ABSTRACT

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.

Keywords: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 ([1] -[3] ) 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 [4] -[7] ), the split feasibility problem (see [8] -[10] ) and image recovery and signal processing (see [6] ).

In 1953, Mann [11] 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 [12] and Takahashi et al. [13] introduce some hybrid iterative process. Motivated by Suzuki’s result [14] and Nakajo-Takahashi’s results [12] .

On the other hand, Tada and Takahashi [15] 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 and4) for all, is continuous.

Takahashi and Chen [16] proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in themathematical programming. Recently Saejung [17] improved the result in [16] .

Takahashi’s result gives us new idea that a finite family of uniformly asymptotically nonexpansive semigroups 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 [15] , Takahashi et al. [13] , He and Chen [16] and Saejung [17] .

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 [18] ). Let be a closed convex subset of. Given and a point. Then if and only if for all.

Lemma 2.2 (see [12] ). Let be a closed convex subset of. Then for all and we have

.

Lemma 2.3 (see [18] ). Let be a real Hilbert space, there hold the following identities:

1), for all and.

2), for all.

Lemma 2.4 (see [19] ) Let be a real Hilbert space. For,

with.

For solving the equilibrium problem, let us assume the following conditions for a bifunction (see [1] ):

1), for all.

2), for all.

3) For each,

4) is convex and lower semicontinuous for each.

Lemma 2.5 (see [1] ) 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 [20] ) about nonexpansive semigroups is the special case of our results. Our results improve chang’s result in [21] .

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 asymptotically 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

(1)

Since

(2)

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

(3)

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

(4)

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

(5)

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 [17] , 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

(6)

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).

References

NOTES

^*Corresponding author.