ON PSEUDO-CONTRACTIVE MAPPINGS

Let X be a Banach space, D QX. A mapping U:D-^>X is said to be pseudo-contractive if for all u,v(ED and all r>0, (I** — »H = ll(.i+r)(tt—v)— r(U(u) — U(v))\\. This concept is due to F. E. Browder, who showed that U:X-+X is pseudo-contractive if and only if I — Uis 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. Let X be a Banach space and DEX. A mapping U'.D—^X is said to be pseudo-contractive (Browder [4]) if for all u, vED and all r>0, \\u v\\ g ||(1 + r)(u v) r(U(u) U(v))\\. This class of mappings is easily seen to be more general than the class of nonexpansive mappings; that is, mappings Ofor which \\Uix) U(y)\\ ^||x-y||, x, y £ D. However, the main interest in pseudo-contractive mappings stems from the firm connection which exists between these mappings and the important class of accretive mappings; namely, U is pseudocontractive if and only if 7— U is accretive [4, Proposition l]. Thus the mapping theory for accretive mappings is closely related to fixedpoint theory of pseudo-contractive mappings. Using highly analytic techniques, and relying on this connection, Browder has proved the following theorem. Theorem 1 [4]. Let X be a uniformly convex Banach space, B a closed ball in X, G an open set containing B. Let U be a pseudo-contractive mapping of G into X such that U maps the boundary of B into B. Suppose also that U is demicontinuous and that either (a) U is uniformly continuous in the strong topology on bounded subsets of X, or (b) X* is uniformly convex. Then U has a fixed point in B. The object of this note is to give an elementary geometric proof of a theorem which is a slight variation of the "(a) version" of the above. Received by the editors August 16, 1968. A MS Subject Classifications. Primary 4785.