## A bound sets technique for Dirichlet problem with an upper-Carathéodory right-hand side.(English)Zbl 1237.34024

Consider the vector Dirichlet problem $\ddot{x}(t)\in F(t,x(t),\dot{x}(t))\quad \text{a.e.}\,\, t\in [0,T], \tag{1}$
$x(0) = x(T)=0, \tag{2}$ where $$F(t,x(t),\dot{x}(t))$$ is a subset of $$\mathbb{R}^{n}.$$ By a solution of problem (1), (2) is meant a function $$x: [0,T]\to\mathbb{R}^{n}$$ with absolutely continuous first derivative satisfying (1), (2). It is proved that the problem (1), (2) has a solution if some conditions are fulfilled.

### MSC:

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

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