Multiparameter Eigenvalue Problems and Their Applications in Electrodynamics

Abstract

A nonlinear n-parametric eigenvalue problem called the problem P is considered. In addition to n spectral parameters, the problem P depends on n 2 numerical parameters; for zero values of these parameters, the problem splits into n linear problems P 0 i , i = 1, n. To the problem P , one can assign n nonlinear problems Pi, which, in particular, have solutions that are not related to the solutions of the problems P 0 i . The problems Pi are treated in this work as "nonperturbed" problems. Using the properties of eigenvalues of the problems Pi, we prove the existence of eigenvalues of the problem P ; some of these eigenvalues are not related to solutions of the problems P 0 i . Keywords and phrases: nonlinear Sturm-Liouville-type problem, multiparameter eigenvalue problem, perturbation method, method of integral dispersion equations.