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Abstract and Applied Analysis
Volume 2014, Article ID 896941, 8 pages
http://dx.doi.org/10.1155/2014/896941
Research Article

On a Class of Variational-Hemivariational Inequalities Involving Upper Semicontinuous Set-Valued Mappings

1School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
2Department of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

Received 24 January 2014; Accepted 17 March 2014; Published 27 April 2014

Academic Editor: Fu-quan Xia

Copyright © 2014 Guo-ji Tang 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

This paper is devoted to the various coercivity conditions in order to guarantee existence of solutions and boundedness of the solution set for the variational-hemivariational inequalities involving upper semicontinuous operators. The results presented in this paper generalize and improve some known results.

1. Introduction

Let be a nonempty, closed, and convex subset in . Let be a set-valued mapping and let be a convex and lower semicontinuous function such that , where is the effective domain of . Let be a bounded open set in and be a function. Let be a linear and continuous mapping, where . We shall denote and denote by Clarke’s generalized directional derivative of a locally Lipschitz mapping at the point with respect to the direction , where . In this paper, we consider the following variational-hemivariational inequality problems:(P)find and such that which is studied by some researchers (see, for example, [1, 2]). Problem (P) includes some models as special cases.

Case 1. In the case when is single-valued, problem (P) becomes the following variational-hemivariational inequality problem: find such that In 2000, by using Mosco’s Theorem, Motreanu and Rǎdulescu [3] proved that if the operator is monotone and hemicontinuous, then problem (2) admits a solution (see Theorem 2 of [3]).

Case 2. If , where is the indicator function over the set , that is, if and otherwise, then problem (P) reduces to the following hemivariational inequality: find and such that which is studied recently by Zhang and He [4, 5]. In 2011, by introducing the notion of stable quasimonotonicity and applying KKM theorem, Zhang and He [4] obtained some existence results of the hemivariational inequality (3).

Case 3. If is single-valued and , then problem (P) reduces to the following hemivariational inequality of finding such that which is introduced and named as Hartman-Stampacchia type hemivariational inequality by Panagiotopoulos et al. [6] and further studied by Costea and Rǎdulescu [7]. Under some suitable assumptions, the authors obtained corresponding existence theorems.

Case 4. If , then problem (P) is equivalent to finding and such that which is called the generalized mixed variational inequality problem and intensively studied by many researchers (see, e.g., [812]). Further, if is single-valued and , then problem (5) reduces to well-known formulation of variational inequality: find such that

The notion of the hemivariational inequality was introduced by Panagiotopoulos (see, e.g., [1316]) in the early 1980s as variational expressions for several classes of mechanical problems with nonsmooth and nonconvex energy superpotentials. The derivative of hemivariational inequality is based on the mathematical notion of the generalized gradient of Clarke (see [17]). The hemivariational inequalities appear in a variety of mechanical problems, for example, the unilateral contact problems in nonlinear elasticity, the problems describing the adhesive and frictional effects, the nonconvex semipermeability problems, the masonry structures, and the delamination problems in multilayered composites; see [14, 16, 18] for detailed descriptions. Extensive attention has been paid to the existence results for some types of hemivariational inequalities by many researchers in recent years. For example, Carl [19], Carl et al. [20, 21], and Xiao and Huang [22] studied the existence of solutions of some kinds of hemivariational inequalities using the method of sub-super solutions. Migórski and Ochal [23] and Park and Ha [24, 25] studied the problem using the regularized approximating method. Goeleven et al. [26] and Liu [27] proved the existence of solutions using the method of the first eigenfunction. For more related works regarding the existence of solutions for hemivariational inequalities, we refer to [1, 3, 6, 1416, 2830] and the references therein.

Due to the presence of a set-valued mapping, problem (P) becomes more difficult than the single-valued case. First, we recall some definitions of continuity for set-valued mapping.

Definition 1. The set-valued mapping is said to be the following:(i)lower semicontinuous at if, for any and sequence with , a sequence can be determined which converges to . If this is true at every , we say that is lower semicontinuous on ;(ii)lower hemicontinuous if the restriction of to every line segment of is lower semicontinuous;(iii)upper semicontinuous if, for all and for any open set satisfying , there exists an open neighborhood of such that for all .

We remark that when is single-valued, both the notion of lower semicontinuity and that of upper semicontinuity coincide with the usual notion of continuity of a map.

When the constrained set is unbounded, in order to obtain existence theorems of problems, various of coercivity conditions usually are required (see, e.g., [2, 4, 5, 3134]).

Recently, Tang and Huang [2] introduced some coercivity conditions for problem (P) involving lower hemicontinuous mappings. Using -quasimonotonicity of mappings, the authors obtained some existence theorems and studied the boundedness of the solution set of problem (P). A natural problem is whether these coercivity conditions are valid for problem (P) involving upper semicontinuous mappings or not. This is the main motivation of this paper. On the other hand, Zhang and He [5] also investigated problem (3) involving upper semicontinuous mappings. How to extend the main results of [5] from problem (3) to problem (P) is another motivation of this work.

Motivated and inspired by the research work mentioned above, in this paper, we investigate various coercivity conditions in order to guarantee existence of solutions and boundedness of the solution set for the variational-hemivariational inequalities involving upper semicontinuous operators. The results presented in this paper generalize and improve some known results.

2. Preliminaries

For a nonempty, closed, and convex subset of a Euclidean space and every , we define Let be a linear compact operator, where and , and let be a bounded open set in . Denote by the conjugated exponent of ; that is, . Let be a function such that the mapping We assume that at least one of the following conditions holds: either there exists such that or and there exists such that

Recall that denotes Clarke’s generalized directional derivative of the locally Lipschitz mapping at the point with respect to the direction , while is the Clarke’s generalized gradient of at (see, e.g., [17]); that is,

Let be an arbitrary locally Lipschitz functional. For each there exists (see, e.g., [17]) such that

Lemma 2 (Proposition 2.1.1 of [17]). Let be Lipschitz of rank near . Then(i)the function    is finite, positively homogeneous, and subadditive on    and satisfies(ii)  is upper semicontinuous as a function of    and, as a function of    alone, is Lipschitz of rank    on  ;(iii).

Lemma 3 (Propositions 2.1.2 and 2.1.5 of [17]). Let be Lipschitz of rank near . Then(i)  is a nonempty, convex, and  -compact subset of    and    for every    in  ;(ii)for every    in  , one has(iii)  is upper semicontinuous at  .

Lemma 4 (Theorem 2.7.5 of [17]). If and satisfies the conditions (8) and (9) or (8) and (10)-(11), then is uniformly Lipschitz on bounded subsets, and one has Further, if is regular at then is regular at and equality holds.

Lemma 5 (Proposition 24.1 of [35]). Let be Banach spaces and let be an upper semicontinuous set-valued mapping with nonempty compact values from to . Then for any compact subset of , is compact.

Lemma 6 (Theorem 4.1 of [10]). Assume that is a reflexive, strictly convex, and smooth Banach space with the dual space and has property weakly and imply that . Suppose that is nonempty closed convex set and is proper, convex, and lower semicontinuous. Let for all , where is normalized duality mapping and is a compact mapping with compact convex values. Suppose that there exists a vector such that and the set is bounded (possibly empty). Then there exists and such that

Theorem 7. Let be a closed and convex subset of  . Let be an upper semicontinuous set-valued mapping with compact convex values and let be a convex and lower semicontinuous function such that . Suppose that there exists a vector such that and the set is bounded (possibly empty). Then there exists and such that

Proof. For any bounded set , we have . Since is an upper semicontinuous mapping with compact convex values, by Lemma 5, we obtain that is a compact set and so is compact. Hence is a compact mapping with compact convex values. Taking for any and applying Lemma 6, we obtain the conclusion.

Remark 8. Theorem 7 generalizes Theorem 3.1 of Qiao and He [36] (see also Corollary 4.2 of [10]) from set-valued variational inequalities to set-valued mixed variational inequalities. Moreover, it also generalizes the corresponding result of Facchinei and Pang [33] (see Proposition of [33]). This theorem plays a crucial role in analysis of the next section.

3. Coercivity Conditions and Applications to Existence Theorems

First, we consider another type of variational-hemivariational inequality problem:(P′)find and such that where is a locally Lipschitz functional.

We denote by (resp. ) the solution set of problem (P) (resp., (P′)).

Proposition 9. Let be the function And let be a linear compact operator, where , , and is a bounded open set in . Assume that is a nonempty, closed, and convex subset of . Let be a set-valued mapping and let be a convex and lower semicontinuous function such that . Further, suppose that satisfies the conditions (8) and (9) or (8) and (10)-(11). Then . Moreover, if is regular at for all and , then .

Proof. For any , there exists such that Since , and satisfies the conditions (8) and (9) or (8) and (10)-(11), by Lemma 4, we have Then we have That is, .
If is regular at , for any and , by Lemma 4, we have Hence, it follows that  . This completes the proof.

Theorem 10. Let be the function And let be a linear compact operator, where , , and is a bounded open set in . Assume that is a nonempty, closed, and convex subset of . Let be a convex and lower semicontinuous function such that and let be an upper semicontinuous set-valued mapping with compact convex values. Further, suppose that satisfies the conditions (8) and (9) or (8) and (10)-(11). If the following coercivity condition holds:(C1)there exists a vector such that and the set is bounded (possibly empty), then problem (P′) has at least one solution.

Proof. For the sake of convenience, denote a set-valued mapping as follows: Then is an upper semicontinuous mapping with compact convex values on . In fact, since satisfies the conditions either (8) and (9) or (8) and (10)-(11), by Lemma 4, is uniformly Lipschitz on , and by item (iii) of Lemma 3, we obtain that is upper semicontinuous, by the assumption of being a linear compact operator and being its adjoint operator, it follows that is upper semicontinuous. Since the sum of upper semicontinuous mappings is also upper semicontinuous, by the assumption of being upper semicontinuous, we have that is upper semicontinuous. By item (i) of Lemma 3 and the assumptions of being a linear compact operator and having compact convex values, we know that has compact convex values. Thus, has compact convex values.
Hence, from Theorem 7, we know that there exist   and such that It follows from the definition of that there exist and such that . Then we have that By Lemma 3(ii), we have that Hence, we have that there exist and such that That is, problem (P′) has at least one solution. This completes the proof.

From Theorem 10 and Proposition 9, we get an existence result of problem (P).

Theorem 11. Let be the function And let be a linear compact operator, where , and is a bounded open set in . Assume that is a nonempty, closed, and convex subset of . Let be a convex and lower semicontinuous function such that and be an upper semicontinuous set-valued mapping with compact convex values. Further, suppose that satisfies the conditions (8) and (9) or (8) and (10)-(11). If coercivity condition (C1) holds, then problem (P) has at least one solution.

Corollary 12. Assume that is a nonempty, bounded, closed, and convex subset of . Let be a convex and lower semicontinuous function and be an upper semicontinuous set-valued mapping with compact convex values. Further, suppose that satisfies the conditions (8) and (9) or (8) and (10)-(11). Then problem (P) has at least one solution.

Proof. Let be the function By the lower semicontinuity of , we know that there exists a vector such that . It follows from the boundedness of that is bounded (possibly empty). Thus, from Theorem 15, we know that problem (P) has at least one solution. This completes the proof.

Remark 13. Corollary 12 generalizes and improves some recent results in the following aspects.(i)If , then Corollary 12 reduces to Theorem 3.2 of Zhang and He [5].(ii)Compared with Theorem 4.1 of Tang and Huang [2], the main difference lies in the assumptions on . Theorem 4.1 of Tang and Huang [2] asked to be a lower hemicontinuous and -quasimonotone mapping.

Proposition 14 (Proposition 4.1 of [2]). Consider the following coercivity conditions.(C2) There exists a nonempty subset    contained in a weakly compact subset    of    such that the setis weakly compact or empty.(C3) There exist    and    such that for every  , there exists some    with    such that(C4) There exist    and    such that for every  , there exists some   with such thatThen we have(i)(C2)(C3), if    is stably  -quasimonotone with respect to the set  ;(ii)(C4)(C3), if  ,    satisfies the conditions (8) and (9) or (8) and (10)-(11).

Theorem 15. Let be the function And let be a linear compact operator, where , , and is a bounded open set in . Assume that is a nonempty, closed, and convex subset of and is a convex and lower semicontinuous function such that . Let be an upper semicontinuous set-valued mapping with compact convex values. If the condition (C3) holds, then the problem (P) admits at least one solution.

Proof. Take and . From Proposition 9 and Corollary 12, we conclude that there exist and such that (i)If , then . Since the condition (C3) holds, there is some with such that Let be arbitrarily fixed. Since , there is such that . Note that is a linear mapping and is convex. It follows from (41), (42) and Lemma 2(i) that Therefore, this together with implies that (ii)If , then for any , there is some such that . Note that is a linear mapping and is a convex function. It follows from (41) and item (i) of Lemma 2 that Therefore, this together with implies that (44) also holds.
Since and satisfies the conditions (8) and (9) or (8) and (10)-(11), by Lemma 4, we have and so If , then and thus the inequality in (47) holds automatically. This together with (47) shows that is a solution of problem (P).

Remark 16. Theorem 15 generalizes and improves some recent results in the following aspects.(i)If , then Theorem 15 reduces to Theorem 3.3 of [5].(ii)Compared with Theorem 4.2 of Tang and Huang [2], the main difference lies in the assumptions on . Theorem 4.2 of Tang and Huang [2] asked to be a lower hemicontinuous and -quasimonotone mapping.
If the constraint set is bounded, then the solution set of the problem (P) is obviously bounded. In the case when the constraint set is unbounded, the solution set of the problem (P) may be unbounded. In the sequel, we provide a sufficient condition to the boundedness of the solution set of the problem (P), when is unbounded. The following theorem also generalizes Theorem 4.1 of [5].

Theorem 17. Let be the function and be a linear compact operator, where , , and is a bounded open set in . Assume that is a nonempty, closed, and convex subset of and is a convex and lower semicontinuous function such that . Let be an upper semicontinuous set-valued mapping with compact convex values. Further, we suppose satisfies the conditions (8) and (9) or (8) and (10)-(11). If condition (C4) holds, then the solution set of problem (P) is nonempty and bounded.

Proof. From Proposition 14, we have (C4)(C3). By Theorem 15, we know that the solution set of the problem (P) is nonempty. If the solution set is unbounded, then there exist and such that and Since , by the condition (C4), there exists with such that which contradicts (49). Hence, the solution set is bounded.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

Guo-ji Tang was partially supported by Guangxi Natural Science Foundation (2013GXNSFBA019015), Scientific Research Foundation of Guangxi University for Nationalities (2012QD015), and Open fund of Guangxi key laboratory of hybrid computation and IC design analysis (HCIC201308). Hong-ling Zhang was partially supported by the Key Program of the Fundamental Research Funds for the Central Universities (A0920502051202-111).

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