A note on contractive mappings

Abstract

then there exists a unique fixed point of A, i.e. a point x such that Ax=x. The problem of defining a family of functions F = { a(x, y) } satisfying 0 <a(x, y) <1, sup a(x, y) = 1 and such that Banach's theorem holds when the constant a is replaced with any a(x, y) CF, was suggested to the author by Professor H. Hanani, and two theorems to that effect are proved in the present paper. In the first, additional conditions requiring compactness of the mapping are imposed; the second holds for any complete metric space, irrespective of the compactness of the mapping. DEFINITION 1. A mapping A of a metric space X into itself is said to be contractive if for every two distinct points x, y GX