On the completeness of elementary solutions of a class of second-order differential equations with operator coefficients
type: Item
instance of: scholarly article
title: On the completeness of elementary solutions of a class of second-order differential equations with operator coefficients
zbMATH Open document ID: 1201.34133
DOI: 10.1134/S0001434609110170
author: F. A. Gulieva
author: S. S. Mirzoev
published in: Mathematical Notes
publication date: 2010-02-05T00:00:00Z
review text: Consider the boundary value problem \(d^2u(t)/dt^2+(pA+A_1)(du/dt)+(qA^2+A_2)u(t)=0\), \(t\in\mathbb R_+\), \(u(0)=\varphi\), where \(u:\mathbb R_+\to H\), \(\varphi\in H\), \(H\) is a given Hilbert space, and the coefficients satisfy: (1)~\(A\) is a positive definite self-adjoint operator with completely continuous inverse; (2)~\(p,q\in\mathbb R\), \(q<0\); (3)~\(B_1=A_1A^{-1}\) and \(B_2=A_2A^{-2}\) are linear bounded operators on~\(H\). Derivatives are regarded in the sense of distributions. The authors indicate conditions on the coefficients of the given problem which ensure its regular solvability, the completeness in the trace space \(H_{3/2}\) of the system of eigenvectors and associated vectors corresponding to the eigenvalues from the left half-plane, and the completeness of the decreasing elementary solutions in the space of regular solutions of the given problem. No proofs are given.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 5666573
zbMATH Keywords: second-order differential equation
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: operator pencil
zbMATH Keywords: entire function
MaRDI profile type: Publication
cites work: Q5614166
description: scientific article; zbMATH DE number 5666573