## 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

### Citations:

Zbl 0289.34007; Zbl 0627.34073
Full Text:

### References:

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