  A nonempty closed convex bounded subset $C$ of a Banach space is said to have
the weak approximate fixed point property if for every continuous map $f:C\to
C$ there is a sequence $\{x_n\}$ in $C$ such that $x_n-f(x_n)$ converge weakly
to 0. We prove in particular that $C$ has this property whenever it contains no
sequence equivalent to the standard basis of $\ell_1$. As a byproduct we obtain
a characterization of Banach spaces not containing $\ell_1$ in terms of the
weak topology.
