Approximative Method of Fixed Point for Φ-Pseudocontractive Operator and an Application to Equation with Accretive Operator

Yixin Wen,Yuguang Xu,Aifang Feng

doi:10.4236/jamp.2014.21004

Yixin Wen, Yuguang Xu + Show 1 more

Open Access

https://doi.org/10.4236/jamp.2014.21004

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Abstract

In this paper, Φ-pseudo-contractive operators and Φ-accretive operators, more general than the strongly pseudo-contractive operators and strongly accretive operators, are introduced. By setting up a new inequality, authors proved that if is a uniformly continuous Φ-pseudo-contractive operator then T has unique fixed point q and the Mann iterative sequence with random errors approximates to q. As an application, the iterative solution of nonlinear equation with Φ-accretive operator is obtained. The results presented in this paper improve and generalize some corresponding results in recent literature.

Highlights

Introduction and PreliminariesIn 1994, Chidume [1] solved a problem dealt with the fixed point for the class of Lipschitz strictly pseudo-contractive operators in uniformly smooth Banach space X
He proved that the Ishikawa iterative sequence converges strongly to the unique fixed point of T in K where K ⊂ X and T : K → K is Lipschitz strictly pseudo-contractive
The objective of this paper is to introduce Φ-pseudo-contractive operators—a class of operators which are more general than the φ -strongly pseudo-contractive operators and to study the problems of existence, uniqueness and the iterative approximate method of fixed point by setting up a new inequality in arbitrary Banach space

Summary

Introduction

Introduction and PreliminariesIn 1994, Chidume [1] solved a problem dealt with the fixed point for the class of Lipschitz strictly (strongly) pseudo-contractive operators in uniformly smooth Banach space X. In 1994, Chidume [1] solved a problem dealt with the fixed point for the class of Lipschitz strictly (strongly) pseudo-contractive operators in uniformly smooth Banach space X . He proved that the Ishikawa iterative sequence converges strongly to the unique fixed point of T in K where K ⊂ X and T : K → K is Lipschitz strictly (strongly) pseudo-contractive. The iterative solution of nonlinear equation with Φ-accretive operator is obtained.

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