General Viscosity Implicit Midpoint Rule For Nonexpansive Mapping

TL;DR

This paper introduces a new viscosity implicit midpoint iterative scheme for nonexpansive mappings in Hilbert spaces, proving strong convergence and demonstrating applications to various mathematical problems.

Contribution

It proposes a novel iterative method with convergence guarantees for nonexpansive mappings, extending existing approaches in the literature.

Findings

Proves strong convergence of the iterative scheme.

Shows the scheme's applicability to variational inequalities.

Provides solutions to integral and evolution equations.

Abstract

In this work, we suggest a general viscosity implicit midpoint rule for nonexpansive mapping in the framework of Hilbert space. Further, under the certain conditions imposed on the sequence of parameters, strong convergence theorem is proved by the sequence generated by the proposed iterative scheme, which, in addition, is the unique solution of the variational inequality problem. Furthermore, we provide some applications to variational inequalities, Fredholm integral equations, and nonlinear evolution equations. The results presented in this work may be treated as an improvement, extension and refinement of some corresponding ones in the literature.