Smallest and greatest fixed points of quasimonotone increasing mappings

Abstract

In a Banach space E (pre)ordered by a cone we consider a mapping f : [v,w] → E (v,w ∈ E, v ≤ w) which satisfies v ≤ f(v) and f(w) ≤ w. We show that f has a smallest and a greatest fixed point, if it is continuous, quasimonotone increasing and condensing, in essence. Finally we admit discontinuities with upward jumps.