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

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 $f:C\to C$thereisasequence${x_n}$in$C$suchthat$x_n-f(x_n)$ converge weakly to 0. We prove in particular that $C$ has this property whenever it contains no sequence equivalent to the standard basis of ${\mathrm{\ell }}_1$. As a byproduct we obtain a characterization of Banach spaces not containing ${\mathrm{\ell }}_1$ in terms of the weak topology.