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

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. Gornicki, A remark on fixed point theorems for lipschitzian mappings , J. Math. Anal. Appl. 183 (1994), 495–508], [J. Gornicki, The methods of Hilbert spaces and structure of the fixed-point set of lipschitzian mapping , Fixed Point Theory and Applications, Hindawi Publ. Corporation, 2009, Article ID 586487].