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The regions of solvability for some three point problem. (English) Zbl 1275.34024

The author investigates the solvability of the following three-point boundary problem \[ \begin{gathered} -x'' = \mu x^{+} - \lambda x^{-} + h(t, x, x'), \\ x(0) = 0,\quad x(1) = c x(0.5), \end{gathered} \] where \(\mu, \lambda, c \in\mathbb{R}\), \(h\) is a bounded continuous function and \(x^{+} = \max\{x, 0\}\), \(x^{-} = \max\{-x, 0\}\). Firstly, the problem with \(h\equiv 0\) is studied in detail, the corresponding Fučík spectrum is described with respect to the value of the parameter \(c\in\mathbb{R}\). Secondly, the author provides sufficient conditions on \((\mu, \lambda)\), \(c\) and \(h\) which guarantee the existence of a solution to the problem. The point \((\mu, \lambda)\) has to be in one of the “good” regions for solvability (with respect to the Fučík spectrum) determined by \[ (z_{+}(1) - c z_{+}(0.5))(z_{-}(1) - c z_{-}(0.5)) < 0, \] where \(z_{\pm}\) solves the Cauchy problem \[ -z'' = \mu z^{+} - \lambda z^{-}, \quad z(0) = 0, \quad z'(0)=\pm 1. \]

MSC:

34B08 Parameter dependent boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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