### 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.

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.

#### Keywords

Amenable semigroup; uniformly Lipschitzian mapping; Hölder continuous retraction; fixed point

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