On the oscillation of eigenfunctions of a fourth-order spectral problem
The authors consider the spectral problem for the fourth-order ordinary differential equation \[ l(y)(x):= (p(x)y''(x))'' - (q(x)y'(x))' + r(x)y(x\,) = \lambda \tau (x)y(x),\quad 0 < x < l, \] with boundary conditions \[ \begin{aligned} (py'')(0) - C_0 Ty(0) - C_1 y'(0) = 0, \\ y(0) + C_2 Ty(0) - C_0 y'(0) = 0, \\ (py'')(l) + D_0 Ty(l) + D_1 y'(0) = 0, \\ y(l) - D_2 Ty(0) + D_0 y'(l) = 0, \end{aligned} \] where \(\lambda \in \mathbb C\) is a spectral parameter, \( Ty \equiv (py'')' - qy',\) the functions \(p,r\) are strictly positive and continuous on \([0,l]\), the function \(p\) has an absolutely continuous derivative on \([0,l]\), and \(C_{i}, D_i \geq 0\) for \(i=0,1,2\) are real constants. This paper studies the structure of root subspaces and the number of zeros in \((0,l)\) of the eigenfunctions.
scientific article; zbMATH DE number 6106460