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Nonlinear spectra: the Neumann problem. (English) Zbl 1177.34105

The authors consider nonlinear equations of the form
\[ x'' = - \lambda f (x^{+}) + \mu g (x^{-}), \]
where \(\lambda\) and \(\mu\) are parameters, \(f,g : \mathbb{R}_{+} \longrightarrow \mathbb{R}_{+}\) are locally Lipschitz continuous functions such that \(f (0) = g (0) = 0, \;\;\;x^{+} : = \max \{x,0\}, \;\;x^{-} : = \max \{-x,0\}.\) Boundary conditions of the Neumann type, \(x'(a) = 0, \;\;x'(b) = 0\), are assumed. The authors describe values \((\lambda,\mu)\) for which such a problem has nontrivial solutions. The set of those pairs \((\lambda, \mu)\) are called the spectrum of the problem.

MSC:

34L05 General spectral theory of ordinary differential operators
34B08 Parameter dependent boundary value problems for ordinary differential equations
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