Geodesic
Fixed points of periodic and firmly lipschitzian mappings in Banach spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 4, pp. 573-579.

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W.A. Kirk in 1971 showed that if $T\colon C\to C$, where $C$ is a closed and convex subset of a Banach space, is $n$-periodic and uniformly $k$-lipschitzian mapping with $k$, then $T$ has a fixed point. This result implies estimates of $k_0(n)$ for natural $n\geq 2$ for the general class of $k$-lipschitzian mappings. In these cases, $k_0(n)$ are less than or equal to 2. Using very simple method we extend this and later results for a certain subclass of the family of $k$-lipschitzian mappings. In the paper we show that $k_0(3)>2$ in any Banach space. We also show that $\operatorname{Fix}(T)$ is a Hölder continuous retract of $C$.
Classification : 47H09, 47H10
Keywords: lipschitzian mapping; firmly lipschitzian mapping; $n$-periodic mapping; fixed point; retractions 
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Pupka, Krzysztof. Fixed points of periodic and firmly lipschitzian mappings in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 4, pp. 573-579. http://geodesic.mathdoc.fr/item/CMUC_2012__53_4_a5/