A New Modified Fixed-Point Iteration Process

Chanchal Garodia,Izhar Uddin,Afrah A N Abdou

doi:10.3390/math9233109

Chanchal Garodia, Izhar Uddin + Show 1 more

Open Access

https://doi.org/10.3390/math9233109

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Abstract

In this paper, we present a new modified iteration process in the setting of uniformly convex Banach space. The newly obtained iteration process can be used to approximate a common fixed point of three nonexpansive mappings. We have obtained strong and weak convergence results for three nonexpansive mappings. Additionally, we have provided an example to support the theoretical proof. In the process, several relevant results are improved and generalized.

Highlights

Nonlinear analysis is a natural mixture of Topology, Analysis and Linear Algebra.Fixed-point theory is a very challenging and rapidly growing area of nonlinear functional analysis
The early findings in fixed-point theory revolve around generalization of Banach Contraction Principle
It is well known Banach Contraction Principle does not hold good for nonexpansive mappings i.e. nonexpansive mapping need not admit a fixed point on complete metric space

Summary

Introduction

Nonlinear analysis is a natural mixture of Topology, Analysis and Linear Algebra. Fixed-point theory is a very challenging and rapidly growing area of nonlinear functional analysis. Picard iteration need not be convergent for a nonexpansive map in a complete metric space. This led to the beginning of a new era of fixed-point theory for. Approximation of fixed points in different domains for nonlinear mappings using the different iterative processes is the thrust of fixed-point theory. Owing to its importance, fixed-point theory is attracting young researchers across the world and in the last few years many iterative processes have been obtained in different domains. Motivated and inspired by the research going on in this direction, we introduce a new iteration process for approximating common fixed point of three nonexpansive mappings. The aim of this paper is to prove some weak and strong convergence results involving the iteration process (2) for three nonexpansive mappings. Our results extend the corresponding results of [17]

Preliminaries

Convergence Results

Numerical Example

Conclusions