Remarks on pseudo-contractive mappings

Let X be a Banach space, D CX. A mapping U:D-*X is said to be pseudo-contractive if for all u,vED and all r>O, |Iu-vII<?1(l+r)(u-v)-r(U(u)-U(v))jj. This concept is due to F. E. Browder, who showed that U:X--*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 Lip- schitzian 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.