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Strictly localized bounding functions for vector second-order boundary value problems. (English) Zbl 1185.34016
Summary: The solvability of the second-order Floquet problem
\[ \begin{aligned} \ddot x(t)+A(t)\dot x(t)&+B(t)x(t)\in F(t,x(t),\dot x(t)),\quad \text{a.a. }t\in[0,T],\\ &x(T)=Mx(0),\quad \dot x(T)=N\dot x(0),\end{aligned}\tag{S} \]
where 9mm
\(A,B:[0,T]\to \mathbb R^{n\times n}\) are measurable matrix functions such that \(|A(t)|\leq a(t)\) and \(|B(t)|\leq b(t)\), for all \(t\in [0,T]\) and suitable integrable functions \(a,b:[0,T]\to [0,\infty)\),
\(M\) and \(N\) are \(n\times n\) matrices, \(M\) is nonsingular,
\(F:[0,T]\times \mathbb R^n\times \mathbb R^n\,\diagrbar\circ\,\mathbb R^n\) is an upper-Carathéodory multivalued mapping
in a given set is established by means of \(C^{2}\)-bounding functions for vector upper-Carathéodory systems. The applied Scorza-Dragoni type technique allows us to impose related conditions strictly on the boundaries of bound sets. An illustrating example is supplied for a dry friction problem.

34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
47H04 Set-valued operators
Full Text: DOI
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