Fixed points of holomorphic maps in Banach spaces

Abstract

This paper is concerned with the problem of existence of fixed points of continuous maps of the closed unit ball of a complex Banach space into itself which are holomorphic on the open unit ball. We show that if the Banach space is separable and reflexive and F is the map in question that for a.e. 9 in (0,2n the map e1' F has a fixed point. This result does not hold in general; hence, additional conditions are imposed which insure the existence of fixed points in every Banach space. Fixed points of some linear fractional maps are explicitly computed.