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Continuous branches of multivalued mappings with nonconvex images and functional-differential inclusions. (Russian) Zbl 0614.34016
Let F be a multifunction from \([a,b]\times R^ m\) to closed subsets of \(R^ n\) and satisfy a strong Caratheodory condition (convexity is not assumed), \(\theta: C^ n[a,b]\to L^ n[a,b]\) be continuous. The author considers the problem \(\dot x(t)\in F(t,(\theta x(t))\), \(t\in [a,b]\), \(x(a)=x_ 0\), and develops the method of H. A. Antosiewicz and A. Cellina [J. Differ. Equations 19, 386-398 (1975; Zbl 0279.54007)] to study existence of solutions, their continuability, boundedness, density in the solution set of ”convexed inclusion” and continuous dependence on right-hand sides. The basic result is existence of continuous selector for multivalued mapping \(\phi\) whose values are closed subsets of \(L^ n_ 1[a,b]\), such that for \(E_ 1,E_ 2\subset [a,b]\), \(E_ 1\cup E_ 2=[a,b]\), \(E_ 1\cap E_ 1=\emptyset\), \(v_ 1,v_ 2\in \phi (x)\) one has \(\chi (E_ 1)v_ 1+\chi (E_ 2)v_ 2\in \phi (x),\chi\) is a characteristic function of a set.
Reviewer: V.Tsalyuk

34A60 Ordinary differential inclusions