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

Let $K$ be a nonempty subset of a Banach space $X$. A mapping $T\text{\hspace{0.17em}⁣}:K\to K$ is called $\mathfrak{c}\mathfrak{m}$-nonexpansive if for any sequence $(u_i{)}_{i=1}^{\mathrm{\infty }}$ and $y$ in $K$, ${\mathrm{lim\hspace{0.17em}sup}}_{i\to \mathrm{\infty }}{\sup }_{A\subset \{1,\dots ,n\}}\mathrm{\parallel }{\sum }_{k\in A}(T{u}_{i+k}-Ty)\mathrm{\parallel }\le {\mathrm{lim\hspace{0.17em}sup}}_{i\to \mathrm{\infty }}{\sup }_{A\subset \{1,\dots ,n\}}\mathrm{\parallel }{\sum }_{k\in A}(u_{i+k}-y)\mathrm{\parallel }$ 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 $\mathfrak{c}\mathfrak{m}$-nonexpansiveness, ${\mathrm{\ell }}_1$ 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 $\mathfrak{c}\mathfrak{m}$-nonexpansive maps.