The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. I

Abstract

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 $R: L_2(G) \to L_2(G)$ 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 $V(L).$