Remarks on pseudo-contractive mappings

Let $X$ be a Banach space, $D \subset X$. A mapping $U:D \to X$ is said to be pseudo-contractive if for all $u,v \in D$ and all $r > 0$, $||u - v|| \leqq ||(1 + r)(u - v) - r(U(u) - U(v))||$. This concept is due to F. E. Browder, who showed that $U:X \to X$ is pseudo-contractive if and only if $I - U$ is accretive. In this paper it is shown that if $X$ is a uniformly convex Banach, $B$ a closed ball in $X$, and $U$ a Lipschitzian pseudo-contractive mapping of $B$ into $X$ which maps the boundary of $B$ into $B$, then $U$ has a fixed point in $B$. This result is closely related to a recent theorem of Browder.