- Research Article
- Open Access

# On the System of Nonlinear Mixed Implicit Equilibrium Problems in Hilbert Spaces

- YeolJe Cho
^{1}and - Narin Petrot
^{2}Email author

**2010**:437976

https://doi.org/10.1155/2010/437976

© Y. J. Cho and N. Petrot. 2010

**Received:**22 December 2009**Accepted:**10 January 2010**Published:**17 January 2010

## Abstract

We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems (SMIE) in Hilbert spaces. The algorithm for finding a solution of the problem (SMIE) is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.

## Keywords

- Variational Inequality
- Nonexpansive Mapping
- Lipschitz Mapping
- Variational Inequality Problem
- Unique Fixed Point

## 1. Introduction and Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and respectively. Let be given two bi-functions satisfying for all and . Let be a nonlinear mapping. Let be a nonempty closed convex subset of . In this paper, we consider the following problem.

Find such that

The problem of type (1.1) is called the *system of nonlinear mixed implicit equilibrium problems*.

We denote by SMIE the set of all solutions of the problem (1.1).

- (I)
If where is a maximal monotone mapping for then the problem (1.1) becomes the following problem.

Find such that

which is called the system of *variational inclusion problems*. In particular, when
and
the problem (1.2) is reduced to the problem, so-called the *generalized variational inclusion problem*, which was studied by Kazmi and Bhat [1].

- (II)
If for all , where is a real valued function for each Then the problem (1.1) reduces to the following problem.

Find such that

- (III)
For each , let be a nonlinear mapping and fixed positive real numbers. If and for all , then the problem (1.3) reduces to the following problem.

Find such that

*system of nonlinear mixed variational inequalities problems*. A special case of the problem (1.4), when and , has been studied by He and Gu [3].

- (IV)
If for all , where is the indicator function of defined by

then the problem (1.4) reduces to the following problem.

Find such that

which is called the *system of nonlinear variational inequalities problems*. Some corresponding results to the problem (1.6) were studied by Agarwal et al. [4], Chang et al. [5], Cho et al. [6], J. K. Kim and D. S. Kim, [7] and Verma [8, 9].

- (V)
If , and is a univariate mapping, then the problem (1.6) reduces to the following problem.

Find such that

which is known as the *classical variational inequality* introduced and studied by Stampacchia [12] in 1964. This shows that a number of classes of variational inequalities and related optimization problems can be obtained as special cases of the system (1.1) of mixed equilibrium problems.

Inspired and motivated by the recent research going on in this area, in this paper, we use the Wiener-Hopf equations and the Yosida approximation notion to suggest and prove the existence and uniqueness of solutions for the problem (1.1). We also discuss the convergence criteria and stability of the iterative algorithm. The results presented in this paper improve and generalize many known results in the literature.

In the sequel, we need the following basic concepts and lemmas.

Definition 1.1 (Blum and Oettli [13]).

A real valued bifunction is said to be:

Definition 1.2.

*lower semicontinuous*at if, for all there exists a constant such that

*lower semicontinuous*on if it is lower semicontinuous at every point of .

Lemma 1.3 (Combettes and Hirstoaga [14]).

Let be a nonempty closed convex subset of and be a bifunction of into satisfying the following conditions:

(C1) is monotone and upper hemicontinuous;

(C2) is convex and lower semi-continuous for all .

Then is a single-valued mapping.

Definition 1.4.

Remark 1.5.

Definition 1.4 is an extension of the Yosida approximation notion introduced in [15]. The existence and uniqueness of the solution of the problem (1.15) follow from Lemma 1.3.

Definition 1.6.

Let be a set-valued mapping.

*maximal*if is not properly contained in any other monotone operators.

Example 1.7 (Huang et al. [16]).

where is the Yosida approximation of and we recover the classical concepts.

Using the idea as in Huang et al. [16], we have the following result.

Lemma 1.8.

Proof.

This implies that is a nonexpansive mapping. This completes the proof.

Now, for solving the problem (1.1), we consider the following equation: let and be fixed positive real numbers. Find such that

Lemma 1.9.

Proof.

The proof directly follows from the definitions of and .

In this paper, we are interested in the following class of nonlinear mappings.

Definition 1.10.

## 2. Existence of Solutions of the Problem (1.1)

In this section, we give an existence theorem of solutions for the problem (1.1). Firstly, in view of Lemma 1.9, we can obtain the following, which is an important tool, immediately.

Lemma 2.1.

Now, we are in position to prove the existence theorem of solutions for the problem (1.1).

Theorem 2.2.

Then is a singleton.

Proof.

By the condition (2.3), we have , which implies that is a contraction mapping. Hence, by Banach contraction principle, there exists a unique such that This completes the proof.

## 3. Convergence and Stability Analysis

In view of Lemma 2.1, for the fixed point formulation of the problem (2.1), we suggest the following iterative algorithm.

### 3.1. Mann Type Perturbed Iterative Algorithm (MTA)

For any , compute approximate solution given by the iterative schemes:

where is a sequence of real numbers such that and

In order to consider the convergence theorem of the sequences generated by the algorithm (MTA), we need the following lemma.

Lemma 3.1.

where with and . Then

Now, we prove the convergence theorem for a solution for the problem (1.1).

Theorem 3.2.

If all the conditions of the Theorem 2.2 hold, then the sequence in generated by the algorithm (3.1) converges strongly to the unique solution for the problem (1.1).

Proof.

Thus the sequence in converges strongly to a solution for the problem (1.1). This completes the proof.

### 3.2. Stability of the Algorithm (MTA)

Consider the following definition as an extension of the concept of stability of the iterative procedure given by Harder and Hicks [17].

Definition 3.3 (Kazmi and Khan [18]).

If
implies that
, then the iterative procedure
is said to be
*-stable* or *stable* with respect to
.

Theorem 3.4.

where is a sequence defined in (3.1). If is defined as in (2.1), then the iterative procedure generated by (3.1) is -stable.

Proof.

This completes the proof.

## Declarations

### Acknowledgments

The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050). The second author was supported by the Commission on Higher Education and the Thailand Research Fund (project no. MRG5180178).

## Authors’ Affiliations

## References

- Kazmi KR, Bhat MI: Convergence and stability of iterative algorithms for some classes of general variational inclusions in Banach spaces.
*Southeast Asian Bulletin of Mathematics*2008, 32(1):99–116.MathSciNetMATHGoogle Scholar - Kassay G, Kolumbán J: System of multi-valued variational inequalities.
*Publicationes Mathematicae Debrecen*2000, 56(1–2):185–195.MathSciNetMATHGoogle Scholar - He Z, Gu F: Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces.
*Applied Mathematics and Computation*2009, 214(1):26–30. 10.1016/j.amc.2009.03.056MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Cho YJ, Li J, Huang NJ: Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in -uniformly smooth Banach spaces.
*Journal of Mathematical Analysis and Applications*2002, 272(2):435–447. 10.1016/S0022-247X(02)00150-6MathSciNetView ArticleMATHGoogle Scholar - Chang SS, Joseph Lee HW, Chan CK: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces.
*Applied Mathematics Letters*2007, 20(3):329–334. 10.1016/j.aml.2006.04.017MathSciNetView ArticleMATHGoogle Scholar - Cho YJ, Fang YP, Huang NJ, Hwang HJ: Algorithms for systems of nonlinear variational inequalities.
*Journal of the Korean Mathematical Society*2004, 41(3):489–499.MathSciNetView ArticleMATHGoogle Scholar - Kim JK, Kim DS: A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces.
*Journal of Convex Analysis*2004, 11(1):235–243.MathSciNetMATHGoogle Scholar - Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities.
*Computers & Mathematics with Applications*2001, 41(7–8):1025–1031. 10.1016/S0898-1221(00)00336-9MathSciNetView ArticleMATHGoogle Scholar - Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods.
*Journal of Optimization Theory and Applications*2004, 121(1):203–210.MathSciNetView ArticleMATHGoogle Scholar - Nie H, Liu Z, Kim KH, Kang SM: A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings.
*Advances in Nonlinear Variational Inequalities*2003, 6(2):91–99.MathSciNetMATHGoogle Scholar - Verma RU: General convergence analysis for two-step projection methods and applications to variational problems.
*Applied Mathematics Letters*2005, 18(11):1286–1292. 10.1016/j.aml.2005.02.026MathSciNetView ArticleMATHGoogle Scholar - Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes.
*Comptes Rendus de l'Académie des Sciences: Paris*1964, 258: 4413–4416.MathSciNetMATHGoogle Scholar - Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*1994, 63(1–4):123–145.MathSciNetMATHGoogle Scholar - Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces.
*Journal of Nonlinear and Convex Analysis*2005, 6(1):117–136.MathSciNetMATHGoogle Scholar - Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In
*Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar - Huang N-J, Lan H, Cho YJ: Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with non-monotone set-valued mappings.
*Journal of Computational and Applied Mathematics*2006, 196(2):608–618. 10.1016/j.cam.2005.10.014MathSciNetView ArticleMATHGoogle Scholar - Harder AM, Hicks TL: Stability results for fixed point iteration procedures.
*Mathematica Japonica*1988, 33(5):693–706.MathSciNetMATHGoogle Scholar - Kazmi KR, Khan FA: Iterative approximation of a unique solution of a system of variational-like inclusions in real -uniformly smooth Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(3):917–929. 10.1016/j.na.2006.06.049MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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