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We introduce a k-strictly pseudononspreading multivalued in Hilbert spaces more general than the class of nonspreading multivalued. We establish some weak convergence theorems of the sequences generated by our iterative process. Some new iterative sequences for finding a common element of the set of solutions for equilibrium problem was introduced. The results improve and extend the corresponding results of Osilike Isiogugu [1] (Nonlinear Anal.74 (2011)) and others.

Throughout this paper, we denote by

for

The multivalued mapping

The multivalued mapping

Iterative process for approximating fixed points (and common fixed points) of nonexpansive multivalued mappings have been investigated by various authors (see [

Recently, Kohsaka and Takahashi (see [

Lemoto and Takahashi [

Now, inspired by [

Definition 1.1 The multivalued mapping

By Takahashi [

Infact,

Definition 1.2 The multivalued mapping

Observe that suppose

Clearly every nonspreading multivalued mapping is k-strictly pseudononspreading multivalued mapping. The following example shows that the class of k-strictly pseudononspreading mappings is more general than the class of nonspreading mappings.

Example (see [

The equilibrium problem for

Numerous problems in physics, optimization, and economics can be reduced to find a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem see, for instance, Blum and Oettli [

In this paper, inspired by [

In the sequel, we begin by recalling some preliminaries and lemmas which will be used in the proof.

Lemma 2.1 Let

(i)

(ii)

(iii) If

Let

It is known that for each

Lemma 2.2 (see [

We present the following properties of a k-strictly pseudononspreading multivalued mapping.

Lemma 2.3 Let

Proof. Let

we have

Next let

Thus

Lemma 2.4 Let

Proof. Let

Since

Then from Lemma 2.1 we obtain

and so

In addition,

We obtain

Theorem 3.1 Let

Then, the sequences

Proof. Let

First, We claim that

Indeed, if

this implies

By (1.3) and (3.1), we obtain

Observe also that for each

hence

Since

it follows from (3.3) and (3.4) that

Summing (3.5) from n = 1 to n, and dividing by n we obtain

Since

As

Since

from which it follows that

From Lemma 2.2,

Next we show that

Since

Summing (3.9) from

Sine

Hence

This work is supported by the Doctoral Program Research Foundation of Southwest University of Science and Technology (No.11zx7129) and the National Natural Science Foundation of China (No.71071102).

The authors are very grateful to the referees for their helpful comments and valuable suggestions.