TY - JOUR
AU - Baranetskij, Ya.O.
AU - Kalenyuk, P.I.
AU - Kolyasa, L.I.
AU - Kopach, M.I.
PY - 2018/01/02
Y2 - 2024/05/18
TI - The nonlocal problem for the differential-operator equation of the even order with the involution
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 9
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.9.2.109-119
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1454
SP - 109-119
AB - <p>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.</p><p>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.</p><p>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 $L_{0,k}.$</p><p>The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.</p>
ER -