A topological characterization of Banach contractions

Solomon Leader

Vol. 69 (1977), No. 2, 461–466

DOI: 10.2140/pjm.1977.69.461

Abstract

A continuous operator T on a metric space (X,ρ) is a Banach contraction with fixed point p under some metric σ topologically equivalent to ρ if, and only if, every orbit Tⁿx converges to p and the convergence is uniform on some neighborhood of p. For σ to be bounded we demand that the convergence be uniform on X. The latter condition with T uniformly continuous characterizes the case for σ bounded and uniformly equivalent to ρ.

Mathematical Subject Classification 2000

Primary: 54E40

Secondary: 54H25

Milestones

Received: 3 December 1974

Published: 1 April 1977