an:05649938 Zbl 1185.34016 Andres, Jan; Malaguti, Luisa; Pavla??kov??, Martina Strictly localized bounding functions for vector second-order boundary value problems EN Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, 6019-6028 (2009). 00256559 2009
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34A60 34B15 47H04 vector second-order Floquet problem; strictly localized bounding functions; solutions in a given set; Scorza-Dragoni technique; evolution systems; dry friction problem; coexistence of periodic and anti-periodic solutions 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 {\parindent9mm \begin{itemize}\item[{$$(1_i)$$}] $$A,B:[0,T]\to \mathbb R^{n\times n}$$ 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 $$a,b:[0,T]\to [0,\infty)$$, \item[{$$(1_{ii})$$}] $$M$$ and $$N$$ are $$n\times n$$ matrices, $$M$$ is nonsingular, \item[{$$(1_{iii})$$}] $$F:[0,T]\times \mathbb R^n\times \mathbb R^n\,\diagrbar\circ\,\mathbb R^n$$ is an upper-Carath??odory multivalued mapping \end{itemize}} in a given set is established by means of $$C^{2}$$-bounding functions for vector upper-Carath??odory 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.