On some boundary value problems with a shift for a mixed type equation

An important stage in the development of the theory of boundary value problems was the proposed by A.M. Nakhushev in 1969, non-local problems of a new type, which were later called in our country boundary value problems with a shift, and abroad – Nakhushev problems (problems). They are a generalization of the Tricomi problem, and also contain a wide class of well-posed self-adjoint problems. These problems immediately aroused great interest of many authors. In recent years, studies of problems with a shift for equations of mixed type have been carried out especially intensively. But in these works, the boundary conditions, as a rule, contain classical operators, while non-local boundary value problems contain operators of a more complex structure and operators of fractional integro-differentiation. In this paper, we study the unique solvability of problems with mixing for an equation of mixed elliptic-hyperbolic type. Under constraints of unequal type on known functions and different orders of fractional differentiation operators in the boundary condition, uniqueness theorems are proved. The existence of a solution to the problems is proved by reducing the problems to Fredholm equations of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problems.