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Boundary value problems for nonconvex differential inclusions. (English) Zbl 0817.34009
A relaxation theorem for a Sturm-Liouville-type of a boundary value problem for differential inclusions is proved. Namely, the author considers the following two problems: (1) $x''(t)\in F(t, x(t), x'(t))$, $x(0)= x\sb 0$, $x(1)= x\sb 1$; (2) $x''(t)\in \text{ext }F(t, x(t), x'(t))$, $x(0)= x\sb 0$, $x(1)= x\sb 1$. Here $F$ is a set-valued map and $\text{ext }F$ denotes the set of extreme points of $F$. Using Green function it is proved that solutions of (2) do exist provided $F$ is measurable in $t$ and continuous in $(x,x')$ and, moreover, if $F$ is Lipschitz continuous they are dense in the solution set of (1). A special condition on the Green function which gives a priori bounds on the solutions is assumed throughout the paper. An application to control systems is given.

34A60Differential inclusions
34B15Nonlinear boundary value problems for ODE
34B24Sturm-Liouville theory
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