Local invariance via comparison functions
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Date
2004-04-06
Authors
Carja, Ovidiu
Necula, Mihai
Vrabie, Ioan I.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We consider the ordinary differential equation u'(t) = ƒ(t, u(t)), where ƒ : [a, b] x D → ℝn is a given function, while D is an open subset in ℝn. We prove that, if K ⊂ D is locally closed and there exists a comparison function ω : [a, b] x ℝ+ → ℝ such that
limh↓0 inf 1/ h [d(ξ + hƒ(t, ξ); K) - d(ξ; K)] ≤ ω(t, d(ξ; K))
for each (t, ξ) ∈ [a, b] x D, then K is locally invariant with respect to ƒ. We show further that, under some natural extra condition, the converse statement is also true.
Description
Keywords
Viable domain, Local invariant subset, Exterior tangency condition, Comparison property, Lipschitz retract
Citation
Cârjă, O., Necula, M., & Vrabie, I. I. (2004). Local invariance via comparison functions. <i>Electronic Journal of Differential Equations, 2004</i>(50), pp. 1-14.