ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2024, Vol. 45, No. 11, pp. 5413–5423. c  Pleiades Publishing, Ltd., 2024. On a Boundary Value Problem for a Third Order Elliptic-hyperbolic Equation with Superposition Operators of the First and Second Orders in a Rectangular Domain B. I. Islomov 1* and G. K. Kylyshbaeva 1** (Submitted by E. K. Lipachev) 1 National University of Uzbekistan, Department of Differential Equations and Mathematical Physics, Tashkent, 100174 Uzbekistan Received June 24, 2024; revised September 7, 2024; accepted September 15, 2024 Abstract—This article proposes a method for solving a Dirichlet type problem for a third-order equation of elliptic-hyperbolic type with a superposition operators of first and second orders in a rectangular domain. It is shown that the correctness of the formulated problem significantly depends on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. An example is given in which the well posed problem with homogeneous conditions has a nontrivial solution. A solution to the problem is constructed in the form of a sum of a series of eigenfunctions of the corresponding one-dimensional spectral problem. A criterion for the uniqueness of a solution is established. When justifying the uniform convergence of a series, the problem of small denominators arises. In this connection, estimates of small denominators about the distance from zero with the corresponding asymptotic have been established. These estimates made it possible to prove the convergence of the series in the class of regular solutions to this equation. Estimates on the stability of the solution from given boundary functions are proved. DOI: 10.1134/S1995080224606386 Keywords and phrases: Third order equation, Dirichlet type problem, small denominators, uniqueness, existence and stability of the solution 1. INTRODUCTION Well posed boundary value problems for equations of elliptic-hyperbolic and parabolic -hyperbolic types of third order were first studied in the works of A. Bitsadze and M.S. Salakhitdinov [1], M.S. Salakhitdinov [3], T.D. Dzhuraev [5], T.D. Juraeva, A. Sopuev, and M. Mamazhonova [7]. In this work, a mixed domain consist of characteristic triangles and a rectangle (or semicircle). The solution was found in the class of functions representable in the form u(x, y)= υ(x, y)+ ω(x) or u(x, y)= υ(x, y)+ ω(y) υ(x, y)—arbitrary regular solution to a second order equation Lυ =0, ω(x), ω(y) is arbitrary functions from the class C [0, 1] ∩ C 2 (0, 1). This representation is important for equations composed of a product of permutation differential operators. But for a mixed type equation with a generalized operator containing lower terms, this method is not always correct [10]. Currently, the theory of boundary value problems for equations of mixed type third order is one of the developing branches of the theory of partial differential equations. Numerous theoretical studies have been published [6, 13, 16, 21], and [27]. In these works, a number of correct local and nonlocal boundary value problems of a new type were studied. Many third-order equations arise in the theory of elasticity of materials with memory, in nonlinear viscoelastic media, when modeling moisture transfer processes, and when studying the hydrodynamics * E-mail: islomovbozor@yandex.com ** E-mail: kalbaevna85@mail.ru 5413