Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces

Shenghua Wang

doi:10.1186/1687-1812-2013-314

Abstract

Recently, Iemoto and Takahashi considered a weak convergence iterative scheme for a nonspreading mapping and a nonexpansive mapping in Hilbert spaces. In this paper, we suggest two hybrid iterative algorithms by modifying Iemoto and Takahashi’s iterative scheme for a countable family of nonspreading mappings and a nonexpansive mapping in Hilbert spaces.MSC:47H05, 47H09.

Highlights

Introduction and preliminaries LetH be a Hilbert space and C be a nonempty closed convex subset of H
Where T is a nonexpansive mapping of C into itself and {αn} is a sequence in (, )
The following is an important result on a family of mappings {Tn}∞ n= satisfying the AKTT-condition

Summary

Recall that T is said to be nonexpansive if

For approximating the fixed point of a nonexpansive mapping in a Hilbert space, Mann [ ] in introduced the famous iterative scheme as follows:. Theorem IT ([ ]) Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. ) by a hybrid iterative scheme and obtain the strong convergence theorems for a family of nonspreading mappings and a nonexpansive mapping in a Hilbert space. The following is an important result on a family of mappings {Tn}∞ n= satisfying the AKTT-condition. ([ ]) Let K be a nonempty and closed subset of a Banach space E, and let {Tn}∞ n= be a family of mappings of K into itself which satisfies the AKTT-condition. Let {Tn} be a family of nonspreading mappings of C into itself, and assume that limn→∞ Tnx exists for each x ∈ C.

Tn y

Cj for all n