Geodesic
Local invariance via comparison functions
Electronic journal of differential equations, Tome 2004 (2004)
We consider the ordinary differential equation

$ \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.
Classification : 34A12, 34A34, 34C05, 34C40, 34C99
Keywords: viable domain, local invariant subset, exterior tangency condition, comparison property, Lipschitz retract 
@article{EJDE_2004__2004__a128,
     author = {Carja,  Ovidiu and Necula,  Mihai and Vrabie,  Ioan I.},
     title = {Local invariance via comparison functions},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1058.34063},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a128/}
}
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Carja,  Ovidiu; Necula,  Mihai; Vrabie,  Ioan I. Local invariance via comparison functions. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a128/