This is an announcement for the paper “Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps” by C. S. Barrosohttps://arxiv.org/find/math/1/au:+Barroso_C/0/1/0/all/0/1, V. Ferreirahttps://arxiv.org/find/math/1/au:+Ferreira_V/0/1/0/all/0/1.
Abstract: In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-(s) subsequence $(x_n)$ which admits an equivalent convex basic sequence. This fact is used to characterize weak-compactness of bounded, closed convex sets in terms of the generic fixed point property ($\mathcal{G}$-FPP) for the class of affine bi-Lipschitz maps. This result generalizes a theorem by Benavides, Jap'on Pineda and Prus previously proved for the class of continuous maps. We also introduce a relaxation of this notion ($\mathcal{WG}$-FPP) and observe that a closed convex bounded subset of a Banach space is weakly compact iff it has the $\matcal{WG}$-FPP for affine $1$-Lipschitz maps. Related results are also proved. For example, a complete convex bounded subset $C$ of a Hlcs $X$ is weakly compact iff it has the $\mathcal{G}$-FPP for the class of affine continuous maps $f: C\rightarrow X$ with weak-approximate fixed point nets.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.05642