Comparison theorems for linear dynamic equations on time scales.
type: Item
instance of: scholarly article
title: Comparison theorems for linear dynamic equations on time scales.
zbMATH Open document ID: 1034.34042
DOI: 10.1016/S0022-247X(02)00390-6
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The authors study comparison theorems for second-order linear dynamic equations on a time scale \[ \begin{aligned} [p(t)x^{\Delta}(t)]^{\Delta} &+ q(t)x^{\sigma}(t)=0, \tag{1}\\ [p(t)y^{\Delta}(t)]^{\Delta} &+ a^{\sigma}(t)q(t)y^{\sigma}(t)=0,\tag{2}\\ [p(t)z^{\Delta}(t)]^{\Delta} &+ a(t)q(t)z^{\sigma}(t)=0,\tag{3} \end{aligned} \] where \(p(t)>0\) and \(p,q,a\) are right-dense continuous on \(\mathbb{T}\). Three different comparison theorems are presented along with their coresponding corollaries, and, by examples, it is shown that they are all independent. A typical result is the following theorem: If \(a\in C^1_{rd}\), \(\liminf_{t\to\infty} \int_T^t q(s)\,\Delta s\geq 0\) but not identically zero for large \(T\), \(\int^\infty\frac{\Delta s}{p(s)}=\infty\) and \(0<a(t)\leq 1\), \(a^{\Delta}(t)\leq 0\). Then (1) is nonoscillatory implies (3) is nonoscillatory. The obtained results extend comparison theorems for the continuous case and provide some new results in the discrete case.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 1866693
zbMATH Keywords: linear dynamic equation
zbMATH Keywords: oscillation
zbMATH Keywords: nonoscillation
zbMATH Keywords: comparison theorems
author: Lynn Erbe
author: Allan Peterson
author: Pavel Rehak
reviewed by: Zuzana Došlá
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00390-6
OpenAlex ID: W2040538401
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description: scientific article; zbMATH DE number 1866693