On generic convergence of successive approximations of mappings with convex and compact point images

TL;DR

This paper investigates the typical convergence behavior of successive approximation methods for set-valued nonexpansive mappings with convex, compact images in Banach spaces, showing that convergence to fixed points is generic.

Contribution

It demonstrates that for most such mappings and points, the iterative sequence of successive approximations converges uniquely to a fixed point, establishing generic convergence results.

Findings

Successive approximations typically converge to fixed points in the studied setting.

Convergence is unique for most mappings and points.

The results hold in separable Banach spaces for nonexpansive set-valued mappings.

Abstract

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.