Structure of the fixed-point set of mappings with lipschitzian iterates

Abstract

We prove, by asymptotic center techniques and some inequalities
in Banach spaces, that if $E$ is $p$-uniformly convex Banach space,
$C$ is a nonempty bounded closed convex
subset of $E$, and $T\colon C\rightarrow C$ has lipschitzian iterates
(with some restrictions), then the set of fixed-points is not only
connected but even a retract of $C$. The results presented in this
paper improve and extend some results in
[J. Górnicki, < i> A remark on fixed point theorems for lipschitzian mappings< /i> , J. Math.
Anal. Appl. < b> 183< /b> (1994), 495–508],
[J. Górnicki, < i> The methods of Hilbert spaces and structure of the fixed-point set of
lipschitzian mapping< /i> , Fixed Point Theory and Applications, Hindawi Publ. Corporation,
2009, Article ID 586487].