Massera's theorem for monotone dynamical systems in three dimensions (Q1614681)

The author studies the dynamical behaviour of increasing mappings \(f: \mathbb{R}^3\to \mathbb{R}^3\), where \(\mathbb{R}^3\) is ordered by a solid cone. It is proved that, if \(f\) is an increasing and orientation preserving homeomorphism, then either \(f\) has a fixed-point or each sequence with \(x_{n+1}= f(x_n)\) tends to infinity. This result is applied to ODEs \(x'= f(t,x)\), where \(f\) is \(T\)-periodic in \(t\) and quasimonotone increasing or decreasing in \(x\) with respect to the coordinate cone.