Open AccessThe nonlocal problem for the differential-operator equation of the even order with the involution
Author(s) -
Ya. О. Baranetskij,
P. І. Kalenyuk,
Lubov Kolyasa,
М. І. Kopach
Publication year - 2018
Publication title -
karpatsʹkì matematičnì publìkacìï
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.63
H-Index - 4
eISSN - 2313-0210
pISSN - 2075-9827
DOI - 10.15330/cmp.9.2.109-119
Subject(s) - mathematics , eigenfunction , boundary value problem , differential operator , mathematical analysis , operator (biology) , semi elliptic operator , ordinary differential equation , compact operator , linear map , differential equation , pure mathematics , ladder operator , eigenvalues and eigenvectors , physics , quantum mechanics , biochemistry , chemistry , repressor , computer science , transcription factor , extension (predicate logic) , gene , programming language
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.