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Existence of solutions for nonconvex second order differential inclusions. (English) Zbl 1034.34018

The author considers the second-order differential inclusion \[ x''\in F(x,x') + f(t,x,x'),\quad x(0)=x_{0},\quad x'(0)=y_{0}, \tag{1} \] where \(F:\Omega\subset \mathbb{R}^{2m}\to 2^{\mathbb{R}^{m}}\) is an upper semicontinuous, compact-valued multifunction, such that \(F(x,y)\subset\partial V(y)\) for some convex lower semicontinuous function \(V:\mathbb{R}^{m}\to \mathbb{R},\) \(f:\mathbb{R}\times \mathbb{R}^{m} \times \mathbb{R}^{m}\to \mathbb{R}^{m}\) is a Carathéodory function. Here, \(\Omega\) is an open set. An existence theorem for the problem (1) is proved.

MSC:

34A60 Ordinary differential inclusions
47H04 Set-valued operators
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