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Oscillation properties for the equation of vibrating beam with irregular boundary conditions. (English) Zbl 1179.34033

The author studies oscillatory properties for the eigenfunctions of some fourth-order eigenvalue problems when the boundary conditions are irregular (one endpoint \(0\) is a clamped end, and the other is confined by \(y'(1)\cos\gamma +(py'')(1)\sin\gamma=0\), and \(y(1)\cos \delta-((py'')'-qy')(1)\sin \delta=0\)). The author studies properties of eigenvalues and corresponding eigenfunctions. He shows that there exists a sequence of simple eigenvalues tending to infinity, and there are at most two negative; the number of zeros of the \(n\)-th eigenfunction is between \(n-2\) and \(n-1\), and, if \(\gamma\) is sufficiently close to \(\pi\), the number is \(n-2\), whereas if \(\gamma\) is sufficiently close to \(\pi/2\), the number is \(n-1\).

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B24 Sturm-Liouville theory
34B09 Boundary eigenvalue problems for ordinary differential equations
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References:

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