Flow-invariant sets and differential inequalities in normed spaces†

Abstract

A set M in a Banach space B is said to be flow-invariant with respect to the ordinary differential equation x(t)=f(t,x) (t real, xεB,f(t,x)εB), if for each solution x(i) of this equation x(0) ε M implies x(t) ε M for t >0. In this paper, several theorems on flow-invariance are given. These theorems on differential inequalities in ordered Banach spaces. In particular, they apply to the important case when the interior of the positive cone of the Banach space is empty. Finally it is shown that the basic assumption for the validity of a theorem on differential inequalities, namely the quasimonotonicity property as given by Volkmann [11], is equivalent to the tangent condition of Brezis [3]with respect to the positive cone.