Let $D$ be a bounded closed convex subset of a Banach space, and let $T : D\times D \to D$ be a continuous mapping which satisfies for all $x, y, z, t \in D$, 
$$\|T(x,y)-T(z,t)\|\leq \max \{\|x-z\|,\|y-t\|\}$$
with strict inequality holding when $\|x-z\|\not=\|y-t\|$. Suppose $T$ condensing in the sense that
$$\gamma(T(U,V)) < \max \{\gamma(U),\gamma(V)\}$$
for subsets $U, V$ of $D$ for which  $\gamma(U\backslash V ) > 0$ (where $\gamma$ denotes the usual Kuratowski set-measure of noncompactness). A projection-iteration method is shown to converge to a solution of $x = T (x, x)$. The significance of this result is that it holds in arbitrary spaces.