Oscillation properties for the equation of vibrating beam with irregular boundary conditions (Q837111)

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 \(n-1\).