H\"{o}lder 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 ${T_{t}: t \in S}$ 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.