# A New System of Generalized Nonlinear Mixed Variational Inclusions in Banach Spaces

- JianWen Peng
^{1}Email author

**2010**:908490

**DOI: **10.1155/2010/908490

© Jian Wen Peng. 2010

**Received: **5 July 2009

**Accepted: **14 September 2009

**Published: **11 October 2009

## Abstract

We introduce and study a new system of generalized nonlinear mixed variational inclusions in real -uniformly smooth Banach spaces. We prove the existence and uniqueness of solution and the convergence of some new -step iterative algorithms with or without mixed errors for this system of generalized nonlinear mixed variational inclusions. The results in this paper unify, extend, and improve some known results in literature.

## 1. Introduction

Variational inclusion problems are among the most interesting and intensively studied classes of mathematical problems and have wide applications in the fields of optimization and control, economics and transportation equilibrium, as well as engineering science. For the past years, many existence results and iterative algorithms for various variational inequality and variational inclusion problems have been studied. For details, see [1–25] and the references therein.

Recently, some new and interesting problems, which are called to be system of variational inequality problems, were introduced and studied. Pang [1], Cohen and Chaplais [2], Bianchi [3], and Ansari and Yao [4] considered a system of scalar variational inequalities, and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Ansari et al. [5] considered a system of vector variational inequalities and obtained its existence results. Allevi et al. [6] considered a system of generalized vector variational inequalities and established some existence results with relative pseudomonoyonicity. Kassay and Kolumbán [7] introduced a system of variational inequalities and proved an existence theorem by the Ky Fan lemma. Kassay et al. [8] studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. Peng [9], Peng and Yang [10] introduced a system of quasivariational inequality problems and proved its existence theorem by maximal element theorems. Verma [11–15] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of system of variational inequalities in Hilbert spaces. J. K. Kim and D. S. Kim [16] introduced and studied a new system of generalized nonlinear quasivariational inequalities in Hilbert spaces. Cho et al. [17] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities.

As generalizations of system of variational inequalities, Agarwal et al. [18] introduced a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces. Peng and Zhu [19] introduce a new system of generalized nonlinear mixed quasivariational inclusions in -uniformly smooth Banach spaces and prove the existence and uniqueness of solutions and the convergence of several new two-step iterative algorithms with or without errors for this system of generalized nonlinear mixed quasivariational inclusions. Kazmi and Bhat [20] introduced a system of nonlinear variational-like inclusions and proved the existence of solutions and the convergence of a new iterative algorithm for this system of nonlinear variational-like inclusions. Fang and Huang [21], Verma [22], and Fang et al. [23] introduced and studied a new system of variational inclusions involving -monotone operators, -monotone operators and -monotone operators, respectively. Yan et al. [24] introduced and studied a system of set-valued variational inclusions which is more general than the model in [21]. Peng and Zhu [25] introduced and studied a system of generalized mixed quasivariational inclusions involving -monotone operators which contains those mathematical models in [11–16, 21–24] as special cases.

Inspired and motivated by the results in [1–25], the purpose of this paper is to introduce and study a new system of generalized nonlinear mixed quasivariational inclusions which contains some classes of system of variational inclusions and systems of variational inequalities in the literature as special cases. Using the resolvent technique for the -accretive mappings, we prove the existence and uniqueness of solutions for this system of generalized nonlinear mixed quasivariational inclusions. We also prove the convergence of some new -step iterative sequences with or without mixed errors to approximation the solution for this system of generalized nonlinear mixed quasivariational inclusions. The results in this paper unifies, extends, and improves some results in [11–16, 19] in several aspects.

## 2. Preliminaries

where is a constant. In particular, is the usual normalized duality mapping. It is known that, in general, , for all , and is single-valued if is strictly convex.

Note that is single-valued if is uniformly smooth.

Xu [26] and Xu and Roach [27] proved the following result.

Lemma 2.1.

Definition 2.2 (see [28]).

Let be a multivalued mapping:

(ii) is said to be -accretive if is accretive and holds for every (equivalently, for some) , where is the identity operator on .

Remark 2.3.

It is well known that, if is a Hilbert space, then is -accretive if and only if is maximal monotone (see, e.g., [29]).

We recall some definitions needed later.

Definition 2.4 (see [28]).

is called the resolvent operator associated with and .

Remark 2.5.

It is well known that is single-valued and nonexpansive mapping (see [28]).

Definition 2.6.

Let be a real uniformly smooth Banach space, and let be a single-valued operator. However, is said to be

Remark 2.7.

Lemma 2.8 (see [30]).

where , , for all , , . Then .

## 3. System of Generalized Nonlinear Mixed Variational Inequalities

In this section, we will introduce a new system of generalized nonlinear mixed variational inclusions which contains some classes of system of variational inclusions and systems of variational inequalities in literature as special cases.

which is called the system of generalized nonlinear mixed variational inclusions, where are constants.

- (i)
If , then problem (3.1) reduces to the system of nonlinear mixed quasivariational inclusions introduced and studied by Peng and Zhu [19].

If is a Hilbert space and , then problem (3.1) reduces to the system of nonlinear mixed quasivariational inclusions introduced and studied by Agarwal et al. [18].

- (ii)If is a Hilbert space, and for each , for all , where is a proper, convex, lower semicontinuous functional, and denotes the subdifferential operator of , then problem (3.1) reduces to the following system of generalized nonlinear mixed variational inequalities, which is to find such that(3.2)
where are constants.

- (iii)If , then (3.2) reduces to the problem of finding such that(3.3)
Moreover, if , then problem (3.3) becomes the system of generalized nonlinear mixed variational inequalities introduced and studied by J. K. Kim and D. S. Kim in [16].

- (iv)For , if (the indicator function of a nonempty closed convex subset ) and , then (3.2) reduces to the problem of finding , such that(3.4)

If and , then (3.5) reduces to the problem introduced and researched by Verma [11–13].

Lemma 3.1.

where is the resolvent operators of for .

Proof.

It is easy to know that Lemma 3.1 follows from Definition 2.4 and so the proof is omitted.

## 4. Existence and Uniqueness

In this section, we will show the existence and uniqueness of solution for problems (3.1).

Theorem 4.1.

then (3.1) has a unique solution .

Proof.

that is, is a solution of problem (3.1).

This completes the proof.

- (i)If is a 2-uniformly smooth space, and there exist ( ) such that(4.13)
Then (4.1) holds. We note that the Hilbert spaces and (or ) spaces ( ) are 2-uniformly smooth.

- (ii)
Let , by Theorem 4.1, we recover [19, Theorem 3.1]. So Theorem 4.1 unifies, extends, and improves [19, Theorem 3.1, Corollaries 3.2 and 3.3], [16, Theorems 2.1–2.4] and the main results in [13].

## 5. Algorithms and Convergence

This section deals with an introduction of some -step iterative sequences with or without mixed errors for problem (3.1) that can be applied to the convergence analysis of the iterative sequences generated by the algorithms.

Algorithm 5.1.

Theorem 5.2.

Let , , and be the same as in Theorem 4.1, and suppose that the sequences are generated by Algorithm 5.1. If the condition (4.1) holds, then converges strongly to the unique solution of the problem (3.1).

Proof.

where

where .

From the assumption (5.2), we know that satisfy the conditions of Lemma 2.8.

Thus , that is, . It follows from (5.6)–(5.8) that So for . That is, converges strongly to the unique solution of (3.1).

For , let and , by Algorithm 5.1 and Theorem 5.2, it is easy to obtain the following Algorithm 5.3 and Theorem 5.4.

Algorithm 5.3.

where , is a sequence in , satisfying

Theorem 5.4.

Let , , and be the same as in Theorem 4.1, and suppose that the sequences are generated by Algorithm 5.3. If (4.1) holds, then converges strongly to the unique solution of (3.1).

Remark 5.5.

Theorem 5.4 unifies and generalizes [19, Theorems 4.3 and 4.4] and the main results in [11, 12]. So Theorem 5.2 unifies, extends, and improves the corresponding results in [11–14, 16, 19].

## Declarations

### Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009) and the Research Project of Chongqing Normal University (Grant 08XLZ05).

## Authors’ Affiliations

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