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On solutions of a differential inclusion with lower semicontinuous nonconvex right-hand side in a Banach space. (English. Russian original) Zbl 0588.34012
Math. USSR, Sb. 53, 203-231 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 2, 199-230 (1984).
Three differential inclusions in a Banach space (1) $$x'\in \Gamma (t,x)$$, (2) $$x'\in \overline{co} \Gamma(t,x)$$, (3) $$x'\in \overline{ext} \overline{co} \Gamma (t,x)$$ are considered. $$\overline{ext} A$$ denotes the closure of the extremal point set of the set A, $$\Gamma$$ is the multivalued mapping with compact values. For a compact set M let us denote by $$H(i)$$ the set of solutions x(t), $$t\in [0,a]$$ of (i) with x(0)$$\in M$$. The main results of the paper concern the existence of the solutions for inclusions (1)-(3), compactness properties of the sets $$H(i)$$ and the relations $$H(3)\subset H(1)\subset H(2)$$, $$H(2)=\overline{H(1)}=\overline{H(3)}$$. Assumptions in general do not involve the existence of the Caratheodory type selector of the multifunction $$\Gamma$$, that makes the difference between this paper and earlier works. The main instruments of research are the theorems stating the existence of the set U of absolutely continuous and almost everywhere differentiable functions x(.) such that $\{y(.):\quad y(t)=x_ 0+\int^{t}_{0}v(s)ds,\quad x_ 0\in M,\quad v(s)\in \overline{co} \Gamma (s,U(s))\}\subset U$ and $$U(0)=M$$. $$(U(s)=^{def}\{x(s): x(.)\in U\}.)$$
Reviewer: V.Tsalyuk

##### MSC:
 34A60 Ordinary differential inclusions 34G20 Nonlinear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
##### Keywords:
Banach space; compactness properties
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