Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 50, pp. 1-14.
Title: Local invariance via comparison functions
Authors: Ovidiu Carja (Al. I. Cuza Univ., Romania)
Mihai Necula (Al. I. Cuza Univ., Romania)
Ioan I. Vrabie (Al. I. Cuza Univ., Romania)
Abstract:
We consider the ordinary differential equation $u'(t)=f(t,u(t))$,
where $f:[a,b]\times D\to \mathbb{R}^n$ is a given function,
while $D$ is an open subset in $\mathbb{R}^n$. We prove that, if
$K\subset D$ is locally closed and there exists a comparison
function $\omega:[a,b]\times\mathbb{R}_+\to \mathbb{R}$ such that
$$
\liminf_{h\downarrow 0}\frac{1}{h}\big[d(\xi+hf(t,\xi);K)-d(\xi;K)\big]
\leq\omega(t,d(\xi;K))
$$
for each $(t,\xi)\in [a,b]\times D$, then $K$ is locally invariant with
respect to $f$. We show further that, under some natural extra condition,
the converse statement is also true.
Submitted June 18, 2003. Published April 6, 2004.
Math Subject Classifications: 34A12, 34A34, 34C05, 34C40, 34C99.
Key Words: Viable domain; local invariant subset; exterior tangency
condition; comparison property; Lipschitz retract.