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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) $$x''(t)\in F(t, x(t), x'(t))$$, $$x(0)= x_ 0$$, $$x(1)= x_ 1$$; (2) $$x''(t)\in \text{ext }F(t, x(t), x'(t))$$, $$x(0)= x_ 0$$, $$x(1)= x_ 1$$. 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.

##### MSC:
 34A60 Ordinary differential inclusions 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory
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