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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.

Recall the following equilibrium problem. Let

the set of solutions is denoted by

A mapping

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 [

In 1953, Mann [

where the initial point

On the other hand, Tada and Takahashi [

A family

1)

2)

3)

4) for all

Takahashi and Chen [

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

Definition 1.1 A family

1)

2)

3)

4) for all

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 [

Throughout the paper, we denote weak convergence of

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

Example 2.1 As an example, we consider the nonempty closed convex subset

For every point

that is,

To prove our result, we recall the following Lemma.

Lemma 2.1 (see [

Lemma 2.2 (see [

Lemma 2.3 (see [

1)

2)

Lemma 2.4 (see [

For solving the equilibrium problem, let us assume the following conditions for a bifunction

1)

2)

3) For each

4)

Lemma 2.5 (see [

Lemma 2.6 Let

Then, the following holds:

1)

2)

3)

4)

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 [

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

Theorem 3.1 Let

Proof. Let

for

We obtain

Let

Since

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

Hence, we have

Theorem 3.2 Let

bifunction of

where

1)

2)

3)

4)

then, the sequences

Proof. 1) First, we prove

Indeed,

Since

which implies that

2) Next, we prove that

Since

Since

It follows that the sequence

3) Now we show that

Infact, from Lemma 2.2 we have

witch implies that we get

from condition (C1), so we have

this implies

that is,

Using

that is,

which implies

Without loss of generality, as in Saejung’s article [

where

4) Now we prove that

First, since

and hence

Since

If

which gives

which is

For

Since

5) Now we prove that

Since

Since

which implies

From Theorem 3.1, taking

Corollary 3.1 Let

where

1)

2)

3)

then, the sequences

The authors declare that they have no competing interests.

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).