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We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

Throughout this paper, we assume that

A family

for all

We denote by

Let

Let the set-valued mapping

Let

In 2006, Martinez-Yanes and Xu [

In 2008, Zhang et al. [

Recently, Shehu [

In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Martinez-Yanes and Xu [

Let

In order to prove our main results, we need the following Lemmas.

Let

the fixed-point set

define a mapping

A family of mappings

Let

For each

Let

for every

a mapping

Let

Each Hilbert space

Let

Lemma

Let

Let

Let C be a nonempty bounded closed-convex subset of H, let

For solving the generalized mixed equilibrium problem for

for each

for each

for each

for each

C is a bounded set,

then one has the following lemma.

Let C be a nonempty closed-convex subset of H. Let

for each

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Let

then

First, we show that

We show that

We claim that

We claim that the following statements hold:

For

We show that

(1) First, we prove that

(2) Next, we show that

(3) Now, we prove that

(4) At last, we show that

Noting that

Using Theorem

Let _{n} be the

then

From Theorem

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:

Let

From Theorem

Let

The authors thank the referees for their appreciation, valuable comments, and suggestions. They would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support. Furthermore, they would like to thank the Faculty of science (KMUTT) and the National Research Council of Thailand. This work was completed with the support of the National Research Council of Thailand (NRCT 2010-2011).