The nonlocal problem for the differential-operator equation of the even order with the involution
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 2
Abstract
In this paper, the problem with boundary non-self-adjoint conditions for 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 the operator $L$ of the studied problem is a Riesz basis is established.
Authors and Affiliations
Ya. Baranetskij, P. Kalenyuk, L. Kolyasa, M. Kopach
Keywords
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- DOI 10.15330/cmp.9.2.109-119
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