Carja, OvidiuNecula, MihaiVrabie, Ioan I.2021-04-192021-04-192004-04-06Cârjă, O., Necula, M., & Vrabie, I. I. (2004). Local invariance via comparison functions. <i>Electronic Journal of Differential Equations, 2004</i>(50), pp. 1-14.1072-6691https://hdl.handle.net/10877/13387We 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.Text14 pages1 file (.pdf)enAttribution 4.0 InternationalViable domainLocal invariant subsetExterior tangency conditionComparison propertyLipschitz retractArticle