Remarks on Pseudo-Contractive Mappings

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