×

zbMATH — the first resource for mathematics

On a scalar equation with discontinuous right-hand side and the uniqueness theorem. (English. Russian original) Zbl 1038.34010
Differ. Equ. 38, No. 10, 1435-1445 (2002); translation from Differ. Uravn. 38, No. 10, 1348-1357 (2002).
In the first part of the paper, the author proves some results concerning the properties of the set \(D(F)\) of solutions to the (discontinuous) scalar differential inclusion, \(y'(t)\in F(t,y(t))\), where \(F(.,.)\) is assumed to have a Luzin-type “approximate upper semicontinuity” property and such that there exist \(\epsilon_0>0\), an integrable function \(g_1(.)\) and a measurable function \(g_2(.)\) such that \(F(t,y)\subseteq [\epsilon_0, g_1(t)g_2(y)]\).
In the last part of the paper, the author proves some extensions of the classical uniqueness theorems of Kamke and Osgood, respectively, to discontinuous differential equations of the form \(y'=f(t,y), \;y(t_0)=y_0\).
One may note that \(\omega(.,.)\) in Theorem 4.1 cannot be a “multimapping” since the basic hypothesis \(\| f(t,y_1)-f(t,y_2)\| \leq \omega(t,\| y_1-y_2\| )\) may have sense only if \(\omega(.,.)\) is a (single-valued) function.
One may note also that the text of this paper is very difficult to follow since the author uses a large number of unspecified and noncommon notations, definitions and results from his previous books and articles.

MSC:
34A36 Discontinuous ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A60 Ordinary differential inclusions
PDF BibTeX XML Cite
Full Text: DOI