TY - JOUR
AU - Baranetskij, Ya.O.
AU - Ivasiuk, I.Ya.
AU - Kalenyuk, P.I.
AU - Solomko, A.V.
PY - 2018/12/31
Y2 - 2024/11/01
TI - The nonlocal boundary problem with perturbations of antiperiodicity conditions for the elliptic equation with constant coefficients
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 10
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.10.2.215-234
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1484
SP - 215-234
AB - <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>
ER -