1. Introduction
Throughout this paper, we denote by 
and 
the sets of positive integers and real numbers, respectively. Let 
be a nonempty closed subset of a real Hilbert space
. Let 
and 
denote the family of nonempty subsets and nonempty closed bounded subsets of
, respectively. The Hausdorff metric on 
is defined by
for
, where
. An element 
is called a fixed point of a multivalued mapping 
if
. The set of fixed points of a multivalued mapping 
is represented by
.
The multivalued mapping 
is called nonexpansive if
The multivalued mapping 
is called quasi-nonexpansive if 
and
Iterative process for approximating fixed points (and common fixed points) of nonexpansive multivalued mappings have been investigated by various authors (see [2] -[5] ).
Recently, Kohsaka and Takahashi (see [6] [7] ) introduced an important class of mappings which they called the class of nonspreading mappings. Let 
be a subset of Hilbert space
, they called a mapping 
nonspreading if
Lemoto and Takahashi [8] proved that 
is nonspreading if and only if
Now, inspired by [6] and [7] , we propose a definition as follows.
Definition 1.1 The multivalued mapping 
is called nonspreading if
(1.1)
By Takahashi [8] , We get also the multivalued mapping 
is nonspreading if and only if
(1.2)
Infact,
Definition 1.2 The multivalued mapping 
is called 
-strictly pseudononspreading if there exists 
such that
(1.3)
Observe that suppose 
is k-strictly pseudononspreading with
, and
, then
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 [1] page 1816 Example 1), Let 
denote the reals with the usual norm. Let 
be defined for each 
by
The equilibrium problem for 
is to find 
such that
,
. The set of such solution is denoted by
. Given a mapping
, let 
for all
. The 
if and only if 
is a solution of the variational inequality 
for all
.
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 [9] , Combettes and Hirstoaga [10] , Li and Li [11] , Giannessi, Maugeri, and Pardalos [12] , Moudafi and Thera [13] and Pardalos, Rassias and Khan [14] , Ceng et al. [15] . In the recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points of single-valued nonexpansive mappings in the framework of Hilbert spaces has been intensively studied by many authors.
In this paper, inspired by [1] we propose an iterative process for finding a common element of the set of solutions of equilibrium problem and the set of common fixed points of k-strictly pseudononspreading multivalued mapping in the setting of real Hilbert spaces. We also prove the strong and weak convergence of the sequences generated by our iterative process. The results presented in the paper improve and extend the corresponding results in [1] and others.
2. Preliminaries and Lemma
In the sequel, we begin by recalling some preliminaries and lemmas which will be used in the proof.
Lemma 2.1 Let 
be a real Hilbert space, for all 
and
, then the following well known results hold:
(i) 
(ii) 
(iii) If 
is a sequence in 
which converges weakly to 
then
Let 
be a nonempty closed convex subset of a real Hilbert space
. The nearest point projection 
defined from 
onto 
is the function which assigns to each 
its nearest point denoted by 
in
. Thus 
is the unique point in 
such that
It is known that for each 
Lemma 2.2 (see [5] ) Let 
be a nonempty closed convex subset of a real Hilbert space
. Let 
be the metric projection of 
onto
. Let 
be a sequence in 
and let 
for all
. Then 
converges strongly.
We present the following properties of a k-strictly pseudononspreading multivalued mapping.
Lemma 2.3 Let 
be a nonempty closed convex subset of a real Hilbert space
, and let 
be a k-strictly pseudononspreading multivalued mapping. If
, and
, then it is closed and convex.
Proof. Let 
and 
(as
). Since 
and
we have 
(as
). Hence
.
Next let
, where 
and
, we have
Thus 
and hence
. This complete the proof of Lemma 2.3 Lemma 2.4 Let 
be a nonempty closed convex subset of a real Hilbert space
, and let 
be a k-strictly pseudononspreading multivalued mapping. If
, and
, then 
is demiclosed at 0.
Proof. Let 
be a sequence in 
which 
and 
(as
).
Since
, it is bounded. For each 
define 
by
Then from Lemma 2.1 we obtain
and so 
(where
).
In addition,
We obtain
. Thus 
and hence
. This complete the proof of Lemma 2.4. ,
3. Main Results
Theorem 3.1 Let 
be a nonempty closed convex subset of a real Hilbert space
, and let 
be a k-strictly pseudononspreading multivalued mapping with 
and
. Let 
and 
be a real sequence in 
such that
. Let 
and 
be sequences generated initially by an arbitrary element 
and then by
Then, the sequences 
converge weakly to
, where 
Proof. Let 
First, We claim that
.
Indeed, if
, then
this implies 
and 
Next, for 
we have
(3.1)
By (1.3) and (3.1), we obtain
(3.2)
Observe also that for each 
hence 
is bounded. By Lemma 2.1 and (3.2), we obtain
(3.3)
Since
(3.4)
it follows from (3.3) and (3.4) that
(3.5)
Summing (3.5) from n = 1 to n, and dividing by n we obtain
(3.6)
Since 
is bounded,then 
is also bounded. Thus there exists a subsequence 
of 
such that 
(as
). we also have
(3.7)
As 
we obtain from (3.7) that
(3.8)
Since 
was arbitrary, setting 
in (3.8) we have
from which it follows that
. Since 
is closed and convex by Lemma 2.3, thus we can define the projection
.
From Lemma 2.2, 
converges strongly. Let
.
Next we show that
.
Since 
and 
are bounded, there exists 
such that 
, then we obtain by 
(3.9)
Summing (3.9) from 
to
, and dividing by 
we obtain
(3.10)
Sine 
as
, and
, we have
Hence
, so, the sequences 
converge weakly to
, where
. This complete the proof of Theorem 3.1. ,
Acknowledgments
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.
NOTES
*Corresponding author.