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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.

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