A New System of Multivalued Mixed Variational Inequality Problem

Abstract

We consider a new system of multivalued mixed variational inequality problem, which includes some known systems of variational inequalities as special cases. Under suitable conditions, the existence of solutions for the system of multivalued mixed variational inequality problem and the convergence of iterative sequences generated by the generalized -projection algorithm are proved. A perturbational algorithm for solving a special case of multivalued mixed variational inequality problem is formally constructed. The results concerned with the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm are also given. Some known results are improved and generalized.

1. Introduction

Variational inequalities are known to be very useful tool to formulate and investigate various network equilibrium problems arising in economic, management, and engineering. An important and useful generalization of the variational inequality is called the mixed variational inequality. This problem was originally considered by Lescarret [1] and Browder [2] in connection with its numerous applications. Konnov and Volotskaya [3] considered rather broad classes of general economic equilibrium problems and oligopolistic equilibrium problems which can be formulated as mixed variational inequality problems.

Recently, some interesting and important problems related to variational inequalities and mixed variational inequalities have been considered by many authors. Chang et al. [4] introduced a generalized system for relaxed cocoercive variational inequalities in Hilbert spaces and established some algorithms for the system. Petrot [5] studied a generalized system for relaxed cocoercive mixed variational inequality problem in Hilbert spaces and found the common solutions for the system using a resolvent operator technique. For more details related to variational inequalities and mixed variational inequalities, we refer to [6–12] and the references therein.

It is well known that projection methods have represented an important tool for solving variational inequalities. In 1994, Alber [13] introduced the generalized projections in uniformly convex and uniformly smooth Banach spaces and studied their properties in detail. Recently, Wu and Huang [14] introduced a new generalized -projection operator in a Banach space, which was a useful tool for solving mixed variational inequality problems. They extended the definition of the generalized projection operators introduced by Alber [13, 15] and proved some properties of the generalized -projection operator. Fan et al. [16] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces by using iterative schemes. Very recently, Li et al. [17] proved some stability results for the generalized -projection operators with perturbations of constraint sets in Banach spaces.

Inspired and motivated by the previous works mentioned above, in this paper, we introduce a new system of multivalued mixed variational inequality problem in Hilbert spaces. This class of systems includes some known systems of variational inequalities as special cases. We construct a new generalized -projection algorithm for solving the system of multivalued mixed variational inequality problem. The existence of solutions and the convergence of iterative sequences generated by the algorithm are presented in this paper. We also construct a perturbational algorithm for solving a special case of multivalued mixed variational inequality problem and give the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm. Our results improve and generalize some known corresponding ones.

2. Preliminaries

Let be a real Hilbert space with scalar product and norm denoted by and , respectively. We recall the concept of the generalized -projector operator, together with its properties. Let , let be a closed convex subset, and let be a proper convex and lower semicontinuous functional. Let be a functional defined as follows: where , and is a positive number.

Definition 1 (see [14]). Let be a real Hilbert space, and let be a nonempty closed and convex subset of . Let be a proper, convex, and lower semicontinuous functional. One says that is a generalized -projection operator if

Lemma 2 (see [14, 16]). Let be a real Hilbert space, and let be a nonempty closed and convex subset of . Let be a proper, convex, and lower semicontinuous functional. Then the following statements hold: (i) is a single-valued mapping with nonempty values;(ii)for all if and only if (iii) is continuous.

Lemma 3 (see [18]). Let be a real Hilbert space, and let be a nonempty closed and convex subset of . Let be a proper, convex, and lower semicontinuous functional. Then

Let and be closed convex subsets in . It is known that the Hausdorff distance between and is defined as follows:

Lemma 4 (see [17]). Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a convex and uniformly continuous mapping. Let be a family of nonempty closed convex subsets such that is proper on each and as . Then, for any ,

Lemma 5 (see [19]). Let and be two nonnegative real sequences satisfying with and . Then

In order to obtain our results, the following definitions are crucial to us.

Definition 6. Let be a mapping. is said to be (i)-Lipschitz continuous with respect to the first argument, if there exists a constant such that (ii)-strongly monotone with respect to the first argument, if there exists a constant such that

Definition 7 (see [20]). Let be a multivalued mapping. is said to be --Lipchitz continuous, if there exists a constant such that

3. The Multivalued Mixed Variational Inequality System

In this section, we will introduce a new system of multivalued mixed variational inequality in a Hilbert space. Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be two single-valued mappings, let be a proper, convex, and lower semicontinuous mapping, and let be two set-valued mappings. We consider the following system of multivalued mixed variational inequality problem: find , , and such that

If , where is the identity mapping, then the problem (11) reduces immediately to the following mixed variational inequality system: find such that

If , , , and for all , where , then the problem (11) is equivalent to finding such that which was studied by Petrot [5].

If , , , and for all , where and , then the problem (11) is equivalent to finding such that The system (14) was studied by Chang et al. [4].

In brief, the system (11) of multivalued mixed variational inequality is more general and includes many systems of variational inequalities as special cases (see [6–9, 21]).

The property of generalized -projection operator plays an important role in solving the system of multivalued mixed variational inequality problem. From Lemma 2, it is easy to see that the following lemma holds trivially.

Lemma 8. The problem (11) is equivalent to finding , , and such that where .

For any given , we choose and . By Nadler Jr. [22], for any and , there exist and such that

Based on (15) and (16), we can construct the following algorithm for solving the problem (11).

Algorithm 9. Assume that , and are the same as in the problem (11). For any given , , and , we compute , , , and as follows: where are constants.

Let be a family of nonempty closed convex subsets in . Now we construct the following perturbational algorithm for the problem (12).

Algorithm 10. Assume that , and are the same as in the problem (11). For any given , we compute and as follows:

4. Existence and Convergence Theorems

In this section, we will prove the existence of solutions for the problem (11) and the unique existence of solutions for the problem (12), respectively. In addition, we will provide the convergence results of Algorithms 9 and 10, respectively.

Theorem 11. Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be a proper, convex, and lower semicontinuous mapping. Let be -strongly monotone and -Lipschitz continuous with respect to the first variable and -Lipschitz continuous with respect to the second variable. Let be -strongly monotone and -Lipschitz continuous with respect to the first variable and -Lipschitz continuous with respect to the second variable. Let be --Lipschitz continuous; let be --Lipschitz continuous, where denotes the collection of all closed subsets of . Suppose that , and satisfy Then generated by Algorithm 9 converges strongly to as ; moreover, is a solution of system (11).

Proof. From Algorithm 9 and Lemma 3, we have Since is -strongly monotone and -Lipschitz continuous with respect to the first variable, we obtain The --Lipschitz continuity of yields that It follows from inequalities (20)–(22) that where and .
By the assumptions of and , following very similar arguments from (20)–(23), we have Hence, where and .
Now (23) and (26) imply that where Since and , we know that , where It is clear that due to (19). Consequently, there exist and such that for all . It follows from (27) that for all . Letting in (30), we obtain and , and so and are Cauchy sequences in . Therefore, there exist such that and .
From inequalities (22) and (24), we know that and are both Cauchy sequences in , and so there exist such that and . Since and , we have Thus, and . By the continuity of , , and , it follows from (17) that Lemma 8 shows that is a solution of system (11). This completes the proof.

Theorem 12. Let be a real Hilbert space and let be a nonempty closed convex subsets of . Let be a convex and uniformly continuous mapping and let be a family of nonempty closed convex subsets such that is proper on each and as . Let be the same as in Theorem 11. Suppose that are such that Then generated by Algorithm 10 converges strongly to as ; moreover, is the unique solution of system (12).

Proof. Define the norm on product space by It is easy to see that is a Banach space. Let be defined by
For any , it follows from Lemma 3 that By the assumptions imposed on and , we obtain From (36)-(37), we have where . It follows assumption (33) that . This shows that is a contractive operator, and so there exists a unique such that . Thus, is the unique solution of system (12).
Now we prove that and as . In fact, it follows from (18) that Following very similar arguments from (36)-(38), we have It follows from (39)-(40) that where An application of Lemma 4 yields that , as . Now Lemma 5 implies that , and so and as . This completes the proof.

Remark 13. Theorems 11 and 12 improve and generalize some known corresponding results. (i)If , , , and , for all , then the problem (11) is equivalent to the problem (13) studied by Petrot [5]. We can get the existence and convergence results of solutions for the problem (13) from Theorems 11 and 12.(ii)If , , , and , for all , then the problem (11) is equivalent to the problem (14) studied by Chang et al. [4]. We can get the existence and convergence results of solutions for the problem (14) from Theorems 11 and 12.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11301359). This work was also funded by the key program of Xihua University (Grant no. z1312621) and the Project supported by Scientific Research Fund of Sichuan Provincial Education Department (Grant no. 14ZB0130).