The nonlocal problem for the $2n$ differential equations with unbounded operator coefficients and the involution

  • Ya.O. Baranetskij Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine
  • I.I. Demkiv Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine https://orcid.org/0000-0003-4015-8171
  • I.Ya. Ivasiuk Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
  • M.I. Kopach Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Keywords: operator of involution, differential-operator equation, eigenfunctions, Riesz basis
Published online: 2018-07-03

Abstract


We study a problem with periodic boundary conditions for a $2n$-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system $V$ of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system $V$ is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained.

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How to Cite
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Baranetskij Y., Demkiv I., Ivasiuk I., Kopach M. The Nonlocal Problem for the $2n$ Differential Equations With Unbounded Operator Coefficients and the Involution. Carpathian Math. Publ. 2018, 10 (1), 14-30.

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