  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.
