Solvability of the mixed problem for a hyperbolic equation in the case of incomplete system of eigenfunctions

A mixed problem for a second-order hyperbolic equation with constant coefficients and a mixed partial derivative is considered. We assume that the roots of the characteristic equation are simple and lie on the positive half-line. The coefficients of the equation and the boundary data are constrained by conditions such that the two-fold completeness of eigenfunctions of the corresponding spectral problem for the differential quadratic pencil is absent. The Poincaré–Cauchy contour integral method is used to obtain various sufficient conditions for the solvability of this problem.