The Implicit Midpoint Procedures for Asymptotically Nonexpansive Mappings

TL;DR

This paper analyzes the convergence of implicit midpoint iterative procedures for asymptotically nonexpansive mappings, providing new conditions for convergence and illustrating results with numerical examples.

Contribution

It establishes necessary convergence conditions for implicit midpoint methods applied to asymptotically nonexpansive mappings, extending existing results under relaxed parameters.

Findings

Convergence conditions for implicit midpoint procedures are established.

Numerical example confirms theoretical convergence results.

New algorithms with strong convergence are derived under relaxed conditions.

Abstract

The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.