Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps

C. S. Barroso; V. Ferreira

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arXiv · 1610.05642

Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps

C. S. Barroso·V. Ferreira

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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 $\mathcal{WG}$-$FPP$ 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 $f\colon C\to X$ with weak-approximate fixed point nets.

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BibTeX RIS

C. S. Barroso, V. Ferreira. 2016-10-18. Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps. https://arxiv.org/abs/1610.05642

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