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

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.

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