H\"{o}lder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces

TL;DR

This paper proves that under certain conditions, the fixed points of a uniformly Lipschitzian semigroup in a Hilbert space form a Hölder continuous retract of the set, extending fixed point existence results.

Contribution

It establishes Hölder continuous retractions for fixed points of Lipschitzian semigroups in Hilbert spaces under new bounds on the Lipschitz constant.

Findings

Fixed points form a Hölder continuous retract of the convex set.

The result applies to semigroups with Lipschitz constant less than √2.

Provides a qualitative complement to existing fixed point theorems.

Abstract

Suppose that S is a left amenable semitopological semigroup. We prove that if ${T_{t}: t \in S}$ is a uniformly k-Lipschitzian semigroup on a bounded closed and convex subset C of a Hilbert space and $k<\sqrt{2}$, then the set of fixed points of this semigroup is a H\"{o}lder continuous retract of C. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.