- Jian-Wen Peng
^{1}and - Lai-Jun Zhao
^{2}Email author

**2009**:364615

**DOI: **10.1155/2009/364615

© J.-W. Peng and L.-J. Zhao. 2009

**Received: **25 September 2009

**Accepted: **2 November 2009

**Published: **3 November 2009

## Abstract

We introduce and study a new system of nonlinear variational inclusions involving a combination of -Monotone operators and relaxed cocoercive mappings. By using the resolvent technique of the -monotone operators, we prove the existence and uniqueness of solution and the convergence of a new multistep iterative algorithm for this system of variational inclusions. The results in this paper unify, extend, and improve some known results in literature.

## 1. Introduction

Recently, Fang and Huang [1] introduced a new class of -monotone mappings in the context of solving a system of variational inclusions involving a combianation of -monotone and strongly monotone mappings based on the resolvent operator techniques. The notion of the -monotonicity has revitalized the theory of maximal monotone mappings in several directions, especially in the domain of applications. Verma [2] introduced the notion of -monotone mappings and its applications to the solvability of a system of variational inclusions involving a combination of -monotone and strongly monotone mappings. As Verma point out "the class of -monotone mappings generalizes -monotone mappings. On the top of that, -monotonicity originates from hemivariational inequalities, and emerges as a major contributor to the solvability of nonlinear variational problems on nonconvex settings." and as a matter of fact, some nice examples on -monotone (or generalized maximal monotone) mappings can be found in Naniewicz and Panagiotopoulos [3] and Verma [4]. Hemivariational inequalities—initiated and developed by Panagiotopoulos [5]—are connected with nonconvex energy functions and turned out to be useful tools proving the existence of solutions of nonconvex constrained problems. It is worthy noting that -monotonicity is defined in terms of relaxed monotone mappings—a more general notion than the monotonicity or strong monotonocity—which gives a significant edge over the -monotonocity. Very recently, Verma [6] studied the solvability of a system of variational inclusions involving a combination of -monotone and relaxed cocoercive mappings using resolvent operator techniques of -monotone mappings. Since relaxed cocoercive mapping is a generalization of strong monotone mappings, the main result in [6] is more general than the corresponding results in [1, 2].

Inspired and motivated by recent works in [1, 2, 6], the purpose of this paper is to introduce a new mathematical model, which is called a general system of -monotone nonlinear variational inclusion problems, that is, a family of -monotone nonlinear variational inclusion problems defined on a product set. This new mathematical model contains the system of inclusions in [1, 2, 6], the variational inclusions in [7, 8], and some variational inequalities in literature as special cases. By using the resolvent technique for the -monotone operators, we prove the existence and uniqueness of solution for this system of variational inclusions. We also prove the convergence of a multistep iterative algorithm approximating the solution for this system of variational inclusions. The result in this paper unifies, extends, and improves some results in [1, 2, 6–8] and the references therein.

## 2. Preliminaries

We suppose that is a real Hilbert space with norm and inner product denoted by and , respectively, denotes the family of all the nonempty subsets of . If be a set-valued operator, then we denote the effective domain of as follows:

Now we recall some definitions needed later.

Definition 2.1 (see [2, 6, 7]).

Let be a single-valued operator and let be a set-valued operator. is said to be

(ii) -monotone with a constant if

(b) is maximal monotone for (i.e., , for all ).

Remark 2.2.

If , , then the definition of -monotonicity is that of -monotonicity in [1, 8]. It is easy to know that if ( the identity map on ), then the definition of -monotone operators is that of maximal monotone operators. Hence, the class of -monotone operators provides a unifying frameworks for classes of maximal monotone operators, -monotone operators. For more details about the above definitions, please refer to [1–8] and the references therein.

It follow from [3, Lemma ] we know that if is a reflexive Banach space with its dual, and be -strongly monotone and is a locally Lipschitz such that is -relaxed monotone, then is -monotone with a constant .

Definition 2.3 (see [1, 7, 8]).

Let , be two single-valued operators. is said to be

Definition 2.4 . (see [2]).

We also need the following result obtained by Verma [2].

Lemma 2.5.

One needs the following new notions.

Definition 2.6.

Let be Hilbert spaces and denote the norm of , also let and be two single-valued mappings:

In a similar way, we can define the Lipschitz continuity and the strong monotonicity (monotonicity), relaxed cocoercivity (cocoercivity) of with respect to in the th argument .

## 3. A System of Set-Valued Variational Inclusions

Below are some special cases of (3.1).

However, (3.2) is called a system of set-valued variational inclusions introduced and researched by Fang and Huang [1, 9] and Verma [2, 6].

problem (3.3) is introduced and studied by Fang and Huang [8]. It is easy to see that the mathematical model (2) studied by Verma [7] is a variant of (3.3).

## 4. Existence of Solutions and Convergence of an Iterative Algorithm

In this section, we will prove existence and uniqueness of solution for (3.1). For our main results, we give a characterization of the solution of (3.1) as follows.

Lemma 4.1.

Proof.

Theorem 4.2.

Then, (3.1) admits a unique solution.

Proof.

By Lemma 4.1, is the unique solution of (3.1). This completes this proof.

Corollary 4.3.

Then, problem (3.1) admits a unique solution.

Remark 4.4.

Theorem 4.2 and Corollary 4.3 unify, extend, and generalize the main results in [1, 2, 6–8].

## 5. Iterative Algorithm and Convergence

In this section, we will construct some multistep iterative algorithm for approximating the unique solution of (3.1) and discuss the convergence analysis of these Algorithms.

Let and be two real sequences of nonnegative numbers that satisfy the following conditions:

Algorithm 5.2.

Theorem 5.3.

For , let be the same as in Theorem 4.2. Assume that all the conditions of theorem 4.1 hold. Then generated by Algorithm 5.2 converges strongly to the unique solution of (3.1).

Proof.

It follows from hypothesis (4.3) that .

Therefore, converges to the unique solution of (3.1). This completes the proof.

## Declarations

### Acknowledgment

This study was supported by grants from National Natural Science Foundation of China (project no. 70673012, no. 70741028 and no. 90924030), China National Social Science Foundation (project no. 08CJY026).

## Authors’ Affiliations

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