On the existence of an infinite number of eigenvalues in one nonlinear problem of waveguide theory

A nonlinear Sturm–Liouville-type eigenvalue problem on an interval with a boundary condition of the first kind and an additional local condition at one of the boundaries of the interval is considered. All the parameters of the problem are real. The existence of an infinite number of (isolated) positive eigenvalues is proven, their asymptotic behavior is indicated, a condition for the periodicity of the eigenfunctions is found, the period is calculated, and an explicit formula for the zeros of the eigenfunction is presented. It is shown that methods of perturbation theory are not applicable to the complete study of the nonlinear problem.