Remarks on pseudo-contractive mappings

Abstract

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