On the smoothness of a generalized solution of a semi-nonlocal boundary value problem for a fourth-order mixed type equation of the second kind

In this paper, we investigate the unique solvability of a regular solution and the smoothness of a generalized solution of a semi-nonlocal boundary value problem for a fourth-order mixed-type equation of the second kind in Sobolev spaces.Theorems of uniqueness are proved by the energy integral method, and the existence and smoothness of the solution are proved by the methods of "ε-regularization", a priori estimates, modified Galerkin methods, and the Fourier method.