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.