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 S = {T t : t ∈ S} is a uniformly k-Lipschitzian semigroup on a bounded closed and convex subset C of a Hilbert space and k < √ 2, then the set of fixed points of S is a Hölder continuous retract of C. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.2010 Mathematics Subject Classification. 47H10; 47H20; 54C15.