The nonlocal problem for the differential-operator equation of the even order with the involution

  • 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
  • L.I. Kolyasa 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
Keywords: operator of involution, differential-operator equation, eigenfunctions, Riesz basis
Published online: 2018-01-02

Abstract


In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated.

By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n.

We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$

The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.

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How to Cite
(1)
Baranetskij Y., Kalenyuk P., Kolyasa L., Kopach M. The Nonlocal Problem for the Differential-Operator Equation of the Even Order With the Involution. Carpathian Math. Publ. 2018, 9 (2), 109-119.

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