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 $x,y,z,t\in D$, 
$$\Vert T(x,y)-T(z,t)\Vert \le max\{\Vert x-z\Vert ,\Vert y-t\Vert \}$$
with strict inequality holding when $\Vert x-z\Vert \ne \Vert y-t\Vert$. Suppose $T$ condensing in the sense that
$$\gamma (T(U,V))<max\{\gamma (U),\gamma (V)\}$$
for subsets $U,V$ of $D$ for which  $\gamma (U\mathrm{\setminus }V)>0$ (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.