Electron. J. Diff. Eqns., Vol. 2004(2004), No. 50, pp. 1-14.

Local invariance via comparison functions

Ovidiu Carja, Mihai Necula, & Ioan I. Vrabie

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.

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Ovidiu Carja
Faculty of Mathematics, ``Al. I. Cuza" University
700506 Iasi, Romania
email: ocarja@uaic.ro
Mihai Necula
Faculty of Mathematics, ``Al. I. Cuza" University
700506 Iasi, Romania
email: necula@uaic.ro
Ioan I. Vrabie
Faculty of Mathematics, ``Al. I. Cuza" University
700506 Iasi, Romania
email: ivrabie@uaic.ro

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