APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2014.46033APM-46625ArticlesPHYSICS & MATHEMATICSConvergence Theorem of Hybrid Iterative Algorithm for Equilibrium Problems and Fixed Point Problems of Finite Families of Uniformly Asymptotically Nonexpansive SemigroupsHongboLiu1YiLi1*School of Science, Southwest University of Science and Technology, Mianyang, China* E-mail:liyi@swust.edu.cn(YL);06062014040624425211 April 201411 May 2014 18 May 2014© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Throughout this paper, we introduce a new hybrid iterative algorithm for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. We then prove the strong convergence theorem with respect to the proposed iterative algorithm. Our results in this paper extend and improve some recent known results.

Hybrid Iterative Algorithm Uniformly Asymptotically Nonexpansive Semigroups Equilibrium Problem Common Fixed Point
1. Introduction

Recall the following equilibrium problem. Let be a closed convex subset of a real Hilbert space with inner produce and norm. Let be a bifunction, where is the set of real numbers. The equilibrium problem for is to to find such that

the set of solutions is denoted by.

A mapping of a normed space into itself is said to be nonexpansive if for each. We denote by the set of fixed point of. Given a mapping, let for all.Then if and only if for all, i.e., is a solution of the variational inequality, there are several other problems, for example, the complementarity problem, minimax problems, the Nash equilibrium problem in noncooperative games, fixed point problem and optimization problem, which can also be written in the form of an EP. In other words, the EP is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP; see, for example ( - ) and references therein.

Iterative methods for finding fixed points of nonexpansivemappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem (see  - ), the split feasibility problem (see  - ) and image recovery and signal processing (see  ).

In 1953, Mann  introduced the following iterative process to approximate a fixed point of a nonexpansive single valued mapping in a Hilbert space:

where the initial point is taken in arbitrarily and is a sequence in. However, we note that Mann’s iteration process has only weak convergence. To obtain strong converges for Mann iteration, Nakajo and Takahashi  and Takahashi et al.  introduce some hybrid iterative process. Motivated by Suzuki’s result  and Nakajo-Takahashi’s results  .

On the other hand, Tada and Takahashi  introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H.

A family of mappings on a closed convex subset of a Hibert space is called a nonexpansive semigroup if it satisfies the following conditions:

1) for all;

2) for all;

3) for all and,

4) for all, is continuous.

Takahashi and Chen  proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in themathematical programming. Recently Saejung  improved the result in  .

Takahashi’s result gives us new idea that a finite family of uniformly asymptotically nonexpansive semi- groups is introduced.

Definition 1.1 A family of mappings on a closed convex subset of a Hibert space is called an uniformly asymptotically nonexpansive semigroup with sequence (and) if it satisfies the following conditions:

1) for all;

2) for all;

3) for all, ,

4) for all, is continuous.

In this paper, we introduce a new hybrid iterative process for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. Then we prove some strong convergence theorems of the proposed iterative process. Our results generalize results of Tada and Takahashi  , Takahashi et al.  , He and Chen  and Saejung  .

2. Preliminaries

Throughout the paper, we denote weak convergence of by, and strong convergence by. Let be a closed convex subset of, we use to denote the common fixed points set of the semigroup

. i.e.,.

Next, We present an example of an uniformly asymptotically nonexpansive semigroup.

Example 2.1 As an example, we consider the nonempty closed convex subset of a Hilbert space. define. Observe that is an uniformly asymptotically nonexpansive semigroup.

For every point, there exists a unique nearest point in, denoted by such that

that is,. is called the metric projection of onto. It is well known that is a nonexpansive mapping. It is also known that H satisfies Opial’s condition, i.e., for any sequence with, following the inequality holds:

To prove our result, we recall the following Lemma.

Lemma 2.1 (see  ). Let be a closed convex subset of. Given and a point. Then if and only if for all.

Lemma 2.2 (see  ). Let be a closed convex subset of. Then for all and we have

Lemma 2.3 (see  ). Let be a real Hilbert space, there hold the following identities:

1), for all and.

2), for all.

Lemma 2.4 (see  ) Let be a real Hilbert space. For,

with.

For solving the equilibrium problem, let us assume the following conditions for a bifunction (see  ):

1), for all.

2), for all.

3) For each,

4) is convex and lower semicontinuous for each.

Lemma 2.5 (see  ) Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)-(A4). Let and. Then, there exists such that

Lemma 2.6 Let satisfies (A1)-(A4). For and, define a mapping as follows:

Then, the following holds:

1) is single valued;

2) is firmly nonexpansive, i.e., for any,;

3);

4) is closed and convex.

In 2013, Mohammad, E. introduce a new hybrid iterative process for finding a common element of the set of common fixed points of a finite family of nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. He then prove strong convergence of the proposed iterative process. In this paper, we improve Mohammad’s result, and obtain follwing main results.

Mohammad’s Theorem 3.1 (see  ) about nonexpansive semigroups is the special case of our results. Our results improve chang’s result in  .

3. Main Results

First, we show the following theorem to our main results.

Theorem 3.1 Let be nonempty closed convex subset of. be an uniformly asymptoti- cally nonexpansive semigroups with nonnegative real sequences with and (as), then is a closed and convex subset of.

Proof. Let be a sequence in, such that. Since be an uniformly asymptotically nonexpansive semigroups, we have

for and for all. Therefore,

We obtain. Hence,. So, we have. This implies is closed.

Let and, and put. Next we prove that. Indeed, in view of Lemma 2.3 2), let, we have

Since

Substituting (1) into (2) and simplifying it we have

Hence, we have. This implies that. Since is closed, we have, i.e.,. This completes the proof of theorem 3.1.

Theorem 3.2 Let be a nonempty closed convex subset of a real Hilbert space and be a

bifunction of into satisfying (A1)-(A4). Let be a finite family of uniformly asymptotically semigroups with sequence (and). Assume that. For an initial piont, let and be sequences generated by

where is the metric projection of onto. If, , and satisfying the fol- lowing conditions:

1);

2) (for) and;

3) and;

4), ,

then, the sequences and converge strongly to.

Proof. 1) First, we prove.

Indeed, is obvious. Suppose that, then for and, by Lemma 2.6 we have

Since be a finite family of uniformly asymptotically semigroups,we have

which implies that.Therefore we have for all. Note is closed and convex.this implies that is well defined. From Lemma 2.5, sequence is also well defined.

2) Next, we prove that exists.

Since is closed and convex subset of, there exists a unique such that. From, we have

Since, we get that

It follows that the sequence is bounded and non decreasing, this implies that exists

3) Now we show that,.

Infact, from Lemma 2.2 we have

witch implies that we get is Cauchy. Hence there exists such that. Since, thus. By Lemma 2.4, we have

from condition (C1), so we have

this implies for all. We know that, hence we have

that is,

Using we get that

that is,

which implies. Hence for all we get that

Without loss of generality, as in Saejung’s article  , let. For and,

where denotes the maximal integer that is not larger than. Since for mapping for a fixed and, then.

4) Now we prove that.

First, since and, by (A2) we get that

and hence

Since, and A(4), we get that

If and, let, then. So, from (A1)-(A4) we have

which gives for all. Hence by (A3) we have

which is.

For, we have

Since, then i.e., for all and thus.

5) Now we prove that.

Since and, we get that

Since, we have

which implies. The proof is completed.

From Theorem 3.1, taking and, we obtain

Corollary 3.1 Let be a nonempty closed convex subset of a real Hilbert space and be a bifunction of into satisfying (A1)-(A4). Let be a finite family of uniformly asymptotically semigroups with sequence (and). Assume that. For an initial piont, let and be sequences generated by

where is the metric projection of onto. If, , and satisfying the following conditions:

1);

2) (for) and;

3), ,

then, the sequences converge strongly to.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgements

The authors are very grateful to reviewers for carefully reading this paper and their comments. This work is supported by the Doctoral Program Research Foundation of Southwest University of Science and Technology (No. 11zx7129) and Applied Basic Research Project of Sichuan Province (No. 2013JY0096).

ReferencesBLUM E. , OETTLI W. ,et al. (1994)BLUM, E. AND OETTLI, W. FROM OPTIMIZATION AND VARIATIONAL INEQUALITIES TO EQUILIBRIUM PROBLEMS MATHEMATICS STUDENTS 63, 123-145.FLAM S.D. , ANTIPIN A.S. ,et al. (1997)EQUILIBRIUM PROGRAMMING USING PROXIMAL-LINK ALGOLITHMS MATHEMATICAL PROGRAMMING 78, 29-41.HTTP://DX.DOI.ORG/10.1007/BF02614504MOUDAFI A. , THERA M. ,et al. (1999)MOUDAFI, A. AND THERA, M. PROXIMAL AND DYNAMICAL APPROACHES TO EQUILIBRIUM PROBLEMS LECTURE NOTE IN ECONOMICS AND MATHEMATICAL SYSTEMS 477, 187-201.BAUSCHKE H.H. , BORWEIN J.M. ,et al. (1996)ON PROJECTION ALGORITHMS FOR SOLVING CONVEX FEASIBILITY PROBLEMS SIAM REVIEW 38, 367-426.HTTP://DX.DOI.ORG/10.1137/S0036144593251710BUTNARIU D., CENSOR Y., GURFIL P. , HADAR E. ,et al. (2008)ON THE BEHAVIOR OF SUBGRADIENT PROJECTIONS METHODS FOR CONVEX FEASIBILITY PROBLEMS IN EUCLIDEAN SPACES SIAM JOURNAL ON OPTIMIZATION 19, 786-807.HTTP://DX.DOI.ORG/10.1137/070689127HALE E.T., YIN W. , ZHANG Y. ,et al. (2010)HALE, E.T., YIN, W. AND ZHANG, Y. FIXED-POINT CONTINUATION APPLIED TO COMPRESSED SENSING: IMPLEMENTATION AND NUMERICAL EXPERIMENTS JOURNAL OF COMPUTATIONAL MATHEMATICS 28, 170-194.MARUSTER S. , POPIRLAN C. ,et al. (2008)ON THE MANN-TYPE ITERATION AND THE CONVEX FEASIBILITY PROBLEM JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 212, 390-396.HTTP://DX.DOI.ORG/10.1016/J.CAM.2006.12.012BYRNE C. ,et al. (2004)A UNIFIED TREATMENT OF SOME ITERATIVE ALGORITHMS IN SIGNAL PROCESSING AND IMAGE RECONSTRUCTION INVERSE PROBLEMS 20, 103-120.HTTP://DX.DOI.ORG/10.1088/0266-5611/20/1/006CENSOR Y., ELFVING T., KOPF N. , BORTFELD T. ,et al. (2005)THE MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM AND ITS APPLICATIONS FOR INVERSE PROBLEMS INVERSE PROBLEMS 21, 2071-2084.HTTP://DX.DOI.ORG/10.1088/0266-5611/21/6/017XU H.K. ,et al. (2006)A VARIABLE KRASNOSELSKII-MANN ALGORITHM AND THEMULTIPLE-SET SPLIT FEASIBILITY PROBLEM INVERSE PROBLEMS 22, 2021-2034.HTTP://DX.DOI.ORG/10.1088/0266-5611/22/6/007MANN W.R. ,et al. (1953)MEAN VALUE METHODS IN ITERATION PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 4, 506-510.HTTP://DX.DOI.ORG/10.1090/S0002-9939-1953-0054846-3NAKAJO K. , TAKAHASHI W. ,et al. (2003)STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE MAPPINGS AND NONEXPANSIVE SEMIGROUPS JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 279, 372-379.HTTP://DX.DOI.ORG/10.1016/S0022-247X(02)00458-4TAKAHASHI W., TAKEUCHI Y. , KUBOTA R. ,et al. (2008)STRONG CONVERGENCE THEOREMS BY HYBRID METHODS FOR FAMILIES OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 341, 276-286.HTTP://DX.DOI.ORG/10.1016/J.JMAA.2007.09.062SUZUKI T. ,et al. (2003)ON STRONG CONVERGENCE TO COMMON FIXED POINTS OF NONEXPANSIVE SEMIGROUPS IN HILBERT SPACES PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 131, 2133-2136.HTTP://DX.DOI.ORG/10.1090/S0002-9939-02-06844-2TADA A. , TAKAHASHI W. ,et al. (2007)WEAK AND STRONG CONVERGENCE THEOREMS FOR A NONEXPANSIVE MAPPING AND AN EQUILIBRIUM PROBLEM JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 133, 359-370.HTTP://DX.DOI.ORG/10.1007/S10957-007-9187-ZHE, H. AND CHEN, R. (2007) STRONG CONVERGENCE THEOREMS OF THE CQ METHOD FOR NONEXPANSIVE SEMIGROUPS. FIXED POINT THEORY AND APPLICATIONS, 2007, ARTICLE ID 59735.SAEJUNG, S. (2008) STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE SEMIGROUPS WITHOUT BOCHNER INTEGRALS. FIXED POINT THEORY AND APPLICATIONS, 2008, ARTICLE ID 745010.MARINO G. , XU H.K. ,et al. (2007)WEAK AND STRONG CONVERGENCE THEOREMS FOR STRICT PSEUDO-CONTRACTIONS IN HILBERT SPACE JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 329, 336-346.HTTP://DX.DOI.ORG/10.1016/J.JMAA.2006.06.055CHOLAMJIAK, W. AND SUANTAI, S. (2010) AHYBRID METHOD FOR A COUNTABLE FAMILY OF MULTIVALUED MAPS, EQUILIBRIUM PROBLEMS, AND VARIATIONAL INEQUALITY PROBLEMS. DISCRETE DYNAMICS IN NATURE AND SOCIETY, 2010, ARTICLE ID: 349158.MOHAMMAD E. ,et al. (2013)HYBID METHOD FOR EQUILIBRIUM PROBLEMS AND FIXED PIONT PROBLEMS OF FINITE OF NONEXPANSIVE SEMIGROUPS REVISTA SERIE A MATEMÁTICAS 107, 299-307.CHANG S.S., WANG L., TANG Y.K., WANG B. , QIN L.J. ,et al. (2012)STRONG CONVERGENCE THEOREMS FOR A COUNTABLE FAMILY OF QUASI-Ψ-ASYMPTOTICALLY NONEXPANSIVE NONSELF MAPPINGS APPLIED MATHEMATICS AND COMPUTATION 218, 7864-7870.HTTP://DX.DOI.ORG/10.1016/J.AMC.2012.02.002