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.


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