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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.

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