Existence of eigenvalues of a boundary value problem with shift for an equation of parabolic-hyperbolic type

In this work a spectral problem for an operator of parabolic-hyperbolic type of first kind with non-classical boundary conditions is considered. The problem is considered in a standard domain. The parabolic part of the space is a rectangle. And the hyperbolic part of the space coincides with a characteristic triangle. We consider the problem with the local boundary condition in the domain of parabolicity and with the boundary condition with displacement in the domain of hyperbolicity. We prove the strong solvability of the considered problem. The main aim of the paper is to research spectral properties of the problem. The existence of eigenvalues of the problem is proved.