Weak compactness and fixed point property for affine bi-Lipschitz maps

Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pe\l czy\'nski's property $(u)$, then either $C$ is weakly compact, contains an $\ell_1$-sequence or a $\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We also introduce a generalized form of property $(u)$, called {it property $(\mathfrak{su})$}, and use it to prove that if $X$ has property $(\mathfrak{su})$ then either $C$ is weakly compact or contains a wide-$(s)$ sequence which is uniformly shift equivalent. In this case, weak compactness in such spaces can also be characterized in terms of the $\mathcal{G}$-FPP for affine uniformly bi-Lipschitz mappings. It is also proved that every Banach space with a spreading basis has property $(\mathfrak{su})$, thus property $(\mathfrak{su})$ is stronger than property $(u)$. These results yield a significant strengthening of an important theorem of Benavides, Jap\'on-Pineda and Prus published in 2004.