## 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
Full Text: