Modified Krasnoselskii–Mann iterative algorithm for nonexpansive mappings in Banach spaces

Yekini Shehu

doi:10.1007/s40065-013-0066-1

Abstract

In this paper, we prove that the sequence {xn} generated by modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al. [J Appl Math Comput 29:383–389, 2009] converges strongly to a fixed point of a nonexpansive mapping T in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Furthermore, we present an example that illustrates our result in the setting of a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. The results of this paper extend and improve several results presented in the literature in the recent past. Open image in new window

Highlights

Let E be a real Banach space and C a nonempty, closed and convex subset of E
Though simple in form, the Krasnoselskii–Mann iteration is remarkably useful for finding fixed points of a nonexpansive mapping and provides a unified framework for many algorithms from various different fields
The purpose of this paper is to prove a strong convergence theorem for approximation of fixed point of a nonexpansive mapping using a modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al [37] in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm

Summary

Introduction

Let E be a real Banach space and C a nonempty, closed and convex subset of E. Though simple in form, the Krasnoselskii–Mann iteration is remarkably useful for finding fixed points of a nonexpansive mapping and provides a unified framework for many algorithms from various different fields In this respect, the following result is basic and important. Yao et al [37] introduced a modified Krasnoselskii–Mann iterative algorithm for nonexpansive mappings in the framework of a real Hilbert space and proved the following theorem. The purpose of this paper is to prove a strong convergence theorem for approximation of fixed point of a nonexpansive mapping using a modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al [37] in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Our theorem extends Theorem 1.4 from Hilbert spaces to a more general uniformly convex Banach space with uniformly Gâteaux differentiable norm

Preliminaries

Main results

Application