Solvability of a mixed problem for a hyperbolic equation with splitting boundary conditions in the case of incomplete system of eigenfunctions

In this paper, we consider a mixed problem for a second-order hyperbolic equation with constant coefficients and a mixed partial derivative. We assume that the boundary conditions are splitted (i.e., one condition is posed at the left endpoint of the main interval and the other at the right endpoint) and the roots of the characteristic equation are simple and lie on the positive half-line. The coefficients of the equation and the boundary conditions are constrained by conditions that guarantee the absence of the two-fold completeness of eigenfunctions of the corresponding spectral problem for the differential quadratic pencil. Using the Poincaré–Cauchy contour integral method, we to obtain sufficient conditions for the solvability of this problem.