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

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).

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.

References

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NOTES

*Corresponding author.