On ordinary differential equations with quasimonotone increasing right-hand side

Abstract.

We prove that the initial value problem x ^' (t) = f (t,x (t)), $t \in [0,T]$, x (0) = x ₀, is uniquely solvable in certain ordered Banach spaces if, f is quasimonotone increasing with respect to x, and if f is satisfying a one-sided Lipschitz condition with respect to the order-inducing cone.