Local invariance via comparison functions
Vrabie, Ioan I.
Southwest Texas State University, Department of Mathematics
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.
Viable domain, Local invariant subset, Exterior tangency condition, Comparison property, Lipschitz retract
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.