Comparison of fastness of the convergence among Krasnoselskij, Mann, and Ishikawa iterations in arbitrary real Banach spaces

Gvr Babu,Knvv Vara Prasad

doi:10.1155/fpta/2006/35704

Abstract

Let be an arbitrary real Banach space and a nonempty, closed, convex (not necessarily bounded) subset of . If is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant , then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of .

Highlights

By approximation of fixed points of certain classes of operators which satisfy weak contractive-type conditions that do not guarantee the convergence of Picard iteration [2, Example 2.1, page 76], certain mean value fixed point iterations, namely, Krasnoselskij, Mann, and Ishikawa iteration methods are useful to approximate fixed points
For a certain class of mappings, two or more fixed point iteration procedures can be used to approximate their fixed points, it is of theoretical and practical importance to compare the rate of convergence of these iterations, and to find out, if possible, which one of them converges faster
Suppose that E is a real Banach space with dual E∗, we denote by J, the normalized duality map from E to 2E∗ defined by

Summary

Introduction

By approximation of fixed points of certain classes of operators which satisfy weak contractive-type conditions that do not guarantee the convergence of Picard iteration [2, Example 2.1, page 76], certain mean value fixed point iterations, namely, Krasnoselskij, Mann, and Ishikawa iteration methods are useful to approximate fixed points. For a certain class of mappings, two or more fixed point iteration procedures can be used to approximate their fixed points, it is of theoretical and practical importance to compare the rate of convergence of these iterations, and to find out, if possible, which one of them converges faster. Recent works in this direction are [1, 4, 5]. Chidume and Osilike [7] approximated fixed points of Lipschitzian strongly pseudocontractive maps in Banach spaces, using both Mann and Ishikawa iterations.

Preliminaries and known results

Results on the comparison of fastness of the convergence

Numerical examples