Hölder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces

Abstract

Suppose that $S$ is a left amenable semitopological semigroup. We prove that if $\mathcal{S}=\{ 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 $\mathcal{S}$ is a Hölder continuous retract of $C$. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.