Dirichlet problem for the mixed type equation with two degeneration lines in a half-strip

In this article, the first boundary problem for the mixed type equation with two degeneration lines at a half-strip in the class of the regular and limited in infinity decisions is discussed. The criterion of uniqueness for the stated problem was formulated by the methods of a spectral analysis. The solution of a problem is constructed in the form of a series on eigenfunctions of the corresponding one-dimensional eigenvalues problem. At justification of the uniform convergence of the constructed series there was a problem of small denominators. The estimation for the separation from zero of a small denominator with the corresponding asymptotics was provided in connection with mentioned problem in the present paper. This assessment at some sufficient conditions on boundary function allowed to prove convergence of the constructed series in a class of the regular solutions of this equation. In difference from other works of similar subject is the criterion of uniqueness and existence of the solution of the stated problem to be proved at all positive values of the parameters entering the equation, not necessarily equal. Such fact is an important consequence of the received result that the constructed solution everywhere in the considered area is the solution of the equation. Therefore the line of change-type of the equation as a special one is eliminated.