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
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Baranetskij Y., Kalenyuk P., Kopach M., Solomko A. The Nonlocal Boundary Value Problem With Perturbations of Mixed Boundary Conditions for an Elliptic Equation With Constant Coefficients. I. Carpathian Math. Publ. 2019, 11 (2), 228-239.

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