Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces

Murtala Haruna Harbau,Godwin Chidi Ugwunnadi,Lateef Olakunle Jolaoso,Ahmad Abdulwahab

doi:10.3390/axioms10030147

Murtala Haruna Harbau, Godwin Chidi Ugwunnadi + Show 2 more

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https://doi.org/10.3390/axioms10030147

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Abstract

In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature.

Highlights

Academic Editor: Hari Mohan SrivastavaReceived: 15 February 2021Accepted: 12 April 2021Published: 3 July 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil-Let X be a real Banach space and C a nonempty closed and convex subset of X
In 1978, Bose [12] started the study of iterative methods for approximating fixed points of asymptotically nonexpansive mapping in a bounded closed convex nonempty subset C of a uniformly convex Banach space which satisfies Opial’s condition
We oberve from Definition 1 that every Hilbert space is a uniformly convex Banach space

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil-. The class of asymptotically nonexpansive mappings was first introduced and studied by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings They proved that if C is a nonempty closed convex and bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on C, T has a fixed point. In 1978, Bose [12] started the study of iterative methods for approximating fixed points of asymptotically nonexpansive mapping in a bounded closed convex nonempty subset C of a uniformly convex Banach space which satisfies Opial’s condition. The first to study the following modified Mann iteration process for approximating the fixed point of an asymptotically nonexpansive mapping T on nonempty closed convex and bounded subsets C of both Hilbert space and (resp.) uniformly convex Banach space with. We give some numerical examples to validate the convergence of our algorithm

Preliminaries

Main Results

Numerical Examples

Conclusions