We introduce viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

Let

Let

In 2005, Combettes and Hirstoaga [

In 1953, Mann [

Using the viscosity approximation method, in 2007, S. Takahashi and W. Takahashi [

On the other hand, in 2008, Takahashi et al. [

Very recently, Kimura and Nakajo [

Motivated by these results, we introduce the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

Throughout this paper, we denote by

For every point

For solving the generalized equilibrium problem, let us assume that

for each

for each

Let

Let

Using (ii) in Lemma

For any

For a sequence

If

Let

On the other hand, a mapping

A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.

We have the following results for Meir-Keeler contractions defined on a Banach space by Suzuki [

Let

Let

For each

In this section, using the shrinking projection method by Takahashi et al. [

Let

For

Let

We first show that

Next, we show that

Let

Since

It is obvious from the assumption that

Thus,

Next, we show that

Since

We have from (

So, we have

Since

By Theorem

Setting

Let

Setting

Let

Setting

Let

Next, using the CQ hybrid method introduced by Nakajo and Takahashi [

Let

As in the proof of Theorem

Next, we will show that

Next, we will show that

Next, we will show that

Setting

Let

Setting

Let

Setting

Let

In this section, we present some convergence theorems deduced from the results in the previous section. Recall that a mapping

A mapping

Recently, Kocourek et al. [

Let

Let

Setting

Let

Put

Setting

Let

Setting

Let

Setting

Let

Setting

Let

Put

Setting

Let

Setting

Let

Setting

Let

The authors would like to thank Naresuan University for financial support.