The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

Abstract

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

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

“…This paper is a continuation of the research [6,7] for the case of nonhomogeneous boundary conditions.…”

“…It is well known (see [6,21]), that the operator B 0 has the point spectrum σ (B 0 ) := {φ q ∈ R : φ q = π 2 (2q − 1) 2 , q ∈ N} and the system of eigenfunctions…”

“…Show that w 0 (t) ∈ W 2n 2 ((0, 1), H). Taking into account the theorem on intermediate derivatives [6,14,17], we should show that A 2n w 0 (t) ∈ L 2 ((0, 1), H) and D 2n t w 0 (t) ∈ L 2 ((0, 1), H).…”

“…We show that w(t) ∈ W 2n 2 ((0, 1), H). Taking into account the theorem on intermediate derivatives [6,14,17], we should show that…”

“…Sitnik [17]. The properties of differential operators that have an infinite number of multiple eigenvalues have been studied in the papers [4,6,9,[13][14][15].…”