Remarks on Pseudo-Contractive Mappings

W. A. Kirk
1970 Proceedings of the American Mathematical Society  
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
more » ... is closely related to a recent theorem of Browder. Received by the editors August 16, 1968. A MS Subject Classifications. Primary 4785.
doi:10.2307/2036758 fatcat:eocz5nugevb7ph7d356e7w26h4