The nonlocal problem for the differential-operator equation of the even order with the involution - Baranetskij, Ya.O.; Kalenyuk, P.I.; Kolyasa, L.I.; Kopach, M.I. (Carpathian Mathematical Publications, 2018)
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^∞ 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.