On a Nonclassical Eigenvalue Problem with Nonlinearizable Solutions

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