On solutions of a differential inclusion with lower semicontinuous nonconvex right-hand side in a Banach space. (English. Russian original) Zbl 0588.34012

Tolstonogov, A. A.; Finogenko, I. A.

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