Local invariance via comparison functions

Abstract

We consider the ordinary differential equation $u'(t)=f(t,u(t))$, where $f:[a,b]imes Do mathbb{R}^n$ is a given function, while $D$ is an open subset in $mathbb{R}^n$. We prove that, if $Ksubset D$ is locally closed and there exists a comparison function $omega:[a,b]imesmathbb{R}_+o mathbb{R}$ such that $$ liminf_{hdownarrow 0}frac{1}{h}ig[d(xi+hf(t,xi);K)-d(xi;K)ig] leqomega(t,d(xi;K)) $$ for each $(t,xi)in [a,b]imes D$, then $K$ is locally invariant with respect to $f$. We show further that, under some natural extra condition, the converse statement is also true.

Keywords

• Viable domain
• local invariant subset
• exterior tangency condition
• comparison property
• Lipschitz retract.