Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators

Hudson Akewe,Godwin Amechi Okeke

doi:10.1186/s13663-015-0315-4

Abstract

AbstractIn this paper we use the general class of contractive-like operators introduced by Bosede and Rhoades (J. Adv. Math. Stud. 3(2):1-3, 2010) to prove strong convergence and stability results for Picard-Mann hybrid iterative schemes considered in a real normed linear space. We establish the strong convergence and stability of the Picard iterative scheme as a corollary. Our results generalize and improve a multitude of results in the literature, including the recent results of Chidume (Fixed Point Theory Appl. 2014:233, 2014).

Highlights

1 Introduction and preliminary definitions Fixed point iterative schemes are designed to be applied in solving equations arising in physical formulation but there is no systematic study of the numerical aspects of these iterative schemes
Akewe et al [ ] introduced the Kirk-multistep type iterative schemes and proved strong convergence and stability results, with some numerical examples to back up their work
Motivated by the work of Khan [ ], we prove the strong convergence of the Picard-Mann iterative scheme for a general class of operators in a real normed space

Summary

Introduction

Introduction and preliminary definitionsFixed point iterative schemes are designed to be applied in solving equations arising in physical formulation but there is no systematic study of the numerical aspects of these iterative schemes. Akewe et al [ ] introduced the Kirk-multistep type iterative schemes and proved strong convergence and stability results, with some numerical examples to back up their work. He introduced the following Picard-Mann hybrid iterative scheme for a single nonexpansive mapping T.

Results

Conclusion