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Two countable families of hemirelatively nonexpansive mappings are considered based on a hybrid projection algorithm. Strong convergence theorems of iterative sequences are obtained in an uniformly convex and uniformly smooth Banach space. As applications, convex feasibility problems, equilibrium problems, variational inequality problems, and zeros of maximal monotone operators are studied.

Throughout this paper, we always assume that

Let

The set of solutions to (

(1) If

(2) If

(3) If

(4) If

(5) If

(6) If

The problem (

A Banach space

The modulus of convexity of

For each

if

if

if

if

if

Let

Let

Let

a mapping

a mapping

From the definitions, it is obvious that a relatively nonexpansive mapping is a weak relatively nonexpansive mapping, and a weak relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, but the converse is not true.

Next, we give an example which is a closed hemirelatively nonexpansive mapping.

Let

In 2005, Matsushita and Takahashi [

Let

Since then, algorithms constructed for solving the same equilibrium problem, variational inequality problems, and fixed point of relatively nonexpansive mappings (or weak relatively nonexpansive mappings or hemi-relatively nonexpanisve mappings) have been further developed by many authors. For a part of works related to these problems, please see [

Motivated and inspired by the results in the literature, in this paper we focus our attention on finding a common fixed point of two countable families of hemi-relatively nonexpansive mappings (we shall give the definition of a countable family of hemi-relatively nonexpansive mappings in the next section) by using a simple hybrid algorithm. Furthermore, we will give some applications of our main result in equilibrium problems, variational inequality problems, and convex feasibility problems.

Let

Recall that

A point

A point

Using the definition of (strong) asymptotic fixed point of

Countable family of mappings

Countable family of mappings

Now, we introduce the definition of countable family of hemi-relatively nonexpansive mappings which is more general than countable family of relatively nonexpansive mappings and countable family of weak relatively nonexpansive mappings.

Countable family of mappings

From Definitions

The definitions of relatively nonexpansive mapping, weak relatively nonexpansive mapping, and hemi-relatively nonexpansive mapping are special cases of Definitions

Countable family of hemi-relatively nonexpansive mappings, which do not need the restriction

Next we give an example which is a countable family of hemi-relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

Let

First, it is obvious that

In what follows, we will need the following lemmas.

Let

Let

Let

Now, we give our main results in this paper.

Let

We first show that

Let

Noticing

Since

By using Lemma

Finally, we show that

Theorem

from the class of a countable family of weak relatively nonexpansive mappings to the one of a countable family of hemi-relatively nonexpansive mappings;

from a single countable family of mappings to two countable families of mappings.

When

Let

We notice that if

In this section, we consider the following convex feasibility problem (CFP):

Using Theorem

Let

From Lemma

If we only consider a countable family of nonempty closed convex subset of

Let

Putting

In this section, we apply our main results to prove some strong convergence theorems concerning generalized mixed equilibrium problems in a Banach space

Let

(I)

(II)

For solving the generalized mixed equilibrium problem (

the function

The following result can be found in Blum and Oettli [

Let

Let

For convenience, we set

(I) We show that

(II) We show that

(III) We show that

(IV) We show that the function

For each

Let

Let

Next, we shall apply Theorem

Let

From Lemmas

Let

From Lemmas

If we let

Let

By analysis of special cases for generalized mixed equilibrium problem, we can obtain the corresponding results based on Theorems

Let

A monotone operator

The following result is also well known.

Let

Let

If

First, we give an important lemma for this section and remark that the following lemma can be as example of a countable family of hemi-relatively nonexpansive mappings.

Let

One has

In addition, for any

We consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [

Let

From Lemma

The authors are thankful to an anonymous referee for his useful comments on this paper. This research was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC-KMUTT) (Grant Project no. NRU56000508). The first author is supported by the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J13LI51) and the Foundation of Shandong Yingcai University (Grant no. 12YCZDZR03).