@article{Baranetskij_Ivasiuk_Kalenyuk_Solomko_2018, title={The nonlocal boundary problem with perturbations of antiperiodicity conditions for the elliptic equation with constant coefficients}, volume={10}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/1484}, DOI={10.15330/cmp.10.2.215-234}, abstractNote={<p>In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$</p> <p>Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. 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 $L_{2}(G)$.</p> <p>In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$</p>}, number={2}, journal={Carpathian Mathematical Publications}, author={Baranetskij, Ya.O. and Ivasiuk, I.Ya. and Kalenyuk, P.I. and Solomko, A.V.}, year={2018}, month={Dec.}, pages={215–234} }