# On the Convergence for an Iterative Method for Quasivariational Inclusions

- Yu Li
^{1}and - Changqun Wu
^{2}Email author

**2010**:278973

**DOI: **10.1155/2010/278973

© Y. Li and C.Wu. 2010

**Received: **27 September 2009

**Accepted: **13 December 2009

**Published: **12 January 2010

## Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

## 1. Introduction and Preliminaries

Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm . Let be a nonlinear mapping. In this paper, we use to denote the fixed point set of

Recall the following definitions.

Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. See, for example, [1–6] and the references therein.

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :

where is a linear bounded and strongly positive operator and is a potential function for (i.e., for ).

Recently, Marino and Xu [2] studied the following iterative scheme:

They proved that the sequence generated in the above iterative scheme converges strongly to the unique solution of the variational inequality:

which is the optimality condition for the minimization problem (1.5).

Next, let be a nonlinear mapping. Recall the following definitions.

(6)Recall also that a set-valued mapping
is called *monotone* if for all
,
and
imply
The monotone mapping
is *maximal* if the graph of
of
is not properly contained in the graph of any other monotone mapping.

The so-called quasi-variational inclusion problem is to find a for a given element such that

where and are two nonlinear mappings. See, for example, [7–12]. A special case of the problem (1.13) is to find an element such that

In this paper, we use to denote the solution of the problem (1.14). A number of problems arising in structural analysis, mechanics, and economic can be studied in the framework of this class of variational inclusions.

Next, we consider two special cases of the problem (1.14).

which is said to be the*mixed quasi-variational inequality*. See, for example, [7, 8] for more details.

For finding a common element of the set of fixed points of a nonexpansive mapping and of the set of solutions to the variational inequality (1.16), Iiduka and Takahashi [13] proved the following theorem.

Theorem IT

*Let*

*be a closed convex subset of a real Hilbert space*

*. Let*

*be an*

*-inverse-strongly monotone mapping of*

*into*

*and let*

*be a nonexpansive mapping of*

*into itself such that*

*. Suppose that*

*and*

*is given by*

Recently, Zhang et al. [11] considered the problem (1.14). To be more precise, they proved the following theorem.

Theorem ZLC

*Let*

*be a real Hilbert space,*

*an*

*-inverse-strongly monotone mapping,*

*a maximal monotone mapping, and*

*a nonexpansive mapping. Suppose that the set*

*, where*

*is the set of solutions of variational inclusion ( 1.14 ). Suppose that*

*and*

*is the sequence defined by*

where and is a sequence in satisfying the following conditions:

In this paper, motivated by the research work going on in this direction, see, for instance, [2, 3, 7–21], we introduce an iterative method for finding a common element of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem (1.14) with multivalued maximal monotone mapping and relaxed -cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.

In order to prove our main results, we need the following conceptions and lemmas.

Definition 1.1 (see [11]).

Let be a multivalued maximal monotone mapping. Then the single-valued mapping defined by for all , is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Lemma 1.2 (see [4]).

Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that

Lemma 1.3 (see [22]).

Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all and Then

Lemma 1.4 (see [11]).

Lemma 1.5 (see [11]).

The resolvent operator associated with is single-valued and nonexpansive for all .

Lemma 1.6 (see [23]).

Let be a closed convex subset of a strictly convex Banach space . Let and be two nonexpansive mappings on . Suppose that is nonempty. Then a mapping on defined by , where , for is well defined and nonexpansive and holds.

Lemma 1.7 (see [24]).

Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be a nonexpansive mapping. Then is demiclosed at zero.

Lemma 1.8 (see [25]).

Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each . Then, as , is nonexpansive such that .

## 2. Main Results

Theorem 2.1.

where and are sequences in . Assume that , . If the control consequences and satisfy the following restrictions:

Proof.

The strong monotonicity of (see [2, Lemma ]) implies that and the uniqueness is proved. Below we use to denote the unique solution of (2.2).

Next, we show that the mapping is nonexpansive. Indeed, for all , one see from the condition that

By simple inductions, one obtains that which gives that the sequence is bounded, so are and .

On the other hand, we see from the nonexpansivity of the mappings that

Setting

On the other hand, we have

Next, we prove that where To see this, we choose a subsequence of such that

Finally, we show that as Indeed, we have

From the condition (C2), (2.23), and using Lemma 1.2, we see that This completes the proof.

Letting and , the identity mapping, we can obtain from Theorem 2.1 the following result immediately.

Corollary 2.2.

where and are sequences in . Assume that , . If the control consequences and satisfy the following restrictions:

Remark 2.3.

Corollary 2.2 improves Theorem 2.1 of Zhang et al. [11] in the following sense:

(1)from nonexpansive mappings to strict pseudocontractions;

(2)the analysis technique used in this paper is different from [11]'s: the proof is also more concise than [11]'s;

## Declarations

### Acknowledgments

The authors are extremely grateful to the referee for useful suggestions that improved the contents of the paper. This work was supported by Important Science and Technology Research Project of Henan province, China (092102210134).

## Authors’ Affiliations

## References

- Deutsch F, Yamada I:
**Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.***Numerical Functional Analysis and Optimization*1998,**19**(1–2):33–56.MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu HK:
**A general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2006,**318**(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar - Qin X, Shang M, Su Y:
**Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems.***Mathematical and Computer Modelling*2008,**48**(7–8):1033–1046. 10.1016/j.mcm.2007.12.008MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar - Yamada I, Ogura N, Yamashita Y, Sakaniwa K:
**Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space.***Numerical Functional Analysis and Optimization*1998,**19**(1–2):165–190. 10.1080/01630569808816822MathSciNetView ArticleMATHGoogle Scholar - Noor MA, Noor KI:
**Sensitivity analysis for quasi-variational inclusions.***Journal of Mathematical Analysis and Applications*1999,**236**(2):290–299. 10.1006/jmaa.1999.6424MathSciNetView ArticleMATHGoogle Scholar - Noor MA:
**Generalized set-valued variational inclusions and resolvent equations.***Journal of Mathematical Analysis and Applications*1998,**228**(1):206–220. 10.1006/jmaa.1998.6127MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Sriprad W:
**A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-20.Google Scholar - Peng JW, Wang Y, Shyu DS, Yao J-C:
**Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems.***Journal of Inequalities and Applications*2008,**2008:**-15.Google Scholar - Zhang SS, Lee JHW, Chan CK:
**Algorithms of common solutions to quasi variational inclusion and fixed point problems.***Applied Mathematics and Mechanics*2008,**29**(5):571–581. 10.1007/s10483-008-0502-yMathSciNetView ArticleMATHGoogle Scholar - Zhao LC, Chang SS, Liu M:
**Viscosity approximation algorithms of common solutions for fixed points of infinite nonexpansive mappings and quasi-variational inclusion problems.***Communications on Applied Nonlinear Analysis*2008,**15**(3):83–98.MathSciNetMATHGoogle Scholar - Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods& Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleMATHGoogle Scholar - Chang SS:
**Set-valued variational inclusions in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**248**(2):438–454. 10.1006/jmaa.2000.6919MathSciNetView ArticleMATHGoogle Scholar - Qin X, Su Y:
**Approximation of a zero point of accretive operator in Banach spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho SY, Kang SM:
**Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications.***Journal of Computational and Applied Mathematics*2009,**233**(2):231–240. 10.1016/j.cam.2009.07.018MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.***Journal of Computational and Applied Mathematics*2009,**225**(1):20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications.***Nonlinear Analysis: Theory, Methods and Applications*2010,**72**(1):99–112. 10.1016/j.na.2009.06.042MathSciNetView ArticleMATHGoogle Scholar - Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis: Real World Applications. In pressGoogle Scholar
- Takahashi W, Toyoda M:
**Weak convergence theorems for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2003,**118**(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar - Bruck RE Jr.:
**Properties of fixed-point sets of nonexpansive mappings in Banach spaces.***Transactions of the American Mathematical Society*1973,**179:**251–262.MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces.***Archive for Rational Mechanics and Analysis*1967,**24:**82–90.MathSciNetView ArticleMATHGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar

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