A Generalized System of Nonlinear Variational Inequalities in Banach Spaces

Abstract

We introduce a new generalized system of nonlinear variational inequality problems (GSNVIP) by using the generalized projection method. Moreover, we introduce an iterative scheme for finding a solution to this problem. Moreover, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces under suitable conditions. The results presented in the paper improve and extend some recent results.

1. Introduction

Variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in pure and applied sciences which include work on differential equations, general equilibrium problems in economics and mechanics, control problems, and transportation. In 2005, Verma [1] introduced a general model for two-step projection methods and applied it to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space. Based on the convergence of projection methods, Chang et al. [2] introduced and studied the approximate solvability of a generalized system for relaxed cocoercive nonlinear variational inequalities in Hilbert spaces (see, for instance, [3–5] and the references therein). Recently, Chang et al. [6] introduced a system of generalized nonlinear variational inequalities and an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problems by using the generalized projection method. Moreover, they proved some existence and strong convergence theorems in uniformly smooth and strictly convex Banach spaces.

In this paper, we introduce a generalized system of nonlinear variational inequality problems (GSNVIP) by using the generalized projection approach to introduce an iterative scheme for finding a solution to this problem. Finally, we prove some existence and strong convergence theorems in uniformly smooth and strictly convex Banach spaces under suitable conditions.

2. Preliminaries

Let be a real Banach space with dual space , the dual pair between and , and a nonempty closed convex subset of . The normalized duality mapping is defined by A Banach space is said to be strictly convex if for all with . is said to be uniformly convex if for each there exists such that for all with . is said to be smooth if the limit exists for all . is said to be uniformly smooth if the above limit exists uniformly in .

Remark 1 (see [7]). (i) If is a uniformly smooth Banach space, then the normalized duality mapping is uniformly continuous on each bounded subset of .
(ii) If is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping is a single valued bijective mapping.
(iii) If is a smooth, strictly convex and reflexive Banach space and is the duality mapping in , then , , and .
(iv) If is a strictly convex and reflexive Banach space, then is hemicontinuous; that is, is norm-weak-continuous.
(v) is uniformly smooth if and only if is uniformly convex.
(vi) If is a uniformly smooth and strictly convex Banach space with the Kadec-Klee property (i.e., for any sequence , if and , then ), then both the normalized duality mappings and are continuous.
(vii) Each uniformly convex Banach space has the Kadec-Klee property.

Assume that is a smooth, strictly convex and reflexive Banach space and is a nonempty closed convex subset of ; to denote the Lyapunov functional defined by

Following Alber [8], the generalized projection is defined by , where is the unique solution to the minimization problem The existence and uniqueness of the mapping follow from the property of the function and the strict monotonicity of the mapping .

Lemma 2 (see [8]). Let be a smooth, strictly convex and reflexive Banach space and a nonempty closed convex subset of . Then the following conclusions hold: (a)if and , then (b) is a continuous mapping from onto .

Remark 3. If is a real Hilbert space, then (identity mapping), , and is the metric projection from onto .

Lemma 4 (see [9, 10]). Let be a uniformly convex Banach space, a positive number, and a closed ball of . Then, for any given finite subset and for any given positive numbers with , there exists a continuous, strictly increasing, and convex function with such that for any with the following holds:

Lemma 5 (see [11]). Let be a real reflexive, smooth, and strictly convex Banach space. Then the following inequality holds:

Lemma 6 (see [6]). Let be a real Banach space, a nonempty closed convex subset of with , and the generalized projection. Then for each , one has .

3. Main Results

In this section, we assume that is a real Banach space with dual space and is a nonempty closed convex subset of . Let be nonlinear mappings and a mapping. The generalized system of nonlinear variational inequality problems (GSNVIP) is to find such that for all

If , , and are nonlinear mappings, then the generalized system of nonlinear variational inequality problems (GSNVIP) reduces to the following problem (see [6]) to find , , such that, for all ,

If and are nonlinear mappings and is a mapping, then the generalized system of nonlinear variational inequality problems (GSNVIP) reduces to the following problem to find , such that, for all ,

If are nonlinear mappings and are two mappings. Define by and . Then the generalized system of nonlinear variational inequality problems (GSNVIP) reduces to the following problem to find such that, for all , where and are two positive constants.

Lemma 7. Let be a smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset of . Let be mappings, a bijective mapping, and any positive real numbers. Then is a solution to problem (8) if and only if is a solution to the following system of operator equations:

Proof. By Lemma 2, we have that is a solution of problem (8), for all , for any ,

Algorithm 8. For any given initial points , compute the sequences by the iterative processes where is the generalized projection and , , , are sequences in .

Theorem 9. Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and a nonempty closed and convex subset of with . Let be an isometry mapping, continuous mappings, and the sequences in with satisfying the following conditions: (i)there exist a compact subset and constants such that where , for all , and for all ;(ii). Let be the sequences defined by (16). Then the problem (8) has a solution and the sequences converge strongly to , respectively.

Proof.
Step 1. We first show that the sequences are bounded in . It follows from Lemma 5 where is bijective and condition (18) that Similarly, we note that By Lemma 6, we obtain that Since is an isometry mapping, we have . By the same argument method as given above, we have . Therefore, we note that exist and hence the sequences are bounded in .
Step 2. By Lemmas 4 and 6, where is an isometry mapping and (19), it follows that there exists a continuous strictly increasing and convex function with such that This implies that Since converges for all , it follows by letting in (23), condition , and the property of that as . By (16) and (24), we have as . Similarly, we can prove that as .
Step 3. Since are bounded and there exists a compact subset such that , there exists a subsequence of such that Since is uniformly smooth and strictly convex, it follows by Lemma 2 (b) and Remark 1 that and are continuous. Thus From (24) and (29), we get By (25) and (30), we have Since is strictly convex and reflexive, it follows by Remark 1 (iv) that is norm-weak-continuous. Therefore, from (30) and (31), we note that and By the Kadec-Klee property, we have Since is continuous, it implies that is a subsequence of such that . Therefore as . So, it follows from (16), (30), (34), and condition (ii) that Since is a bijective mapping, we obtain that Similarly, we can prove that for every subsequence of there exist a subsequence of and such that Since is a continuous mapping, we note that Hence , for all . Therefore, we have By Lemma 7, we can conclude that is a solution of (8) and .

Setting and in Theorem 9, we immediately obtain the following result.

Corollary 10 (see [6]). Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and a nonempty closed and convex subset of with . Let be continuous mappings and , , and the sequences in with satisfying the following conditions. (i)There exist a compact subset and constants , , and such that where , for all , and for all .(ii), , and . Let , , and be the sequences defined by Then the problem (9) has a solution and the sequences , and converge strongly to , , and , respectively.

Setting as a real Hilbert space in Theorem 9, we have the following result.

Corollary 11. Let be a real Hilbert space and a nonempty closed and convex subset of . Let be an isometry mapping and continuous mappings and are sequences in with satisfying the following conditions. (i)There exist a compact subset and constants such that where , for all , and for all .(ii), . Let be the sequences defined by where is a metric projection on to . Then the problem (8) has a solution and the sequences converge strongly to , respectively.

Conflict of Interests

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

Acknowledgments

The first author would like to thank the Office of the Higher Education Commission, Thailand, for support under Grant no. CHE-Ph.D-THA-SUP/191/2551, Thailand. Moreover, the authors would like to express their deep thanks to Naresuan University under Grant no. R2557B055.