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In this works, by using the modified viscosity approximation method associated with Meir-Keeler contractions, we proved the convergence theorem for solving the fixed point problem of a nonexpansive semigroup and generalized mixed equilibrium problems in Hilbert spaces.

As you know, there are many problems that are reduced to find solutions of equilibrium problems which cover variational inequalities, fixed point problems, saddle point problems, complementarity problems as special cases. Equilibrium problem which was first introduced by Blum and Oettli [

From now on, we assume that

The solution set of problem (1.1) is denoted by

Recently the so-called generalized mixed equilibrium problem has been investigated by many authors [

where

The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequality problem, minimax problems, the Nash equilibrium problems in noncooperative games and others (see [

Special Cases: The following problems are the special cases of problem (1.2).

1) If

is called mixed equilibrium problems.

2) If

is called mixed variational inequality of Browder type [

3) If

is called generalized equilibrium problems (shortly, (GEP)). We denote GEP(G,A) the solution set of problem (GEP).

4) If

5) Let

It is known that

Let

A mapping

A mapping

Remark 1.1 Every

In 1967, Halpern [

where

Moudafi [

where

A viscosity approximation method with Meir-Keeler contraction was first studied by Suzuki [

where

Takahashi and Takahashi [

where

In this paper, from the recent works [

Let

Let X be a translation invariant subspace of

for each

Let

Assume that

Lemma 2.1 [

Moreover if

We can write

Lemma 2.2 [

1)

2)

3)

4) if

Let

where

A mapping

Definition 2.3 Let

1)

for all

2) Meir-Keeler type mapping if for each

Theorem 2.4 [

Theorem 2.5 [

1)

2) there exists an

Theorem 2.6 [

Proposition 2.7 [

1)

2) For each

Lemma 2.8 [

where

1)

2)

Then

Lemma 2.9 [

where

If

then

Lemma 2.10 [

and

for some

Lemma 2.11 [

For solving the equilibrium problem we assume that bifunction

(A2)

(A3) for each

(A4) for each

Lemma 2.12 [

Further, if

then we have the followings:

1)

2)

3)

4)

Lemma 2.13 [

for all

Theorem 3.1 Let K be a nonempty closed convex subset of a Hilbert space

Let

where

Then the sequence

Proof. We give the several steps for the proof.

Step 1: First we show that

Set

By induction, we can prove that

Hence the sequence

Step 2: We next show that

Observe that

Indeed

Since

From

it follows that

We see that

Combining (3.4) and (3.5) with (3.6), we obtain

Using Lemma 2.13, (3.3),(C1) and (C4), then we have

From this inequality and (C3), it follows from Lemma 2.9 that

It implies that

Step 3: Next we prove that for all

Put

Set

From Corollary 1.1 in [

Since

for all

We observe from Lemma 2.2 (iii) that

Combining (3.10), (3.12) and (3.12), we have for all

Let

Hence

Step 4: We next show that

Using inequality (3.2), we obtain

which implies that

From (C1)-(C4) and (3.8), we obtain

Since

Therefore

Then we have

which yields

Hence, from (C2), (C3) and (3.16) we obtain

Since

On the other hand, by Proposition 2.7 (i), we know that

Step 5: We next show that

To see this, we chose a subsequence

Since

From (A2), we have

Then

Put

From (A4), we have

From (A1)-(A4) and (3.20), we have

It follows that

letting

Hence

Step 6: Now we are in a position to show that

Let

We note that

and

It follows from Lemma 2.10 that

On the other hand, we have

It follows from (3.17) and (3.22) that

Therefore

This is a contradiction. Hence we have

Step 7: We finally show that

Suppose that

So we have

This implies that

Hence

Using (3.21) and (C2), we can conclude by Lemma 2.8 that

This work was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea(2015R1D1A1A09058177).

Kim, J.K., Salahuddin and Lim, W.H. (2017) An Iterative Algorithm for Generalized Mixed Equilibrium Problems and Fixed Points of Nonexpansive Semigroups. Journal of Applied Mathematics and Physics, 5, 276-293. https://doi.org/10.4236/jamp.2017.52025