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On the Nagumo uniqueness theorem. (English) Zbl 1252.34011

The paper discusses a certain generalization of two uniqueness criteria for the Cauchy problem \[ x'= f(t,x),\;x(0)= 0,\;t\in [0,1]. \] The first of the mentioned criteria, due Athanassov, is the following
(1) \(|f(t,x)- f(t,y)|\leq (u'(t)/u(t))|x- y|\) for \(t\in (0,1]\) and \(x,y\in [-1,1]\), where \(u:[0,1]\to \mathbb{R}_+= [0,\infty), u(0)= 0\) and \(u'(t)> 0\) for \(t\in(0,1]\).
Moreover,a second requirement is satisfied,
(2) \(f(t,x)= o(u'(t))\) as \(t\to 0\), uniformly with respect to \(x\in [-1,1]\).
The second criterion, due to Constantin, consists of condition (2) and condition \((1^\prime)\) \(|f(t,x)|\leq (u'(t)/u(t))\omega(|x|)\), where \(\omega: [0,1]\to \mathbb{R}_+\) satisfies some conditions.
The authors did not include a concrete example showing that their criterion is more general than those mentioned above.
Reviewer’s remark: It is a very surprising fact that nobody noticed that Athanassov’s criterion is only a particular case of the uniqueness criteria due to J. Witte [Math. Z. 140, 281–287 (1974; Zbl 0289.34007)] and the reviewer and J. Rivero [Commentat. Math. Univ. Carol. 28, 23–31 (1987; Zbl 0627.34073)]. The details are too involved to be stated here.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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