Abstract <p>In this article, we will be considered the first boundary value problem and a modified boundary value problem (analogous to the Keldysh problem) with periodicity conditions for a mixed type equation with characteristic degeneracy. We have established the criterion of uniqueness of solution of the first problem and showed that the solution of the analogue of the Keldysh problem is unique. The solutions are constructed as the sum of a series according to the system of eigenfunctions of the corresponding one-dimensional spectral problem. In substantiation the convergence of a series representing the solution of the first boundary value problem in the class of regular solutions of this equation, the problem of small denominators arises. An estimate of separation from zero with appropriate asymptotics is established. Based on this estimate, we have found sufficient conditions for boundary functions under which the series defining the solution and its derivatives up to and including the second order converge.</p>