Convergence Theorems for k-Strictly Pseudononspreading Multivalued in Hilbert Spaces

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.


Introduction
Throughout this paper, we denote by  and  the sets of positive integers and real numbers, respectively.Let C be a nonempty closed subset of a real Hilbert space H . Let , : max sup , , sup , , A A CB C ∈ , where ( ) { } 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 C be a subset of Hilbert space H , they called a mapping : Now, inspired by [6] and [7], we propose a definition as follows.Definition 1.1 The multivalued mapping ( ) for , , , .
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 R denote the reals with the usual norm.Let : 2 , 0, The equilibrium problem for : ∈ is a solution of the variational inequality , 0 Tx y ≥ for all y E ∈ .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.

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 H be a real Hilbert space, for all , x y H ∈ and [ ] 0,1 t ∈ , then the following well known results hold:  In addition, ( )  ( ) , ) ( ) Observe also that for each ( ) the family of nonempty subsets and nonempty closed bounded subsets of E , respectively.The Hausdorff metric on

Theorem 3 . 1
Let C be a nonempty closed convex subset of a real Hilbert space H , and let

x
from n = 1 to n, and dividing by n we obtain is bounded,then { } n z is also bounded.Thus there exists a subsequence { }

F
T is closed and convex by Lemma 2.3, thus we can define the projection ( ) a sequence in H which converges weakly to z H [5] C be a nonempty closed convex subset of a real Hilbert space H .The nearest point projection : Lemma 2.2 (see[5]) Let C be a nonempty closed convex subset of a real Hilbert space H .Let:F T ≠ ∅ , and Liu, Y. Li Let C be a nonempty closed convex subset of a real Hilbert space H , and let ,