Multiparameter eigenvalue problems and their applications in electrodynamics

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 them, it splits into $n$ linear problems $P_i^0$, $i=\overline{1,n}$. To the problem $P$, one can assign $n$ other nonlinear problems $P_i$, which, in particular, have solutions that are not related to the solutions of the problems $P_i^0$. The problems $P_i$ are treated in this work as “nonperturbed” problems. Using the properties of eigenvalues of the problems $P_i$, we prove the existence of eigenvalues of the problem $P$; some of these eigenvalues are not related to solutions of the problems $P_i^0$.