On the approximate fixed point property in abstract spaces

Let $X$ be a Hausdorff topological vector space, $X^*$ its topological dual and $Z$ a subset of $X^*$. 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 $Z=X^*$, 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 Fr\'echet-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.