A Generalization of Halpern Iteration. Preliminary Results

TL;DR

This paper introduces a generalized iteration scheme combining viscosity, nonexpansive mappings, and external forces, proving strong convergence to a unique fixed point under broad conditions.

Contribution

It extends the Halpern iteration framework to a more general setting with new convergence results for complex iterative schemes.

Findings

Proves asymptotic strong convergence to a fixed point.

Establishes convergence under asymptotically nonexpansive mappings.

Incorporates external forcing terms into the iteration analysis.

Abstract

A generalization of a viscosity generalized Halpern iteration scheme is analyzed. It is proven that the solution converges asymptotically strongly to a unique fixed point of an asymptotically nonexpansive mapping which drives the iteration together with a contractive self-mapping, a viscosity term and two driving external forcing terms.