The nonlocal boundary problem with perturbations of antiperiodicity conditions for the eliptic equation with constant coefficients

Ya. O. Baranetskij, I. Ya. Ivasiuk, P. I. Kalenyuk, A. V. Solomko

Abstract


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 $RL_0=LR.$

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 $L_{2}(G)$.

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 $V(L).$

Keywords


differential-operator equation, eigenfunctions, Riesz basis

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