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.


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


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