# A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

- Rabian Wangkeeree
^{1}Email author and - Uthai Kamraksa
^{1}

**2009**:369215

https://doi.org/10.1155/2009/369215

© R.Wangkeeree and U. Kamraksa. 2009

**Received: **3 November 2008

**Accepted: **16 January 2009

**Published: **10 February 2009

## Abstract

We introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).

## Keywords

## 1. Introduction

We denote by the set of fixed points of . Recall that a mapping is said to be

(ii) -Lipschitz if there exists a constant such that , for all ;

Remark 1.1.

It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.

The set of solutions of variational inequality (1.3) is denoted by . The variational inequality has been extensively studied in the literature; see, for example, [3, 4] and the references therein.

where is a potential function for (i.e., for ).

where the sequence is in the interval .

where is a contraction, is a nonexpansive mapping, and is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, defined by (1.12) converges strongly to a fixed point of , which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

where is an -inverse strongly monotone mapping, and satisfy some parameters controlling conditions. They showed that if is nonempty, then the sequence generated by (1.14) converges strongly to some .

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17–20] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [21, 22]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [18, 23]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [23, 24]).

where is a nonnegative real sequence with , for all , , form a family of infinitely nonexpansive mappings of into itself. Nonexpansivity of each ensures the nonexpansivity of . Such a is nonexpansive from to and it is called a -mapping generated by and .

where is a mapping defined by (1.15), is a contraction, is strongly positive linear bounded self-adjoint operator, is a -inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences , , , and , the sequences defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

## 2. Preliminaries

It is well known that each Hilbert space satisfies the Opial's condition.

Then is the maximal monotone and if and only if ; see [26].

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1.

Lemma 2.2 (see [27]).

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

Lemma 2.3 (see [28]).

where is a sequence in and is a sequence in such that

Lemma 2.4 (see [9]).

Assume that is a strongly positive linear bounded self-adjoint operator on a Hilbert space with coefficient and . Then .

Throughout this paper, we will assume that , for all . Concerning defined by (1.15), we have the following lemmas which are important to prove our main result.

Lemma 2.5 (see [29]).

Let be a nonempty closed convex subset of a Hilbert space , let be a family of infinitely nonexpansive mapping with , and let be a real sequence such that , for all . Then

(1) is nonexpansive and for each ;

(2)for each and for each positive integer , the limit exists;

is a nonexpansive mapping satisfying and it is called the -mapping generated by and

Lemma 2.6 (see [30]).

## 3. Main Results

Now we are in a position to state and prove the main result in this paper.

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space , let be a contraction of into itself, let be an -inverse strongly monotone mapping of into , and let be a family of infinitely nonexpansive mappings with . Let be a strongly positive linear bounded self-adjoint operator with the coefficient such that . Assume that . Let , , , and be sequences in satisfying the following conditions:

Proof.

which gives that the sequence is bounded, and so are and .

where is a constant such that . Similarly, there exists such that .

Banach's Contraction Mapping Principle guarantees that has a unique fixed point, say . That is, .

This is a contradiction, which shows that .

Since is maximal monotone, we have , and hence .

Using (C1), (3.54), and (3.55), we get . Now applying Lemma 2.3 to (3.58), we conclude that . This completes the proof.

Remark 3.2.

Theorem 3.1 mainly improve the results of Qin and Cho [14] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.

## 4. Applications

In this section, we obtain two results by using a special case of the proposed method.

Theorem 4.1.

where , , , and are sequences in satisfying the following conditions:

Proof.

We have and . Applying Theorem 3.1, we obtain the desired result.

Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings.

Definition 4.2.

The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.

Lemma 4.3.

Let be a Hilbert space, let be a closed convex subset of . For any integer , assume that, for each is a -strictly pseudocontractive mapping for some . Assume that is a positive sequence such that . Then is a -strictly pseudocontractive mapping with .

Lemma 4.4.

Let and be as in Lemma 4.3. Suppose that has a common fixed point in . Then .

This shows that is -inverse-strongly monotone.

Theorem 4.5.

where , , , and are the sequences in satisfying the following conditions:

Proof.

The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.

Remark 4.6.

Theorem 4.5 is a generalization and improvement of the theorems by Qin and Cho [14], Iiduka and Takahashi [16, Thorem 3.1], and Takahashi and Toyoda [15].

## Declarations

### Acknowledgments

The authors would like to thank the referee for the comments which improve the manuscript. R. Wangkeeree was supported for CHE-PhD-THA-SUP/2551 from the Commission on Higher Education and the Thailand Research Fund under Grant TRG5280011.

## Authors’ Affiliations

## References

- 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 - Liu F, Nashed MZ:
**Regularization of nonlinear ill-posed variational inequalities and convergence rates.***Set-Valued Analysis*1998,**6**(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar - Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications*.*Volume 76*. Springer, New York, NY, USA; 2005:501–558.View ArticleGoogle Scholar - Zeng LC, Schaible S, Yao JC:
**Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities.***Journal of Optimization Theory and Applications*2005,**124**(3):725–738. 10.1007/s10957-004-1182-zMathSciNetView ArticleMATHGoogle Scholar - 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 - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society. Second Series*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:
**The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings.**In*Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications, Studies in Computational Mathematics*.*Volume 8*. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar - Marino G, Xu H-K:
**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 - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho Y: Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Computational and Applied Mathematics. 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 - 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 - Aoyama K, Kimura Y, Takahashi W, Toyoda M:
**Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetView ArticleMATHGoogle Scholar - Bauschke HH:
**The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1996,**202**(1):150–159. 10.1006/jmaa.1996.0308MathSciNetView ArticleMATHGoogle Scholar - Shang M, Su Y, Qin X:
**Strong convergence theorems for a finite family of nonexpansive mappings.***Fixed Point Theory and Applications*2007, Article ID 76971**2007:**-9.Google Scholar - Shimoji K, Takahashi W:
**Strong convergence to common fixed points of infinite nonexpansive mappings and applications.***Taiwanese Journal of Mathematics*2001,**5**(2):387–404.MathSciNetMATHGoogle Scholar - Bauschke HH, Borwein JM:
**On projection algorithms for solving convex feasibility problems.***SIAM Review*1996,**38**(3):367–426. 10.1137/S0036144593251710MathSciNetView ArticleMATHGoogle Scholar - Combettes PL:
**The foundations of set theoretic estimation.***Proceedings of the IEEE*1993,**81**(2):182–208.View ArticleGoogle Scholar - Youla DC:
**Mathematical theory of image restoration by the method of convex projections.**In*Image Recovery: Theory and Application*. Edited by: Star H. Academic Press, Orlando, Fla, USA; 1987:29–77.Google Scholar - Iusem AN, De Pierro AR:
**On the convergence of Han's method for convex programming with quadratic objective.***Mathematical Programming Series B*1991,**52**(1–3):265–284.MathSciNetView ArticleMATHGoogle Scholar - Su Y, Shang M, Qin X:
**A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings.***Journal of Applied Mathematics and Computing*2008,**28**(1–2):283–294. 10.1007/s12190-008-0103-yMathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149**(1):75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView 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 - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Takahashi S, Takahashi W:
**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**331**(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleMATHGoogle Scholar - Chang SS:
*Variational Inequalities and Related Problems*. Chongqing Publishing House, Chongqing, China; 2007.Google Scholar - Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.