The topological nature of Krasnoselskii’s cone fixed point theorem

In recent years, the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. In this article we discuss the topological nature of the Krasnoselskii theorem and show that it can be restated in a more general form without reference to a cone structure or the norm of the underlying Banach space. This new perspective brings out a closer relation between the Krasnoselskii Theorem and the classical Brouwer fixed point theorem. It also points to some obvious extensions of the cone theorem.