On Successive Approximations for Compact-Valued Nonexpansive Mappings

TL;DR

This paper demonstrates that for most compact-valued nonexpansive mappings in Banach spaces, there exists a unique converging sequence of successive approximations leading to a fixed point, with most initial points having unique trajectories.

Contribution

It establishes the typical behavior of nonexpansive compact-valued mappings regarding the uniqueness of approximation sequences and trajectories in Banach spaces.

Findings

Unique sequence of successive approximations converges to a fixed point for typical mappings.

Most initial points in separable Banach spaces have a unique trajectory.

The property holds in the sense of Baire category, indicating generic behavior.

Abstract

We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping $F$ has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of $F$. In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have unique trajectory.