extensions of the critical point theory to nondifferentiable functions, in particular locally Lipschitz functions. The nonsmooth critical point theory for locally Lipschitz functions started with the work of Chang (see [8]). He was able to construct a substitute for the pseudogradient vector field of the smooth theory and use it to obtain nonsmooth versions of the Mountain Pass Theorem of Ambrosetti and Rabinowitz (see [9]) and of the Saddle Point Theorem of Rabinowitz (see [10]). Chang used his theory to study semilinear elliptic boundary value problems with a discontinuous nonlinearity. Later, in 2000, Kourogenis and Papageorgiou (see [11]) obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the -Laplacian with discontinuous nonlinearities. Subsequently, many authors also studied the nonsmooth critical point theory (see [2, 1117]), then the nonsmooth critical point theory is also widely used to deal with the nonlinear boundary value problems (see [11, 14, 15, 1726]). A good survey for nonsmooth critical point theory and nonlinear boundary value problems is the book of Gasinski and Papageorgiou [22].

The operator is said to be -Laplacian and becomes -Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearity than -Laplacian, for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications and raise many mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field. Another field of application of equations with variable exponent growth conditions is image processing (see [12, 27]). The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [12, 2833] for an overview on this subject.

In 2003, X. L. Fan and X. Fan (see [34]) studied the ordinary -Laplacian system and introduced a generalized Orlicz-Sobolev space , which is different from the usual space , then Wang and Yuan (see [35]) obtained the existence and mulplicity of periodic solutions for ordinary -Laplacian system (1.1) with a smooth potential under the generalized Ambrosetti-Rabinowitz conditions. In recent years, there are some papers discussing existence and multiplicity of periodic solutions and subharmonic solutions for problem (1.3) and (1.4) when the potential is just locally Lipschitz in the second variable not continuously differentiable. Some results were obtained based on various hypotheses on the potential . Here we only mention [7, 20, 21, 25, 26], and it should be noted that the abstract result of Clarke (see [1, Theorem 2.7.5]) plays an important role in the establishment of corresponding variational structure. However, to the best of our knowledge, few papers investigated the existence of solutions for problem (1.1), because the main difficulty is the abstract result of Clarke cannot be applied to system (1.1) directly. So we have to find a new approach to solve this problem, and our main idea of the new approach comes from the inspiration of the Theorem 2.7.3 and Theorem 2.7.5 in [1].

The main purpose of this paper is to establish the corresponding variational structure for system (1.1), and we get some existence results of periodic solutions for system (1.1) by using nonsmooth critical point theory. Our results are extensions of the results presented in [4, 26], and our results are new even in the case for system (1.1). The paper is divided into four sections. Basic definitions and preliminary results are collected in Section 2. We give the main results and proofs in Section 3. In Section 4, three examples are presented to illustrate our results.

In this paper, we denote by throughout this paper, and we use and to denote the usual inner product and norm in , respectively.

2. Basic Definitions and Preliminary Results

In this section, we recall some known results in nonsmooth critical point theory, and the properties of space are listed for the convenience of readers.

Definition 2.1 (see [35]). Let satisfy the condition (A), define with the norm
For , let denote the weak derivative of , if and satisfies
Define with the norm .

Remark 2.2. If , where is a constant, by the definition of , it is easy to get , which is the same with the usual norm in space .

The space is a generalized Lebesgue space, and the space is a generalized Sobolev space. Because most of the following Lemmas have appeared in [1, 7, 24, 34], we omitted their proofs.

Lemma 2.3 (see [34]). and are both Banach spaces with the norms defined above, when , they are reflexive.

Lemma 2.4 (see [34]). The space is a separable, uniform convex Banach space, its conjugate space is , for any and , we have where .

Lemma 2.5 (see [35]). If we denote , , then(i) (=1;>1) (=1;>1)(ii) , (iii) ; (iv)For and , .

Definition 2.6 (see [2]). with the norm := .

For a constant , using another conception of weak derivative which is called -weak derivative, Mawhin and Willem gave the definition of the space by the following way.

Definition 2.7 (see [2]). Let and , if then is called a -weak derivative of and is denoted by .

Definition 2.8 (see [2]). Define with the norm .

Definition 2.9 (see [34]). Define and let be the closure of in .

Remark 2.10. From Definition 2.8, if , it is easy to conclude that .

Lemma 2.11 (see [34]). (i) is dense in ,(ii) ,(iii)If , then the derivative is also the -weak derivative , that is, .

Remark 2.12. In the following paper, we use instead of for convenience without clear indications.

Lemma 2.13 (see [24]). Assume that , then(i) ,(ii) has its continuous representation, which is still denoted by , ,(iii) is the classical derivative of , if .Since every closed linear subspace of a reflexive Banach space is also reflexive, we have

Lemma 2.14 (see [34]). is a reflexive Banach space if .

Obviously, there are continuous embeddings , , and . By the classical Sobolev embedding theorem we obtain.

Lemma 2.15 (see [34]). There is a continuous embedding when , the embedding is compact.

In order to establish the variational structure for system (1.1), it is necessary to construct some appropriate function spaces. The Cartesian product space is defined by and is also a reflexive and separable Banach space with respect to the norm where .

Lemma 2.16. Define the operator : as follows: then is also a reflexive and separable Banach space with respect to the norm defined in (2.12).

Proof. Let be a Cauchy sequence in , then there exists in such that converge to in . We have by Definition 2.7, then by Lemma 2.4, we conclude as in (2.14). In view of (2.15), is the -weak derivative of , that is, is also in , so is a complete subspace of , which implies is also a reflexive and separable Banach space.

Remark 2.17. We use to denote the norm in defined by (2.12).

Definition 2.18. Let denote the space of essentially bounded measurable functions from into under the norm it is obvious that is a Banach space under the norm defined above.

Remark 2.19. We use and to denote and , respectively.

Lemma 2.20. is a closed subspace of .

Proof. Let be a Cauchy sequence in with respect to the norm defined in (2.16). Then there exists in such that converge to in . By Definition 2.7, we have we conclude that by Lemma 2.4 as in (2.17). In view of (2.18), is the -weak derivative of , that is, is also in , so is a complete subspace of , which implies is a closed subspace of .

Lemma 2.21. Suppose is a bounded linear functional on , if restricted to the space , denoted by , that is, then is a bounded linear functional on .

Proof. It is obvious that is a linear functional on , so we only to show the is continuous on .
Let and , that is, where and , then by Definition 2.1 and (2.16), where . Then we conclude by (2.21) and Definition 2.1.
Furthermore, the norm of in is therefore, combining (2.22) and (2.23), we get the bound and this completes the proof.

Lemma 2.22. The space , where there exists , if , such that

Proof. Let , from Remark 2.10, , from the inequality in classical Sobolev space, there exists a positive constant , such that and this completes the proof.

Lemma 2.23 (see [34]). Each of the following two norms is equivalent to the norm in :(i) , ,(ii) , where .

Proposition 2.24. In space , .

Proof. From Lemma 2.23, there exists a positive constant , such that if , it is easy to get When , we conclude that by Lemma 2.5, it follows (2.29) and (2.30) that which implies that The proof is completed.

Lemma 2.25 (see [34]). If , then the following statements are equivalent to each other(i) , (ii) , (iii) in measure in and .

Definition 2.26 (see [1]). Let be Lipschitz near a given point in a Banach space , and let be any other vector in . The generalized directional derivative of at in the direction , denoted by , is defined as follows: where is also a vector in and is a positive scalar, and we denote by the generalized gradient of at (the Clarke subdifferential).

Lemma 2.27 (see [1]). Let and be points in a Banach space , and suppose that is Lipschitz on open set containing the line segment . Then there exists a point in such that

Definition 2.28 (see [8]). A point is said to be a critical point of a locally Lipschitz if , namely, for all every . A real number is called a critical value of if there is a critical point such that .

Definition 2.29 (see [8]). If is a locally Lipschitz function, we say that satisfies the Palais-Smale condition if each sequence in such that is bounded and has a convergent subsequence. We define , where the minimum exists from the fact that is a -weakly compact convex subset.

Lemma 2.30 (see [8]). Let be a real Banach space, and let be a locally Lipschitz function defined on satisfying the (PS) condition. Suppose with a finite dimensional subspace , and there exist constants and a bounded neighborhood of in such that Then has a critical point.

Lemma 2.31. The functional given by is weakly lower semicontinuous on .

Proof. We divide into two parts, , where it is obvious that is convex and continuous by Lemma 2.25, then is weakly lower semicontinuous by Theorem 1.2 in [2], and is weakly continuous, that is, is the sum of two weakly lower semicontinuous functionals, which implies that is weakly lower semicontinuous.

Lemma 2.32 (see [35]). The functional defined in Lemma 2.31 is continuously differentiable on and is given by and is a mapping of , that is, if weakly in and then has a convergent subsequence on .

Clarke considered the following abstract framework in [1]:(i)let be a -finite positive measure space, and let be a separable Banach space,(ii)let be a closed subspace of , where denotes the space of measure essentially bounded functions mapping to , equipped with the usual supremum norm,(iii)define a functional on via where is a given family of functions,(iv)suppose that the mapping is measurable for each in , and that is a point at which is defined (finitely),(v)suppose that there exist and a function in such that for all and all and in .

Under this conditions described above, is Lipschitz in a neighborhood of and one has Further, if each is regular at for each , then is regular at and equality holds.

We give an example to illustrate Clarke’s abstract framework with the following cast of characters:(i) with Lebesgue measure, and let , which is a separable Banach space,(ii)let , which is a closed subspace of ,(iii)define a functional on via (iv) satisfies the condition (A’).

Under the hypothesis above, we only need to justify the condition (2.42), in fact, by Lebourg’s mean value theorem, where and for a.e. and all and in , where and is a positive constant.

In view of (A’), we get for a.e. and all , in .

We can apply Clarke’s abstract framework to our example, that is, for any such that for all , where is a measurable selection of .

Now we can prove the following result which is fundamental in our paper.

Lemma 2.33. Suppose is given by where and satisfies the condition (A’). The corresponding functionals and on are given by Then is Lipschitz on and one has

Proof. Take an arbitrary element in , and it suffices to prove is Lipschitz on and (2.51) holds for .
is continuously differentiable on , that is, for any in , so is Lipschitz on .
When , we conclude by Lemma 2.15, where is a positive constant. Arguing as in (2.46), for a.e. , where .
By Lemma 2.15 and (2.54), we have so is also Lipschitz on , which implies that as the sum of two Lipschitz functionals is also Lipschitz on .
For any in , one has for any in by Fatou Lemma and it is obvious that for a.e. and all in , then we conclude by (2.56) for any in and (2.58) remains true if we restrict to , which is a closed subspace of by Lemma 2.20.
By Lemma 2.21, we conclude the bounded linear functional on restricted to is also a bounded linear functional, and we use to denote the functional restricted on .
We interpret (2.58) as saying that belongs to the subgradient at (0, 0) of the convex functional which is defined in , where for all in .
In view of condition (A’) and (2.57), we have for a.e. and all , in .
Now we can apply Clarke’s abstract framework to with the following cast of characters:(i) with Lebesgue measure, and let , which is a separable Banach space with the norm ,(ii)let , which is a closed subspace of , and denotes the space of measure essentially bounded functions mapping to , equipped with the usual supremum norm by Definition 2.18,(iii)define a functional on by (2.59),(iv)the mapping is measurable for each in by (2.57), see details in [1], and that is a point at which is defined (finitely),(v)the condition (2.42) in Clarke’s abstract framework is satisfied by (2.60).

By (2.57), We get thus, every can be written as for any in , where for a.e. .

When , it is obvious that and is dense in by Lemma 2.11. So for each , we can choose such that

Combining (2.62) and (2.63), we have for all .

We conclude and this completes the proof.

3. Main Results and Proofs of Theorems

Theorem 3.1. Let satisfy the condition (A’) with , and we suppose the following condition holds where is the same in condition (A).

Then system (1.1) has at least one solution which minimizes in .

If we replace the (A1) in Theorem 3.1 by the following condition: we obtain the following theorem.

Theorem 3.2. Let satisfy the condition (A’) with and (A2). Then system (1.1) has at least one solution in .

Remark 3.3. Theorems 3.1 and 3.2 generalize Theorems 1 and 2, respectively in [3].

Proof of Theorem 3.1. For , let and . From Lemma 2.27, it follows that there exist in such that for a.e. , where .
It follows from (A’), Young inequality and Lemma 2.22 that for all , and some positive constants , , , and .
Hence we have for all , which implies that because of and the Proposition 2.24.
By Lemma 2.31, the functional is weakly lower semicontinuous on , and it follows that has a minimum on by Theorem 1.1 in [2]. Proposition  2.3.2 in [1] implies that , that is, is a critical point for . So, problem (1.1) has at least one solution .

Proof of Theorem 3.2. We will show that defined in Lemma 2.31 satisfies the (PS) condition. Let be a sequence in such that is bounded and as . Using the definition of , it results that for each there exists with
In view of Lemma 2.33, if , it results that there exist such that
It follows Lemma 2.22 and Young inequality that for all and some positive constants , , , and , where is the same as in Lemma 2.22.
Hence, we have for all .
It follows from (2.31) that by (3.8) and (3.9), we have for some positive constants , , and all .
By the proof of (3.2) we have for all .
It follows from the boundness of , (3.10) and (3.11) that for all and some positive constant .
It follows (A2) and (3.12) that is bounded, hence is bounded by (2.31) and (3.10).
The sequence has a subsequence, also denoted by , such that and is bounded by Lemma 2.15, where is a positive constant.
Therefore we have , where is the function from the Palais-Smale condition, and such that as , so where and .
By (3.14) and (3.15), we get , that is, so it follows from Lemma 2.32 that admits a convergent subsequence.
We now prove satisfies the other conditions of Lemma 2.30. Let be the subspace of defined in Lemma 2.22, then we have as in . In fact it follows from (3.2) that for all and some positive constants , , and . for all , which implies (3.17) by Proposition 2.24.
Moreover, we have as in , which follows (A2).
We have proved the functional satisfies all the conditions of Lemma 2.30, so we know that has at least one critical point by Lemma 2.30, which is a periodic solution for system (1.1). The proof is complete.

4. Example

In this section, we give three examples to illustrate our results.

Example 4.1. In system (1.1), let and it is easy to verify that , where for every and all .
By Theorem 3.1, system (1.1) has at least one solution , but it is obvious that the results in the reference cannot be applied to our example.

Example 4.2. In system (1.1), let and , it is easy to verify that , where for every and all .
By Theorem 3.2, system (1.1) has at least one solution , but it is obvious that the results in the reference cannot be applied to our example.

Example 4.3. In system (1.1), let , and where denotes the positive constant .
It is obvious that is continuously differentiable, then the the Clarke subdifferential set reduces to one element