# Generic Convergence of Sequences of Successive Approximations in Banach Spaces

## TL;DR

This paper investigates the typical convergence behavior of successive approximation methods for set-valued mappings in Banach spaces, especially those formed by pairs of nonexpansive mappings, providing new insights into their generic properties.

## Contribution

It establishes the generic convergence of successive approximations for certain set-valued mappings in Banach spaces, answering a previously open question.

## Key findings

- Successive approximation methods generally converge for set-valued mappings in Banach spaces.
- The paper confirms convergence for mappings defined by pairs of nonexpansive mappings.
- It resolves an open question posed by Francesco S. de Blasi.

## Abstract

We study the generic behavior of the method of successive approximations for set-valued mappings in Banach spaces. We consider, in particular, the case of those set-valued mappings which are defined by pairs of nonexpansive mappings and give a positive answer to a question raised by Francesco S. de Blasi.