On a Nonclassical Eigenvalue Problem with Nonlinearizable Solutions

We study a nonlinear eigenvalue problem of a special form arising in electrodynamics. Theproblem is posed for a system of two equations with boundary conditions of the first kind and twoadditional local conditions. There is one spectral parameter in the problem, and two moreparameters, which are subject to an additional constraint, occur in the above-mentioned localconditions. Thus, there are two unknown parameters in the problem: one is the spectralparameter, and the other is some additional parameter, chosen so as to ensure that there exists anontrivial solution of the problem in question. The existence of nonlinearizable solutions of theproblem is proved.