Method of $Y$-mappings for study of multiparameter nonlinear eigenvalue problems

For the study of nonlinear multiparameter eigenvalue problems, a method of $Y$-mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept of a $Y$-mapping, which puts into correspondence to the potential a special nonlinear function of several arguments: eigenfunctions of a linear problem, is defined. The multiparameter nonlinear eigenvalue problem is reduced to the problem of finding fixed points of $Y$-mappings. Using the Schauder theorem, the existence of an infinite set of fixed points of $Y$-mappings and, accordingly, solutions in a nonlinear multiparameter eigenvalue problem for sufficiently small values of the nonlinearity coefficient is proved.