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 $T_t:t\in S$ 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 Hölder continuous retract of C. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.

7 pages, to appear, Topological Methods in Nonlinear Analysis. This is the final version, incorporating the referee's suggestions