Local invariance via comparison functions
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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.
CitationCârjă, O., Necula, M., & Vrabie, I. I. (2004). Local invariance via comparison functions. Electronic Journal of Differential Equations, 2004(50), pp. 1-14.
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