On the approximate fixed point property in abstract spaces

Abstract

Let $X$ be a Hausdorff topological vector space, $X^{\ast }$ its topological dual and $Z$ a subset of $X^{\ast }$. In this paper, we establish some results concerning the $\sigma (X,Z)$-approximate fixed point property for bounded, closed convex subsets $C$ of $X$. Three major situations are studied. First when $Z$ is separable in the strong topology. Second when $X$ is a metrizable locally convex space and $Z=X^{\ast }$, and third when $X$ is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet-Urysohn property for certain sets with regarding the $\sigma (X,Z)$-topology. The support tools include the Brouwer's fixed point theorem and an analogous version of the classical Rosenthal's ${\ell }_1$-theorem for ${\ell }_1$-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.

16 pages; the paper was slightly revised, some more explanations were added

References in corpus (1)

• Spaces not containing ${\ell }_1$ have weak aproximate fixed point property

Cited by in corpus (2)

• Optimal approximate fixed point results in locally convex spaces
• A note on Banach spaces $E$ admitting a continuous map from $C_p(X)$ onto $E_w$