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Bound sets and two-point boundary value problems for second order differential equations. (English) Zbl 1133.34013
The authors study the existence of solutions to a generalized two-point boundary value problem of Floquet type for a vector second order nonlinear differential equation. More precisely, the problem under consideration is the following
$\begin{cases} x''=f(t,x,x'), & t\in [0,1], \\ x(1)=Ax(0), & \\ x'(1)=Bx'(0), & \end{cases}$
where $$f:[0,1]\times \mathbb R^{2m} \to \mathbb R^{m}$$ is a continuous function and $$A$$ and $$B$$ are $$m\times m$$ square matrices, with $$A$$ non-singular. Using a suitable version of Mawhin’s continuation principle the authors prove a general existence result. Applications are given to the existence of solutions which are contained in suitable bound sets with possibly nonsmooth boundary.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 47H10 Fixed-point theorems