`Journal of Applied MathematicsVolume 2013 (2013), Article ID 305068, 9 pageshttp://dx.doi.org/10.1155/2013/305068`
Research Article

## A System of Generalized Variational-Hemivariational Inequalities with Set-Valued Mappings

1Southwest Petroleum University, State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu, Sichuan 610500, China
2Southwest Petroleum University, The First National Gas Plant of ChangQing Oil Field Constituent Company, Xi'an, Shanxi 750001, China
3School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
4Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received 10 July 2013; Accepted 23 August 2013

Copyright © 2013 Zhi-bin Liu et al. 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 using surjectivity theorem of pseudomonotone and coercive operators rather than the KKM theorem and fixed point theorem used in recent literatures, we obtain some conditions under which a system of generalized variational-hemivariational inequalities concerning set-valued mappings, which includes as special cases many problems of hemivariational inequalities studied in recent literatures, is solvable. As an application, we prove an existence theorem of solutions for a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.

#### 1. Introduction

Let be real, separable, reflexive Banach spaces with dual spaces , and let be real reflexive Banach spaces with dual spaces such that there exist linear continuous and compact operators for . We denote by the duality pairing between Banach space and its dual and by , the norms on the space and its dual space , respectively, where . Assume that , are set-valued mappings on the product space of Banach spaces and that is a functional on the product space of Banach spaces , which is locally Lipschitz with respect to each component; that is, for all , the functionals are locally Lipschitz for all fixed , , and ,  are proper, convex, and lower semicontinuous functionals. In this paper, we study a system of generalized variational-hemivariational inequalities concerning set-valued mappings, which is specified as follows.

For all , find and such that where , and is the partial generalized directional derivative (in the sense of Clarke) of the functional , which is locally Lipschitz for each component, with respect to the th component at the point in the direction for all given , , which can be defined by

In the last few years, there are many researchers who dedicated themselves to the study of various types of hemivariational inequalities and systems of hemivariational inequalities, which are a generalization of the variational inequalities, and related problems such as equilibrium problems. In these papers, based on Clarke’s generalized directional derivative and Clarke's generalized gradient for locally Lipchitz functions, the researchers study the existence and uniqueness of solution by mainly using KKM theorems, surjectivity theorems for pseudomonotone and coercive operators, fixed point theorems, critical point theory, and so on. We refer readers for the study of hemivariational inequalities to monographs of Carl et al. [1], Migórski et al. [2], Naniewicz and Panagiotopoulos [3], and Panagiotopoulos [4]. For the system of hemivariational inequalities, Denkowski and Migórski [5] studied a dynamic thermoviscoelastic frictional contact problem which was modeled by a system of evolution hemivariational inequalities. They proved the existence and uniqueness of the weak solution for the problem by using a surjectivity result for operators of pseudomonotone type. In 2011, Repovš and Varga [6] studied the Nash equilibrium point by using the Ky Fan version of the KKM theorem and the Tarafdar fixed point theorem for a class of hemivariational inequality system. It is obvious that some problems studied in literatures are special cases of our system of generalized variational-hemivariational inequalities under some special conditions, such as , are single-valued, or are indicators of some convex subsets for . Although it seems that our problem (P) cannot include the problem studied in [6] as a special case, we remark here that, in essence, the problem (P) is a generalization of the problem in [6] since, when () are single-valued and are the indicators of the convex subsets , the problem (P) reduces to the problem studied by Repovš and Varga [6] with and being incorporated into under the regularity condition. For more information on the research of hemivariational inequalities and systems of hemivariational inequalities, we can refer to [716] and references therein.

It is well known that, by surjectivity theorem of pseudomonotone and coercive operators, there exists solution to each variational-hemivariational inequality in the system (1) for all , under some suitable conditions on the operators , , and . A natural question is whether these conditions are sufficient for the existence of solutions to the system (1) which is combined by solvable variational-hemivariational inequalities. If not, what other stronger conditions do we need to guarantee the solvability of the system (1)? In this paper, we are devoted to these questions by using surjectivity theorem of pseudomonotone and coercive operators rather than the KKM theorem, and the fixed point theorem used by Repovš and Varga in [6] to obtain the existence of the solutions to the problem (P) of a system of generalized variational-hemivariational inequalities concerning set-valued mappings.

As will be seen in the proof of our main theorem in Section 3, the case where for any finite positive integer is a natural generalization of the case where . Therefore, in what follows, We will focus on the problem of a system of two generalized variational-hemivariational inequalities, which can be reformulated as follows. Consider ,  , and such that where , and for .

The paper is structured as follows. In Section 2, we recall some preliminary material. Section 3 gives conditions under which the problem (P) of a system of generalized variational-hemivariational inequalities concerning set-valued mapping is solvable by considering the simple case, the problem () of a system of two generalized variational-hemivariational inequalities. At last, in Section 4, we are concerned with an application of our results to a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.

#### 2. Preliminaries

In this section, we recall some important notations and useful results on nonlinear analysis, nonsmooth analysis, and operators of monotone type, which can be found in [2, 3, 17, 18].

Without confusion of symbols, we suppose, just in this section, that is a Banach space with its dual and duality paring between and , is a proper and convex functional, and is a locally Lipschitz functional with Clarke's generalized directional derivative . We denote by and the subgradient of the convex functional in the sense of convex analysis and Clarke's generalized gradient of the locally Lipschitz functional , respectively. Then,

We have the following basic properties on Clarke's generalized directional derivative and Clarke's generalized gradient (see, e.g., [2, 17]).

Proposition 1. Let be Banach space, and let , and be locally Lipschitz functional defined on . Then one has the following. (1)The function   is finite, positively homogeneous, subadditive, and then convex on ;(2)  is upper semicontinuous as a function of , but as a function of   alone, it is Lipschitz continuous on .(3)  is a nonempty, convex, bounded, and weak*-compact subset of .(4)For every , one has (5)The graph of the Clarke generalized gradient    is closed in    topology. (6)The multifunction    is upper semicontinuous from    into .

Definition 2. A locally Lipschitz functional is said to be regular (in the sense of Clarke) at if (i)for all the directional derivative exists;(ii)for all , , where is directional derivative of at in the direction , which is defined by whenever this limit exists. The functional is regular (in the sense of Clarke) on if it is regular at every point .

Proposition 3. Let and be two Banach spaces. If is locally Lipschitz and either or is regular at , then or equivalently one has where (resp., ) represents the partial generalized subgradient of (resp., ) and (resp., ) denotes the partial generalized directional derivative of (resp., ) at the point (resp., ) in the direction (resp., ), but the converse of inclusion (7) and inequality (8) is not true in general.

Definition 4. Let be real reflexive Banach space with dual . A mapping from into is said to be pseudomonotone if (1)the set is nonempty, bounded, closed, and convex for all ;(2) is upper semicontinuous from each finite dimensional subspace of to endowed with the weak topology;(3) is a sequence in converging weakly to , and is such that then for each element there exists such that

Definition 5. Let be real reflexive Banach space with dual . A mapping from into is said to be generalized pseudomonotone if for any sequences , with , weakly in , weakly in and then one has and .

Proposition 6. Let be real reflexive Banach space with dual and let , be two pseudomonotone mappings from into . Then is pseudomonotone.

Proposition 7. Let be real reflexive Banach space with dual and let be a pseudomonotone mapping from into . Then is a generalized pseudomonotone.

Proposition 8. Let be real reflexive Banach space with dual and let be a bounded, generalized pseudomonotone mapping from into . Assume that, for each , is a nonempty closed convex subset of . Then is pseudomonotone.

Definition 9. Let be real reflexive Banach space with dual . The operator is said to be as follows: (1)monotone if for all , lying in the graph of , one has (2)maximal monotone if it is monotone and if is such that then . (3)quasibounded if for each there exists such that, whenever , , and ; then (4)strongly quasibounded if for each there exists such that for all with and , one has

Definition 10. Let be real reflexive Banach space with dual . A mapping from into is said to be as follows: (1)coercive if there exists a real-valued function on with such that for all , one has (2)coercive with constant if (3)-coercive if there exists a real-valued function on with such that for some and for all , one has

The following theorem is a surjectivity theorem for the sum of a pseudomonotone, coercive operator, and a maximal monotone operator, which is important to the proof of our main results.

Theorem 11 (see [3]). Let be a real reflexive Banach space with dual , let be a maximal monotone mapping from into with , and let be a -coercive, pseudomonotone operator from into . Suppose further that either is quasibounded or is strongly quasibounded, where and the same for . Then .

#### 3. Main Results

In this section, we first give an existence theorem for the solution to the problem () of a system of two generalized variational-hemivariational inequalities. And then, as a natural generalization, an existence theorem for the solution to the problem (P), a system of generalized variational-hemivariational inequalities concerning set-valued mappings is also obtained.

Before we present the main existence theorem, for the simplicity of writing, we define some useful symbols and give a crucial lemma in advance, which establishes the relationship between the problem () of a system of two variational-hemivariational inequalities and a generalized vector variational-hemivariational inequality. Let . Endowed with the norm defined by is a reflexive Banach space with dual . The duality pairing between and is given by

On the Banach space defined above, we further define a set-valued mapping , an operator , and a functional , which are specified as follows. For all , one has

Lemma 12. Assume that ,  are proper, convex, and lower semicontinuous functionals. The functional defined above is also a proper, convex, and lower semicontinuous functional on . Moreover, .

Proof. Since the functionals , are proper and convex, it is easy to show that is also proper and convex by the inequality for all and . As for the lower semicontinuity of the functional , by assuming that in , which implies in and in , we can get from the lower semicontinuity of and that which means that is lower semicontinuous on .
Now, we prove the equality . Assume that , which says that
In particular, for any , let in (23), and then we can get that
Similarly, by letting in (23) for any , we can obtain which together with (24) implies that ; that is, .
Conversely, let . For all ,  , it follows from that By adding the two inequalities (26), we obtain which implies that ; that is, . This completes the proof of Lemma 12.

Now, we consider the following generalized vector variational-hemivariational inequality. Find and such that

We first give a crucial lemma which establishes the relationship between the problem () of a system of two variational-hemivariational inequalities and the problem () of a generalized vector variational-hemivariational inequality.

Lemma 13. Assume that the locally Lipschitz functional is regular on . Then any solution to the problem is always a solution to the problem .

Proof. Assume that solves the problem (), which says that there exists an such that for all , one has
Specially, for any , let in (29), and then we can get from Proposition 3 that
Similarly, by letting in (29) for any , we can obtain that which together with the inequality (30) implies that is a solution to the problem (). This completes the proof of Lemma 13.

Remark 14. It follows from Proposition 3 that, just under regularity condition of the functional , does not hold in general, while the inequality is true. Therefore, without other much stronger conditions on functional , the inverse of the Lemma 13 is not true in general.

We give some assumptions on the operators and in the system (3) of two generalized variational-hemivariational inequalities. The assumption (HA) is as follows.(1) is bounded on and pseudomonotone with respect to the first argument; that is, for all , the operator is pseudomonotone on .(2) is bounded on and pseudomonotone with respect to the second argument; that is, for all , the operator is pseudomonotone on .(3) For all , there exist an element and a constant such that (4) For all , there exist an element and a constant such that

Remark 15. It is clear that the hypotheses (1) and (2) in the assumption (HA) imply that the operator defined in (21) is also bounded on . The hypotheses (3) on the operator and (4) on the operator in the assumption (HA) imply the -coercivity of with respect to the first argument and -coercivity of with respect to the second argument, respectively. Moreover, for , the operator defined in (21) is also -coercive with constant . In fact, for all , one has which implies the -coercivity with constant of operator on .   The assumption (HJ) is as follows.(1) For all , there exist constants such that (2) For all , there exist constants such that

Remark 16. It is clear that the hypotheses in assumption (HJ) imply that and are bounded on and , respectively. Moreover, if is regular on , then is also bounded on . (In the following, let and for simplicity of writing.) In fact, since is regular on , the inclusion holds. It follows from (35) and (36) that with and . This also means that is bounded on .

We are now in a position to give our main result on the existence of solution to the problem (), a system of two generalized variational-hemivariational inequalities.

Theorem 17. Suppose that the set-valued mappings , , which satisfy the assumption (HA), is such that the operator defined in (21) is pseudomonotone on . Let be linear continuous and compact operators, let be a regular, locally Lipschitz functional which satisfies the hypothesis (HJ), let and , be proper, convex, and lower semicontinuous functionals. Then the problem admits at least one solution under the condition where is the norm of the operator defined by (21).

Proof. By Lemma 13, the existence of solution to the problem () of a system of two generalized variational-hemivariational inequalities can be proved as long as the problem () of a generalized vector variational-hemivariational inequality is solvable. Therefore, we consider the following inclusion problem. Find such that where with for all . We will prove the existence of solution to the inclusion problem (39) by the surjectivity theorem (Theorem 11), which implies that the problem () is solvable.
Claim  1 ( is bounded on ). Since the operator is bounded on under assumption (HA) by Remark 15, is also bounded on under the assumption (HJ) by Remark 16, and is linear continuous by the linearity and continuity of the operators , and it is easy to check that is bounded on , which implies that with is quasibounded for any .
Claim  2   ( is pseudomonotone on ). Since and the operator is pseudomonotone, we only need to prove that is pseudomonotone. To this end, firstly, we prove that is generalized pseudomonotone. Let weakly in with weakly in and . There exist such that . Since is bounded on by Remark 16, in by the compactness of the operators , , and , we have the fact that is bounded in . Thus there exists a subsequence, which is also denoted by , such that weakly in with some . By using the equality , it is easy to get that . Since with weakly in and in , we get by the closedness of with topology and the reflexivity of that , and thus . Moreover, it follows from weakly in and in that which together with implies that is generalized pseudomonotone on . Secondly, it is easy to check that is nonempty, convex, and closed in for all since is a nonempty, convex, and closed subset in for all and is linear and continuous on . Thirdly, the operator is bounded on , which has been proved in Claim 1. Consequently, it follows from the Proposition 8 that is pseudomonotone on .
Claim  3  ( is -coercive on ). Let and , and then there exist and such that . By Remarks 15 and 16, we have which together with the condition means that is -coercive on with function .
It is well known that is a maximal monotone operator on since, by Lemma 12, the functional is proper, convex, and lower semicontinuous on (see [18]). We are now in a position to apply Theorem 11 to the set-valued operators and . We deduce that is surjective, which implies that there exist such that
By the definition of the operator , there exist , , and such that
By multiplying the equality (43) by for all , we obtain from the definition of Clarke's generalized subgradient of the functional and subgradient in the sense of convex analysis of the functional that which implies that solves the problem () of a generalized vector variational-hemivariational inequality. As stated at the beginning of our proof, is also a solution to the problem () of a system of two generalized variational-hemivariational inequalities by Lemma 13. This completes the proof of Theorem 17.

Remark 18. The pseudomonotonicity of the operator defined in (21) is necessary for the proof of the existence of solution to the problem () by the surjectivity theorem since the pseudomonotonicity of operator with respect to the first argument and the pseudomonotonicity of operator with respect to the second argument, which are necessary to prove the existence of solution to each generalized variational-hemivariational inequality in problem (), cannot guarantee the pseudomonotonicity of the operator defined in (21) in general. However, some special cases in which the pseudomonotonicity of operator with respect to the first argument and the pseudomonotonicity of operator with respect to the second argument imply the pseudomonotonicity of the operator defined in (21) can be given under some stronger conditions (see [5]).

It is obvious that, by similar arguments as proof of Theorem 17, we have the following results for the existence of solution to each generalized variational-hemivariational inequality in the system (3).

Theorem 19. Suppose that, for , are set-valued mappings satisfying the assumption (HA) and that are linear continuous and compact operators. Let be a regular, locally Lipschitz functional on , which satisfies the hypothesis (HJ), and let , be proper, convex, and lower semicontinuous functionals. Then, the th generalized variational-hemivariational inequality in the system (3) admits at least one solution for all ,  under the condition where is the norm of the operator .

Remark 20. By comparing Theorems 17 with 19, we remark here that, in addition to the pseudomonotonicity of the operator defined in (21), we need strongly condition (38) than (45) to obtain the existence of solution to the problem () of a system of two generalized variational-hemivariational inequalities.

As a natural generalization of Theorem 17 for the existence of solution to the problem () of a system of two generalized variational-hemivariational inequalities, we can obtain the following theorem for the existence of solution to the problem (P) of a system of generalized variational-hemivariational inequalities concerning set-valued mappings.

Theorem 21. Suppose that the following assumptions on the operators in the problem of a system of generalized variational-hemivariational inequalities hold. (1)For , are set-valued mappings satisfying the following. (a) are bounded on and pseudomonotone with respect to the th argument. (b)the operator , which is defined by , is pseudomonotone on . (c)For all , , there exist an element and a constant such that (2) is a regular and locally Lipschitz functional which satisfies that for all and , , there exist constants such that (3)For , are linear continuous and compact operators and are proper, convex, and lower semicontinuous functionals. Then the problem admits at least one solution under the condition where , and is the norm of the operator defined by .

#### 4. An Application

In this section, we are concerned with an application of our results to a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives.

Let be a bounded and open set in , let be real, separable, and reflexive Banach spaces with dual spaces . For , are set-valued mappings, are linear continuous and compact operators on , and are proper, convex, and lower semicontinuous functionals. We consider the following system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives. For all , find and such that where , and is a function satisfying the following assumption: (Hj) is as follows.(1) is locally Lipschitz on for .(2) is a Carathodory function.(3) Either or is regular on for a.e. .(4) For all and a.e. , there exists constant such that

Remark 22. The problem (49) we considered in this section includes the problem studied by Panagiotopoulos et al. [19] by using Brouwer's fixed point theorem as a special case where , is single-valued and is an indicator of a convex subset .

We define a functional on as follows:

It follows from Theorem 3.47 in [2] that, under the assumption (Hj) on the function , defined by (51) is a locally Lipschitz functional on , which satisfies where and .

Now, under the conditions (52), we are in a position to apply our result, Theorem 21, to the problem (49), a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives. We conclude this section with the following theorem, which gives the existence of solution to the problem (49).

Theorem 23. For the problem (49), a system of generalized variational-hemivariational inequalities involving integrals of Clarke's generalized directional derivatives, one assumes the following. (1) For , are set-valued mappings satisfying the following. (a) are bounded on and pseudomonotone with respect to the th argument.(b) the operator , which is defined by , is pseudomonotone on .(c) For all , , there exist an element and a constant such that (2) is a function satisfying the assumption (Hj).(3) For , are linear continuous and compact operators and are proper, convex, and lower semicontinuous functionals. Then the problem (49) admits at least one solution under the condition where , , and is the norm of the operator defined by .

#### Acknowledgments

This work was supposed by the National Natural Science Foundation of China (11101069) and the Open Fund (PLN1104, PLN1102) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

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