Advances in Pure Mathematics
Vol.4 No.7(2014), Article
ID:48174,7
pages
DOI:10.4236/apm.2014.47042
Convergence Theorems for k-Strictly Pseudononspreading Multivalued in Hilbert Spaces
Hongbo Liu*, Yi Li
School of Science, Southwest University of Science and Technology, Mianyang, China
Email: *liuhongbo@swust.edu.cn
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 15 May 2014; revised 15 June 2014; accepted 30 June 2014
Abstract
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.
Keywords:Equilibrium Problem, K-Strictly Pseudononspreading Multivalued Mapping, Common Fixed Point
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.
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NOTES
*Corresponding author.