On solutions of linear differential equations with entire coefficients
type: Item
instance of: scholarly article
title: On solutions of linear differential equations with entire coefficients
zbMATH Open document ID: 1215.34111
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-05-10T00:00:00Z
review text: The first result of the paper concerns the effect of a perturbation of the entire coefficients~\(A_j\) of certain homogeneous linear differential equations \[ y^{(k)}+\sum_{j=0}^{k-2} A_{j}y^{(j)}=0 \quad \text{with } k\geq 2 \tag{1} \] on the oscillation of the solutions~\(y\). In fact, Theorem~1.1 considers small perturbations of the coefficients \(A_j\) (for \(1\leq j\leq k-2\)) of equation~(1) with a solution \(f\) whose zeros have finite exponent of convergence, say \(\lambda(f)\), into some entire coefficients~\(B_j\) (again for \(1\leq j\leq k-2\)) of the equation \[ y^{(k)}+\sum_{j=0}^{k-2}B_{j}y^{(j)}=0 \tag{2} \] and concludes that equation~(2) cannot have any solution whose zeros have finite exponent of convergence, if, for example, either of the following two hypotheses hold: (a) The function \(A_0\) is transcendental of order~\(0\), while \(B_0-A_0\) and the \(A_j\) and \(B_j\) (for \(1\leq j\leq k-2\)) are polynomials, and \(f\) has finitely many zeros. (b) The function \(A_0\) has order, say \(\rho(f)\), less than~\(1/2\), while \(B_0-A_0\) and the \(A_j\) and \(B_j\) (again for \(1\leq j\leq k-2\)) have order less than \(\rho(A_0)\), and \(\lambda(f)<\rho(A_0)\). Let us try to present the statement of Theorem~1.1 without any specific expressions. In this theorem, the authors treat the case when \(A_0,\dots,A_{k-2}\) are entire functions of finite order and the transcendental function \(A_0\) dominates the other functions \(A_j\) in a sense and equation (1) has a certain solution~\(f\) with \(\lambda(f)<\infty\) and, in addition, the distribution of its zeros is negligible in comparison with the growth of the dominant coefficient \(A_0\). Then, they show that any small perturbed equation (2) of (1) with entire coefficients \(B_j\) (for \(0\leq j\leq k-2\)) cannot have a solution \(g\) with \(\lambda(g)<\infty\), provided that \(A_0\) dominates again all the \(B_j\) (for \(1\leq j\leq k-2\)) and the difference \(B_0-A_0\) does not vanish identically bu…
reviewed by: Kazuya Tohge
Mathematics Subject Classification ID: 34M10
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34M03
zbMATH DE Number: 5706215
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: linear differential equations
zbMATH Keywords: zeros
author: Abdullah Alotaibi
author: James K. Langley
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.01.054
OpenAlex ID: W1974519720
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DOI: 10.1016/J.JMAA.2010.01.054
description: scientific article; zbMATH DE number 5706215