- Research Article
- Open Access

# A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces

- Poom Kumam
^{1}and - Chaichana Jaiboon
^{2}Email author

**2010**:361512

https://doi.org/10.1155/2010/361512

© Poom Kumam and Chaichana Jaiboon. 2010

**Received:**2 April 2010**Accepted:**11 June 2010**Published:**4 July 2010

## Abstract

We introduce and analyze a new iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of a system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Furthermore, we prove new strong convergence theorems for a new iterative algorithm under some mild conditions. Finally, we also apply our results for solving convex feasibility problems in Hilbert spaces. The results obtained in this paper improve and extend the corresponding results announced by Qin and Kang (2010) and the previously known results in this area.

## Keywords

- Hilbert Space
- Variational Inequality
- Iterative Algorithm
- Equilibrium Problem
- Nonexpansive Mapping

## 1. Introduction

Let
be a real Hilbert space with inner product
and norm
and let
be a nonempty closed convex subset of
. We denote weak convergence and strong convergence by notations
and
, respectively. Let
be a mapping. In the sequel, we will use
to denote the set of *fixed points* of
, that is,
.

Definition 1.1.

Let be a mapping. Then is called

*contraction* if there exists a constant
such that

*nonexpansive* if

Remark 1.2.

It is well known that if is nonempty, bounded, closed, and convex and is a nonexpansive mapping on then is nonempty; see, for example, [1].

*strongly pseudocontractive* with the coefficient
if

for such a case,
is also said to be a
*-strict pseudocontraction,* and if
, then
is a nonexpansive mapping,

*pseudocontractive* if

The class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of fixed points for strict pseudocontractions.

where is a strict pseudocontraction. Under appropriate restrictions on , it is proved the mapping is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudocontractions.

Let be a proper extended real-valued function and let be a bifunction of into such that , where is the set of real numbers.

We see that is a solution of problem (1.8) implies that

Special Examples

- (3)

- (4)

- (6)

The set of solutions of (1.15) is denoted by .

The generalized mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.8). In 1997, Combettes and Hirstoaga [12] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. Many authors have proposed some useful methods for solving the , and ; see, for instance, [5, 12–23].

Definition 1.3.

Let be a nonlinear mapping. Then is called

(4)
*-inverse-strongly monotone* if there exists a constant
such that

Remark 1.4.

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

where is the metric projection of onto , is a -inverse-strongly monotone mapping, is a sequence in , and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.20) converges weakly to some .

where is a -inverse-strongly monotone mapping, and are sequences in the interval , and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.21) converges strongly to some .

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

where for some and satisfies . Further, they proved that and converge weakly to , where .

where is a -strict pseudocontraction mapping and and are sequences in They proved that under certain appropriate conditions over , , and , the sequences and both converge strongly to some , which solves some variational inequality problems (1.26).

In 2008, Ceng and Yao [5] introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem (1.8) in Hilbert spaces and obtained the strong convergence theorem which used the following condition.

: is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology. We note that the condition for the function : is a very strong condition. We also note that the condition does not cover the case and for each . Very recently, Wangkeeree and Wangkeeree [29] introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a -strict pseudocontraction mapping, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the condition for the sequences generated by these processes.

Then, they proved that under certain appropriate conditions imposed on , , , , , and , the sequence generated by (1.30) converges strongly to , where .

In the present paper, motivated and inspired by Qin and Kang [30], Peng and Yao [21], Plubtieng and Punpaeng [26], and Liu [28], we introduce a new general iterative scheme for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. The results in this paper extend and improve the corresponding recent results of Qin and Kang [30], Peng and Yao [21], Plubtieng and Punpaeng [26], and Liu [28] and many others.

## 2. Preliminaries

The mapping
is called the *metric projection* of
onto

for all .

Lemma 2.1.

Lemma 2.2.

where is the metric projection of onto .

*monotone*if for all , and imply . A monotone mapping is called

*maximal*if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be a monotone map of into , -Lipschitz continuous mappings and let be the

*normal cone*to when , that is,

Then is the maximal monotone and if and only if see [31].

Lemma 2.3.

Let be a Hilbert space, let be a nonempty closed convex subset of and let be -inverse-strongly monotone. It , then is a nonexpansive mapping in

Proof.

So, is a nonexpansive mapping of into .

Lemma 2.4 (see [32]).

Lemma 2.5 (see [25]).

That is, is strongly monotone with coefficient .

Lemma 2.6 (see [25]).

Assume that is a strongly positive linear bounded operator on with coefficient and . Then .

Lemma 2.7 (see [4]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudocontraction mapping with a fixed point. Then is closed and convex. Define by for each . Then is nonexpansive such that .

Lemma 2.8 (see [33]).

Lemma 2.9 (see [34]).

for is well defined and nonexpansive and holds.

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction , the function and the set :

for all

is monotone, that is, for all

for each

for each is convex and lower semicontinuous;

for each is weakly upper semicontinuous;

is a bounded set.

By similar argument as in the proof of Lemma in [35], we have the following lemma appearing.

Lemma 2.10.

for all . Then, the following holds:

(i)for each ;

(ii) is single-valued;

(iv)

(v) is closed and convex.

Remark 2.11.

We remark that Lemma 2.10 is not a consequence of Lemma in [5], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .

Lemma 2.12 (see [36]).

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

Lemma 2.13 (see [37]).

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

(1)

(2) or

Then

Lemma 2.14.

## 3. Main Results

In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of a common solutions of the variational inequalities for inverse-strongly monotone mappings in a real Hilbert space.

Theorem 3.1.

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

,

and

and , where are two positive constants,

, where .

Equivalently, one has

Proof.

Since , it follows that is a contraction of into itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique element such that

Next, we will divide the proof into five steps.

Step 1.

We claim that is bounded.

Hence, is bounded, and so are , , , , , , and .

Step 2.

We claim that and

Step 3.

We claim that the following statements hold:

;

;

;

.

Step 4.

We claim that where is the unique solution of the variational inequality for all

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that We claim that .

Assume also that and .

It follows from Lemma 2.8 that . By (3.55), we have .

Step 5.

We claim that .

and we can see that and . Applying Lemma 2.13 to (3.66), we conclude that converges strongly to in norm. This completes the proof.

If the mapping is nonexpansive, then . We can obtain the following result from Theorem 3.1 immediately.

Corollary 3.2.

Equivalently, one has

If and in Theorem 3.1, then we can obtain the following result immediately.

Corollary 3.3.

where , , , and are sequences in , where , , , and and are positive sequences. Assume that the control sequences satisfy the condition (C1)–(C6) in Theorem 3.1 and . Then, converges strongly to a point , where

If and in Corollary 3.3, then and we get and ; hence we can obtain the following result immediately.

Corollary 3.4.

where , , , and are sequences in . Assume that the control sequences satisfy the conditions (C2) and (C3), in Theorem 3.1, and . Then, converges strongly to a point , where .

where is an integer and each is assumed to be the of solutions of equilibrium problem with the bifunction and the solution set of the variational inequality problem. There is a considerable investigation on in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [38, 39], computer tomography [40], and radiation therapy treatment planning [41].

The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.

Theorem 3.5.

where such that , are positive sequences, and and are sequences in . Assume that the control sequences satisfy the following restrictions:

and

for each ,

, where is some positive constant for each ,

, for each .

Equivalently, one has

## Declarations

### Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper. The first author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute.

## Authors’ Affiliations

## References

- Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle 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 - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C:
**A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.***Journal of Computational and Applied Mathematics*2008,**214**(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleMATHGoogle 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 - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Browder FE:
**Existence and approximation of solutions of nonlinear variational inequalities.***Proceedings of the National Academy of Sciences of the United States of America*1966,**56:**1080–1086. 10.1073/pnas.56.4.1080MathSciNetView ArticleMATHGoogle Scholar - Hartman P, Stampacchia G:
**On some nonlinear elliptic differential-functional equations.***Acta Mathematica*1966,**115:**271–310. 10.1007/BF02392210MathSciNetView ArticleMATHGoogle Scholar - Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity*.*Volume 76*. Edited by: Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558. 10.1007/0-387-23393-8_12View 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 - Combettes PL, Hirstoaga SA:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications.***Nonlinear Analysis*2010,**72**(1):99–112. 10.1016/j.na.2009.06.042MathSciNetView ArticleMATHGoogle Scholar - Gao X, Guo Y:
**Strong convergence of a modified iterative algorithm for mixed-equilibrium problems in Hilbert spaces.***Journal of Inequalities and Applications*2008,**2008:**-23.Google Scholar - Jaiboon C, Kumam P:
**A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings.***Fixed Point Theory and Applications*2009,**2009:**-32.Google Scholar - Jaiboon C, Kumam P:
**Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities.***Journal of Inequalities and Applications*2010,**2010:**-43.Google Scholar - Jaiboon C, Kumam P:
**A general iterative method for addressing mixed equilibrium problems and optimization problems.***Nonlinear Analysis*2010,**73**(5):1180–1202. 10.1016/j.na.2010.04.041MathSciNetView ArticleMATHGoogle Scholar - Jaiboon C, Kumam P, Humphries UW:
**Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems.***Bulletin of the Malaysian Mathematical Sciences Society*2009,**32**(2):173–185.MathSciNetMATHGoogle Scholar - Jung JS:
**Strong convergence of composite iterative methods for equilibrium problems and fixed point problems.***Applied Mathematics and Computation*2009,**213**(2):498–505. 10.1016/j.amc.2009.03.048MathSciNetView ArticleMATHGoogle Scholar - Kumam P, Jaiboon C:
**A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems.***Nonlinear Anal: Hybrid Systems*2009,**3**(4):510–530. 10.1016/j.nahs.2009.04.001MathSciNetMATHGoogle Scholar - Peng J-W, Yao J-C:
**Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems.***Mathematical and Computer Modelling*2009,**49**(9–10):1816–1828. 10.1016/j.mcm.2008.11.014MathSciNetView 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 - Yao Y, Liou Y-C, Yao J-C:
**A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems.***Fixed Point Theory and Applications*2008,**2008:**-15.Google Scholar - Yao Y, Yao J-C:
**On modified iterative method for nonexpansive mappings and monotone mappings.***Applied Mathematics and Computation*2007,**186**(2):1551–1558. 10.1016/j.amc.2006.08.062MathSciNetView ArticleMATHGoogle 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 - Plubtieng S, Punpaeng R:
**A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**336**(1):455–469. 10.1016/j.jmaa.2007.02.044MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Al-Homidan SA, Ansari QH, Yao J-C:
**An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings.***Journal of Computational and Applied Mathematics*2009,**223**(2):967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleMATHGoogle Scholar - Liu Y:
**A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis*2009,**71**(10):4852–4861. 10.1016/j.na.2009.03.060MathSciNetView ArticleMATHGoogle Scholar - Wangkeeree R, Wangkeeree R:
**A general iterative method for variational inequality problems, mixed equilibrium problems, and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-32.Google Scholar - Qin X, Kang SM:
**Convergence theorems on an iterative method for variational inequality problems and fixed point problems.***Bulletin of the Malaysian Mathematical Sciences Society*2010,**33**(1):155–167.MathSciNetMATHGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar - Osilike MO, Igbokwe DI:
**Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations.***Computers & Mathematics with Applications*2000,**40**(4–5):559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Nonlinear operators and nonlinear equations of evolution in Banach spaces.**In*Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill, 1968)*. American Mathematical Society, Providence, RI, USA; 1976:1–308.Google 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 - Peng J-W, Yao J-C:
**A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2008,**12**(6):1401–1432.MathSciNetMATHGoogle 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 - Combettes PL:
**The convex feasibility problem: in image recovery.**In*Advances Imaging and Electron Physics*.*Volume 95*. Edited by: Hawkes P. Academic Press, Orlando, Fla, USA; 1996:155–270.Google Scholar - Kotzer T, Cohen N, Shamir J:
**Images to ration by a novel method of parallel projection onto constraint sets.***Optics Letters*1995,**20:**1172–1174. 10.1364/OL.20.001172View ArticleGoogle Scholar - Sezan MI, Stark H:
**Application of convex projection theory to image recovery in tomograph and related areas.**In*Image Recovery: Theory and Application*. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:155–270.Google Scholar - Censor Y, Zenios SA:
*Parallel Optimization, Numerical Mathematics and Scientific Computation*. Oxford University Press, New York, NY, USA; 1997:xxviii+539.MATHGoogle 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.