On the convergence of the continuous version of the Moudafi's viscosity approximation method

TL;DR

This paper analyzes the long-term behavior of a continuous dynamical system linked to Moudafi's viscosity approximation method, showing its trajectories closely mirror those of the discrete process for fixed point problems.

Contribution

It establishes the asymptotic equivalence between the continuous system's trajectories and the discrete process in fixed point problems.

Findings

Trajectories of the continuous system and discrete process are asymptotically similar.

The continuous dynamical system provides insights into the convergence behavior of the discrete method.

The results extend understanding of viscosity approximation methods for nonexpansive mappings.

Abstract

We study the asymptotic behavior of trajectories of the continuous dynamical system (CDS) associated to the the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi in 2000 [A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl. 241 (2000), 46-55]. We establish that the trajectories $x(t)$ of the system (CDS) and the sequences $(x_{n})$ generated by the the discrete process (DDS) have a very similar asymptotic behaviors.