Uniqueness and nonuniqueness results for ordinary differential equations. (English) Zbl 0882.34003

Bainov, Drumi (ed.) et al., Proceedings of the 5th international colloquium on differential equations, Plovdiv, Bulgaria, August 18–23, 1994. Invited lectures and short communications. Vol. 2. Singapore: SCT Publishing, 140-147 (1995).
Summary: The classical Kamke-type criteria guarantee unique solvability of the initial value problem \(\dot x= f(t,x)\), \(x(t_0)= x_0\), under the assumption that the variation of the function \(f(t,x)\) relative to \(x\) is bounded. A generalized Kamke-condition is presented which allows to estimate changes of the directional field \(f\) including the time variable \(t\). Its advantage can be shown by simple examples.
In the second part of the essay, I will analyze the inverse problem of uniqueness, under which conditions the initial value problem possesses at least two different solutions. Nonuniqueness has been treated only by few authors. The results are based on appropriate modifications of uniqueness criteria. One reason for my treating this subject was that some published results were not quite correct. Inverting the Kamke-condition and the Brauer-Sternberg-condition for the classical and the singular Cauchy problem in order to get applicable nonuniqueness conditions leads to unexpected results.
For the entire collection see [Zbl 0870.00029].


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