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The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-

Throughout this paper, we denote by

Let

We denote the set of solutions of (

Special examples are as follows.

If

which is called the mixed equilibrium problem (MEP) [

If

which is called the mixed variational inequality of Browder type (VI) [

Recently, many authors studied the problems of finding a common element of the set of fixed point for a nonexpansive mapping and the set of solutions for an equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [

Motivated and inspired by the researches going on in this direction, the purpose of this paper is using a hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-

First, we recall some definitions and conclusions.

The mapping

A Banach space

The following basic properties can be found in Cioranescu [

If

If

If

A Banach space

Each uniformly convex Banach space

Next we assume that

It is obvious from the definition of

Following Alber [

Let

if

for

If

Let

(1) A mapping

(1) A mapping

(1) From the definition, it is easy to know that each relatively nonexpansive mapping is closed.

(2) The class of quasi-

Let

Let

For solving the system of generalized mixed equilibrium problems (

the function

Let

(Blum and Oettli [

(Takahashi and Zembayashi [

Let

There exists

If we define a mapping

It follows from Lemma

In this section, we will use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (

Let

Then

We divide the proof of Theorem

We first prove that

In fact, it follows from Lemmas

This implies that

We prove that

This implies that

In view of the structure of

Next, we prove that

Indeed, it is obvious that

Now, we prove that

First, we prove that

In fact, since

Now we prove that

Now we prove that

Hence we have

This implies that

By the similar way as given in the proof of (

From (

Since

For any

Since

From (

In view of conditions (b)

It follows from the property of

Since

Furthermore, by the assumption that for each

This together with (

Next, we prove that

Hence we have

This implies that

By the similar way as given in the proof of (

Since

by the similar way as above, we can also prove that

For

Now, we prove

Let

By the definition of

Let

Since

Theorems

For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property (note that each uniformly convex Banach space must have Kadec-Klee property).

For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or quasi-

We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.