# Around the nonlinear Ryll-Nardzewski theorem

## TL;DR

This paper extends classical fixed point theorems to nonlinear settings involving nonexpansive and distal dynamical systems on convex subsets of dual Banach spaces, with implications for isometries in L-embedded spaces.

## Contribution

It provides a nonlinear extension of the Ryll-Nardzewski theorem and related fixed point results for nonexpansive, distal systems in dual Banach spaces.

## Key findings

- Existence of fixed points in weak* compact convex sets under nonexpansive, distal actions.
- Fixed points form a nonexpansive retract of the set.
- Extension of the Bader-Gelander-Monod theorem to nonlinear contexts.

## Abstract

Suppose that $Q$ is a weak$^{\ast }$ compact convex subset of a dual Banach space with the Radon-Nikod\'{y}m property. We show that if $(S,Q)$ is a nonexpansive and norm-distal dynamical system, then there is a fixed point of $S$ in $Q$ and the set of fixed points is a nonexpansive retract of $Q.$ As a consequence we obtain a nonlinear extension of the Bader-Gelander-Monod theorem concerning isometries in $L$-embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.