### On the structure of fixed point sets of asymptotically regular mappings in Hilbert spaces

DOI: http://dx.doi.org/10.12775/TMNA.2009.049

#### Abstract

The purpose of this paper is to prove the following theorem:

Let $H$ be a Hilbert space, let $C$ be a nonempty bounded closed

convex subset of $H$ and let

$T\colon C\rightarrow C$ be an asymptotically regular mapping. If

$$

\liminf_{n\rightarrow \infty} \|T^n\|< \sqrt{2},

$$

then

$Fix T=\{x\in C:Tx=x\}$ is a retract of $C$.

Let $H$ be a Hilbert space, let $C$ be a nonempty bounded closed

convex subset of $H$ and let

$T\colon C\rightarrow C$ be an asymptotically regular mapping. If

$$

\liminf_{n\rightarrow \infty} \|T^n\|< \sqrt{2},

$$

then

$Fix T=\{x\in C:Tx=x\}$ is a retract of $C$.

#### Keywords

Asymptotically regular mapping; retraction; asymptotic center; fixed point; Hilbert space

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