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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
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