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Solution sets of boundary value problems for nonconvex differential inclusions. (English) Zbl 0785.34018

The following boundary value problem in \(\mathbb{R}^ n\) is considered: (1) \(x''(t) \in F(t,x(t)\), \(x'(t))\), \(x(0)=x(1)=0\). In (1), \(F:[0,1] \times \mathbb{R}^ n \times \mathbb{R}^ n \rightsquigarrow \mathbb{R}^ n\) denotes a set-valued map with nonempty compact values. It is proved that if \(F\) is Lipschitzian then the solution set of (1) is a retract of \(W^{2,1}\). If, moreover, \(F\) has convex values then the solution set is retract of \(C^ 1\). An existence theorem is given for a continuous map \(F\) with nonconvex values.

MSC:

34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
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