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On two point boundary value problems for second order differential inclusions. (English) Zbl 1112.34008
The existence of (almost everywhere) solutions to a second-order differential inclusion
$y''\in F(t, y,y'),\quad y\in\mathbb{R}^n,\quad t\in [0,T],\tag{$$*$$}$ subject to the two-point boundary condtion $$y(0)= A$$, $$y(T)= B$$, where $$A,B\in\mathbb{R}^n$$ and $$F$$ is an upper-Carathéodory set-valued map with compact convex values satisfying a quadratic growth condition. The main result relies on some new estimates leading to a priori bounds for solutions to $$(*)$$ and the usual homotopy continuation method. A discussion of the single-valued counterpart as well as some examples are provided.

##### MSC:
 34A60 Ordinary differential inclusions 34B15 Nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations