On mappings contractive in the sense of Kannan

Abstract:

Let $f:X \to X$ be a continuous compact mapping of a metric space $(X,d)$ into itself with the property that $x,y \in X$ and $x \ne y$ implies $d(f(x),f(y)) < \tfrac {1} {2}[d(x,f(x)) + d(y,f(y))]$. It is shown that under these conditions $f$ has a unique fixed point and, moreover, $f$ is a Banach contraction relative to a suitable remetrization of the space $X$. A similar result concerning condensing mappings is also obtained.