Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods

TL;DR

This paper introduces a viscosity iterative method for solving nonlinear operator equations, proving its convergence and demonstrating its effectiveness for various classes of mappings and integral equations.

Contribution

It presents a new viscosity implicit iterative algorithm with proven convergence for fixed points of nonexpansive mappings in Banach spaces, expanding solution techniques for nonlinear operator equations.

Findings

Convergence of the iterative scheme is established.

Effective for solving equations involving generalized contractions.

Applicable to fixed points of pseudocontractive and monotone mappings.

Abstract

Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of {\lambda}-strictly pseudocontractive mappings, solution of {\alpha}-inverse-strongly monotone mappings, and solution of integral equations of Fredholm type