${\ell }_1$ spreading models and the FPP for Cesàro mean nonexpansive maps

Abstract

Let $K$ be a nonempty subset of a Banach space $X$. A mapping $T:K\to K$ is called $\mathfrak{cm}$-nonexpansive if for any sequence $(u_i{)}_{i=1}^{\infty }$ and $y$ in $K$, ${\mathrm{limsup}}_{i\to \infty }{\sup }_{A\subset \{1,\dots ,n\}}\Vert {\sum }_{k\in A}(T{u}_{i+k}-Ty)\Vert \le {\mathrm{limsup}}_{i\to \infty }{\sup }_{A\subset \{1,\dots ,n\}}\Vert {\sum }_{k\in A}(u_{i+k}-y)\Vert$ for all $n\in \mathbb{N}$. As a subclass of the class of nonexpansive maps, its FPP is well-established in a wide variety of spaces. The main result of this paper is a fixed point result relating $\mathfrak{cm}$-nonexpansiveness, ${\ell }_1$ spreading models and Schauder bases with not-so-large basis constants. As a consequence, we deduce that Banach spaces with the weak Banach-Saks property have the fixed point property for $\mathfrak{cm}$-nonexpansive maps.

In this new version we have improved the results obtained in order to more easily cover non-affine maps. arXiv admin note: text overlap with arXiv:2302.04323

References in corpus (1)

• On the fixed points of nonexpansive mappings in direct sums of Banach spaces