Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces

Suthep Suantai,Prasit Cholamjiak

doi:10.1155/2010/632137

Abstract

We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Frechet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.

Highlights

Let E and E∗ be a real Banach space and the dual space of E, respectively
Motivated and inspired by Marino-Xu 7, Osilike-Udomene 11, Zhou 8, ZhangGuo 13, and Chidume-Shahzad 17, we consider the following Mann-type iteration: x1 ∈ K and xn 1 1 − αn xn αnTnxn, n ≥ 1, 1.16 where αn is a real sequence in 0, 1 and {Tn}∞n 1 is a countable family of strict pseudocontractions on a closed and convex subset K of a real Banach space E
We prove the weak convergence of a Mann-type iteration process 1.16 in a uniformly convex Banach space which has the Frechet differentiable norm for a countable family of strict pseudocontractions under some appropriate conditions

Summary

Introduction

Let E and E∗ be a real Banach space and the dual space of E, respectively. Let K be a nonempty subset of E. Our motivation in this paper is the following: 1 to modify the normal Mann iteration process for finding common fixed points of an infinitely countable family of strict pseudocontractions, 2 to improve and extend the results of Chidume-Shahzad 17 from a real uniformly smooth and uniformly convex Banach space to a real uniformly convex Banach space which has the Frechet differentiable norm.

Results

Conclusion