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

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