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

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.

For all = 1, 2, . . . , , find ∈ and ∈ (u) such that (2) 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 = 1, are single-valued, or are indicators of some convex subsets for = 1, 2, . . . , . 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 ( = 1, 2, . . . , ) 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 ∘ ( ; V − ) being incorporated into ∘ ( ; V − ) under the regularity condition. For more information on the research of hemivariational inequalities and systems of hemivariational inequalities, we can refer to [7][8][9][10][11][12][13][14][15][16] 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 > 2 is a natural generalization of the case where = 2. 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 1 ∈ 1 , 2 ∈ 2 , 1 ∈ 1 ( ) and 2 ∈ 2 ( ) such that 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 variationalhemivariational inequalities concerning set-valued mapping is solvable by considering the simple case, the problem (P ) 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.

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 ∘ ( , V). We denote bŷ( ) :  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 , V ∈ , and be locally Lipschitz functional defined on . Then one has the following. (2) ∘ ( , V) is upper semicontinuous as a function of ( , V), but as a function of V alone, it is Lipschitz continuous on .
(4) For every V ∈ , one has whenever this limit exists. The functional is regular (in the sense of Clarke) on if it is regular at every point ∈ .

Journal of Applied Mathematics
Definition 10. Let be real reflexive Banach space with dual * . A mapping from into 2 * is said to be as follows: (1) coercive if there exists a real-valued function on + with lim → ∞ ( ) = ∞ such that for all ( , * ) ∈ ( ), one has (3) 0 -coercive if there exists a real-valued function on + with lim → ∞ ( ) = ∞ such that for some 0 ∈ 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.

Main Results
In this section, we first give an existence theorem for the solution to the problem (P ) 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 variationalhemivariational 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 (P ) of a system of two variational-hemivariational inequalities and a generalized vector variational-hemivariational inequality. Let = 1 × 2 . 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 : → 2 * , an operator : → 1 × 2 , and a functional : → ∪ {+∞}, which are specified as follows. For all = ( 1 , 2 ) ∈ , one has ( ) = ( 1 ( ) , 2 ( )) , for all , V ∈ and ∈ [0, 1]. As for the lower semicontinuity of the functional , by assuming that → in , which implies 1 → 1 in 1 and 2 → 2 in 2 , we can get from the lower semicontinuity of 1 and 2 that which means that is lower semicontinuous on . Now, we prove the equalitŷ( ) =̂1( 1 ) ×̂2( 2 ). Assume that ∈̂( ) ⊂ * , which says that In particular, for any V 1 ∈ 1 , let V = (V 1 , 2 ) in (23), and then we can get that (24) Similarly, by letting V = ( 1 , V 2 ) in (23) for any V 2 ∈ 2 , we can obtain . For all V ∈ , = 1, 2, it follows from ∈̂( ) that By adding the two inequalities (26), we obtain which implies that ∈̂( ); that is,̂( ) ⊇̂1( 1 ) × Now, we consider the following generalized vector variational-hemivariational inequality. Find = ( 1 , 2 ) ∈ and = ( 1 , 2 ) ∈ ( ) such that We first give a crucial lemma which establishes the relationship between the problem (P ) of a system of two variational-hemivariational inequalities and the problem (P ) of a generalized vector variational-hemivariational inequality.
Lemma 13. Assume that the locally Lipschitz functional : is regular on 1 × 2 . Then any solution = ( 1 , 2 ) ∈ to the problem (P ) is always a solution to the problem (P ).
Proof. Assume that = ( 1 , 2 ) solves the problem (P ), which says that there exists an = ( 1 , 2 ) ∈ ( ) such that for all V ∈ , one has Specially, for any V 1 ∈ 1 , let V = (V 1 , 2 ) ∈ in (29), and then we can get from Proposition 3 that Similarly, by letting V = ( 1 , V 2 ) ∈ in (29) for any V 2 ∈ 2 , we can obtain that which together with the inequality (30) implies that = ( 1 , 2 ) is a solution to the problem (P ). This completes the proof of Lemma 13.

Remark 14. It follows from Proposition 3 that, just under regularity condition of the functional ,
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.
(3) For all 2 ∈ 2 , there exist an element 1 ∈ (̂1) ⊂ 1 and a constant 1 > 0 such that (4) For all 1 ∈ 1 , there exist an element 2 ∈ (̂2) ⊂ 2 and a constant 2 > 0 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 1 and (4) on the operator 2 in the assumption (HA) imply the 1 -coercivity of 1 with respect to the first argument and 2 -coercivity of 2 with respect to the second argument, respectively. Moreover, for = ( 1 , 2 ) ∈ (̂), the operator defined in (21) is also -coercive with constant = min{ 1 , 2 }/2. In fact, for all ∈ ( ), one has which implies the -coercivity with constant of operator on .
The assumption (HJ) is as follows.
We are now in a position to give our main result on the existence of solution to the problem (P ), a system of two generalized variational-hemivariational inequalities.
where ‖ ‖ is the norm of the operator defined by (21).
Proof. By Lemma 13, the existence of solution to the problem (P ) of a system of two generalized variationalhemivariational inequalities can be proved as long as the problem (P ) of a generalized vector variationalhemivariational inequality is solvable. Therefore, we consider the following inclusion problem. Find ∈ such that where : → 2 * 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 (P ) 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 , = 1, 2, and it is easy to check that is bounded on , which implies that 0 : → 2 * with 0 ( ) = ( 0 + ) is quasibounded for any 0 ∈ .
There exist ∈ ( ) such that = * . Since is bounded on by Remark 16, → in by the compactness of the operators , = 1, 2, 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 .
By the definition of the operator , there exist ∈ ( ), ∈ ( ), and ∈̂( ) such that By multiplying the equality (43) by V − for all V ∈ , 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 (P ) of a generalized vector variational-hemivariational inequality. As stated at the beginning of our proof, is also a solution to the problem (P ) of a system of two generalized variationalhemivariational 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 (P ) by the surjectivity theorem since the pseudomonotonicity of operator 1 with respect to the first argument and the pseudomonotonicity of operator 2 with respect to the second argument, which are necessary to prove the existence of solution to each generalized variational-hemivariational inequality in problem (P ), cannot guarantee the pseudomonotonicity of the operator defined in (21) in general. However, some special cases in which the pseudomonotonicity of operator 1 with respect to the first argument and the pseudomonotonicity of operator 2 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).
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 (P ) of a system of two generalized variationalhemivariational inequalities.
As a natural generalization of Theorem 17 for the existence of solution to the problem (P ) 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 variationalhemivariational inequalities concerning set-valued mappings.
Theorem 21. Suppose that the following assumptions on the operators in the problem (P) of a system of generalized variational-hemivariational inequalities hold.

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.
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 = 1, is single-valued and is an indicator of a convex subset .
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 2 (Ω, ), which satisfies wherẽ= √ 2 |Ω| ≥ 0 and̃= √ 2 ≥ 0. 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.