Retraction methods and fixed point free maps with null minimal displacements on unit balls

Abstract

In this paper we consider the class of Lipschitz maps on the unit ball $B_X$ of a Banach space $X$, and the question we deal with is whether for any $\hat{I}»>1$ there exists a $\hat{I}»$-Lipschitz fixed-point free mapping $T:B_X\to B_X$ with $\mathrm{d}(T,B_X)=0$. We also consider its Hölder version. New related results are obtained. We show that if $X$ has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in \cite{Bar}. In the general case, using a recent approach of R. Medina \cite{M} concerning Hölder retractions of $(r_n)$-flat closed convex sets, we show that for any decreasing null sequence $(r_n)\subset \mathbb{R}$ and $\hat{I}\pm \in (0,1)$, there exists a fixed-point free mapping $T$ on $B_X$ so that $\Vert T^{n}x-T^{n}y\Vert \le r_n(\Vert x-y{\Vert }^{\hat{I}}\pm +1)$ for all $x,y\in B_X$ and $n\in \mathbb{N}$.

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