The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. I - Baranetskij, Ya.O.; Kalenyuk, P.I.; Kopach, M.I.; Solomko, A.V. (Carpathian Mathematical Publications, 2019)
In this article we investigate a problem with nonlocal boundary conditions which are multipoint perturbations of mixed boundary conditions in the unit square G using the Fourier method. The properties of a generalized transformation operator R: L_2(G) → L_2(G) that reflects normalized eigenfunctions of the operator L_0 of the problem with mixed boundary conditions in the eigenfunctions of the operator L for nonlocal problem with perturbations, are studied. We construct a system V(L) of eigenfunctions of operator L. Also, we define conditions under which the system V(L) is total and minimal in the space L_2(G), and conditions under which it is a Riesz basis in the space L_2(G). In the case if V(L) is a Riesz basis in L_2(G), we obtain sufficient conditions under which nonlocal problem has a unique solution in form of Fourier series by system V(L).