On extremal fixed points in Schauder's theorem with applications to differential equations (Q1766236)

The author derives necessary and sufficient conditions for the existence of a least and a greatest fixed point of an operator which satisfies the hypothesis of Schauder's fixed point theorem. The main result is the following Theorem. Let \(N\) be an ordered normed space, \(D\subset N\) a non-empty, bounded, closed and convex subset and \(T\colon D\rightarrow D\) a completely continuous operator. Then \(T\) has a greatest (least) fixed point if and only if the set of fixed points of \(T\) is upward (downward) directed. Two applications of the obtained result to the existence of extremal solutions of initial and boundary value problems are presented.