A Problem with Shift for Mixed-Type Equation in Domain, the Elliptical Part of Which Is a Horizontal Strip

In this article, the issue of the unique solvability of a problem with shift in an unbounded domain is investigated; the elliptical part of the domain is a horizontal strip. The uniqueness of the theorem is proven by the method of energy integrals under constraints of unequal type on known functions and different orders of fractional differentiation operators in the boundary condition. Problem in the elliptical part of the mixed domain, which is a horizontal strip, is solved by the Green’s function method. By reduction to the Fredholm equation of the second kind, the existence of a solution to the problem is proved; the unconditional solvability to the problem follows from the uniqueness of the solution to the problem.