Spectral and oscillatory properties of a linear pencil of fourth-order differential operators (Q382353)

The authors consider the spectral problem related to the differential equation \[ (py'')'' - \lambda ( - y'' + cry) = 0 \] with some of the sets of boundary conditions \[ y(0) = y'(0) = y(1) = y'(1) = 0 \] or \[ y(0) = y'(0) = y'(1) = (py'')'(1) - \lambda \alpha y(1). \] Here, the coefficients \(p,r\in C[0,1]\) are uniformly positive, and the physical parameters \(c\) and \(\alpha\) are real. In addition to the above boundary conditions, the solutions \(y\) satisfy the natural restrictions \(y \in C^2 [0,1]\) and \(py'' \in C^2 [0,1]\). The main goal is to study the simplicity of eigenvalues of the problem under consideration and the oscillatory properties of its eigenfunctions.