On a boundary value problem with third kind boundary conditions for a third-order mixed-type equation with an elliptic-hyperbolic principal operator

In this paper, a method for solving a problem with a boundary condition of the third kind for a third-order equation of elliptic-hyperbolic type with a superposition of first- and second-order operators in a rectangular domain is proposed. It is shown that the correctness of the problem statement depends significantly on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. An example is given in which the problem with homogeneous conditions has a nontrivial solution. A solution to the problem is constructed as a sum of a series in eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the uniqueness of the solution is established. When substantiating the uniform convergence of the series, the problem of small denominators arises. In this connection, estimates of small denominators on the distance from zero with the corresponding asymptotics are established. These estimates made it possible to prove the convergence of the series in the class of regular solutions of this equation. Estimates of the stability of the solution from the given boundary functions are proved