Abstract
This paper studies a differential equation representing the differences of two operators. One of the operators is generated by linear differential expressions that depend on time. The second operator is the Ionkin operator with respect to the spatial variable. In this paper, the differential operator with respect to time is generated by two-point Birkhoff regular boundary conditions. At the same time, the elliptic operator with respect to the spatial variable does not satisfy the so-called Agmon conditions. Moreover, the operator with respect to the spatial variable is not self-adjoint. In the beginning, the solvability of the problem is proved. In the final part, the uniqueness of the solution is proved. Direct application of the methods of the authors’ previous works to prove the uniqueness of the solution to the problem is quite problematic. However, the authors managed to modify the reasoning of previous works to prove the uniqueness of the solution to the problem.