Properties of solutions of a class of third-order linear differential equations
type: Item
instance of: scholarly article
title: Properties of solutions of a class of third-order linear differential equations
zbMATH Open document ID: 0705.34032
DOI: 10.1007/BF01848121
author: R. S. Dahiya
author: V. Sree Hari Rao
published in: Periodica Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: Consider the third-order differential equation \((1)\quad (r(ry')')'+(q_ 1y)'+q_ 2y'=0\) and its adjoint equation \((2)\quad (r(rz')')'+(q_ 2z)'+q_ 1z'=0,\) where r, \(q_ i\) for \(i=1,2\), are real valued continuous functions on \(I=[a,\infty)\) with \(r(t)>0\) on I. Equation (1) is said to be \(C_ 1\) (or \(C_ 2)\) on I, if for each \(c\in (a,\infty)\) and nontrivial solution y(t) of (1) satisfying \(y(c)=y'(c)=0\), we have y(t)\(\neq 0\) for \(t\in [a,c)\) (or \(t\in (c,\infty))\). Obviously, \(C_ 1\) implies nonoscillation. The authors give the following results: If \(\int^{\infty}_{a}(r(s))^{-1}ds=\infty,\) \((q_ 2-q_ 1)'\leq 0\) (or \((q_ 2-q_ 1)'\geq 0)\), and \((q_ 2-q_ 1)'\) does not vanish identically on any interval, then equation (1) is \(C_ 1\) (or \(C_ 2)\). Equation (1) is \(C_ 1\) \((C_ 2)\) if and only if equation (2) is \(C_ 2\) \((C_ 1)\). The authors also give sufficient conditions for all solutions of (1) which have a zero on I to be oscillatory and sufficient conditions for all oscillatory solutions of (1) to be bounded on I.
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 4156953
zbMATH Keywords: bounded solutions
zbMATH Keywords: third-order differential equation
zbMATH Keywords: oscillatory solutions
reviewed by: Tsai-Sheng Liu
MaRDI profile type: Publication
cites work: Some oscillation properties of third order linear homogeneous differential equations
cites work: On the question of distribution of zeros of solutions of linear differential equations of third order
cites work: Canonical forms for third-order linear differential equations
cites work: On the Disconjugate Behavior of $y''' + py = 0$
cites work: Integral comparison theorems for third order linear differential equations
cites work: Oscillation criteria for third-order linear differential equations
cites work: Subfunction and second-order ordinary differential inequalities
cites work: Q4119477
cites work: The behavior of solutions of the differential equation \(y'''+p(x)y'+q(x)y = 0\)
cites work: Q5584775
description: scientific article; zbMATH DE number 4156953