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

  • Ya.O. Baranetskij Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine
  • P.I. Kalenyuk Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine
  • M.I. Kopach Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
  • A.V. Solomko Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0002-6213-4130
Keywords: differential equation with partial derivatives, eigenfunctions, Riesz basis
Published online: 2019-12-31

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 $V(L).$

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How to Cite
(1)
BaranetskijY.; KalenyukP.; KopachM.; SolomkoA. The Nonlocal Boundary Value Problem With Perturbations of Mixed Boundary Conditions for an Elliptic Equation With Constant Coefficients. I. Carpathian Math. Publ. 2019, 11, 228-239.

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