Local invariance via comparison functions - Ovidiu Carja; Mihai Necula; Ioan I. Vrabie (Electronic Journal of Differential Equations, 2004)
We consider the ordinary differential equation $u'(t)=f(t,u(t))$, where $f:[a,b]imes Do mathbbR^n$ is a given function, while $D$ is an open subset in $mathbbR^n$. We prove that, if $Ksubset D$ is locally closed and there exists a comparison function $omega:[a,b]imesmathbbR_+o mathbbR$ 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.