On quasimonotone increasing homeomorphisms

Abstract

Let \( f\in C(\Bbb R^n,\Bbb R^n) \) be quasimonotone increasing such that \( \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) \) for a linear and strictly positive functional \( \Psi \) and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of \( x'=f(x) \).