The nonlocal problem for the $2n$ differential equations with unbounded operator coefficients and the involution

Abstract

We study a problem with periodic boundary conditions for a $2n$-order differential equation whose coefficients are non-self-adjoint operators. It is established that the operator of the problem has two invariant subspaces generated by the involution operator and two subsystems of the system of eigenfunctions which are Riesz bases in each of the subspaces. For a differential-operator equation of even order, we study a problem with non-self-adjoint boundary conditions which are perturbations of periodic conditions. We study cases when the perturbed conditions are Birkhoff regular but not strongly Birkhoff regular or nonregular. We found the eigenvalues and elements of the system $V$ of root functions of the operator which is complete and contains an infinite number of associated functions. Some sufficient conditions for which this system $V$ is a Riesz basis are obtained. Some conditions for the existence and uniqueness of the solution of the problem with homogeneous boundary conditions are obtained.

Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument

“…The action of the operator T h on the deviation over the spatial argument T h u(t, x) = u(t, x + h) can be interpreted as the action of the pseudo-differential infinite order operator u(t, x + h) = exp(ih ∂u(t, x)/∂x). Thereby, the results of this paper are closely related to the research conducted in the article [9], which deals with the boundary problems for equations with pseudo-differential operators in infinite domains by spatial coordinates and the nonlocal problems for the differential-operator equations [4,5].…”

A nonlocal problem with multipoint perturbations of strongly regular Birkhoff boundary conditions for an even-order differential operator

“…Для звичайних диференціальних рівнянь, які містять оператор інволюції в працях [16,20]. Нелокальні задачі для диференціально-операторних рівнянь з інволюцією досліджено в статтях [15,17]. Властивості розвязків задач для еліптичних рівнянь зі сталими коефіцієнтами вивчено в роботах [18,19].…”

Nonlocal multipoint problem for a differential equation of $2n$-th order with operator coefficients

“…Nonlocal problems on a finite interval for differential equations with unbounded operator coefficients were studied by Ya.O. Baranetskij et al [3], Yu.A. Dubinskii [10], P.I.…”