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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
$$(1_i)$$
$$A,B:[0,T]\to \mathbb R^{n\times n}$$ 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 $$a,b:[0,T]\to [0,\infty)$$,
$$(1_{ii})$$
$$M$$ and $$N$$ are $$n\times n$$ matrices, $$M$$ is nonsingular,
$$(1_{iii})$$
$$F:[0,T]\times \mathbb R^n\times \mathbb R^n\,\diagrbar\circ\,\mathbb R^n$$ 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.

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