Fixed Point Theorems for Expanding Maps

Abstract

**Please note that the full text is embargoed** ABSTRACT: Suppose (X,d) is a metric space, h > 0 and T: X -> X. We shall use the notation T E E(h) to mean [see pdf for notation] for each x,y E X. If h > 1, then T will be called an expanding map. Clearly T E E(h) implies T is a 1-1 function and [see pdf for notation] for each x,y E T(X). In this paper some conditions are found to insure that an expanding map will have a fixed point. It is shown that each finite dimensional Banach space X has the following property: each continuous and expanding map from X into X has a fixed point. It is also shown that not all infinite dimensional Banach spaces have the above property.