Fixed point results in ordered normed spaces with applications to abstract and differential equations

Abstract

In [S] fixed point results are derived for an increasing operator A: [u, U] -+ [u, 01, where [u, u] is a conical segment in an ordered Banach space with normal order cone P. As the main results, an existence a fixed point of A is proved by assuming that A( [u, u]) is weakly relatively compact and separable. If P is also minihedral, then A is shown to have least and greatest fixed points. No continuity or linearity hypotheses are imposed on A. In this paper we shall show, for instance, that if [u, u] is a conical segment in an ordered normed space E, and if A: [u, u] + [u, u] is increasing and weakly relatively order compact, then A has the least and the greatest fixed point, provided that the order cone of E is normal or A( [u, u]) is separable. Moreover, these fixed points are constructed by countable iteration processes. The so obtained fixed point results are then applied to derive results on existence of extremal or unique solutions of the operator equation