Peculiarities in power type comparison results for half-linear dynamic equations
type: Item
instance of: scholarly article
title: Peculiarities in power type comparison results for half-linear dynamic equations
zbMATH Open document ID: 1291.34164
DOI: 10.1216/RMJ-2012-42-6-1995
published in: Rocky Mountain Journal of Mathematics
publication date: 2013-04-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1361800616
review text: This paper deals with the nonoscillatory properties of the second order half-linear dynamic equation \[ \big(r(t)\,\Phi_p(y^\Delta)\big)^\Delta+c(t)\,\Phi_p(y^\sigma)=0, \tag{\(1_p\)} \] where \(\Phi_p(y):=|y|^{p-1}\mathrm{sgn} y\) and \(p>1\). The function \(r(t)\) is positive and \(1/r(t)\) is rd-continuous. The author derives criteria for two such equations (1\(_p\)) and (1\(_\alpha\)) which guarantee that the nonoscillation of equation (1\(_p\)) is preserved when \(p\) is changed to \(\alpha\). Two separate cases are considered, namely \[ \int^\infty r^{1-q}(t)\,\Delta t=\infty \quad\text{and}\quad \int^\infty r^{1-q}(t)\,\Delta t<\infty, \] where \(q\) is the conjugate exponent to \(p\) (i.e., \(1/p+1/q=1\)). The author shows how the statements depend on the behavior of the graininess \(\mu(t)\) for large \(t\) (``small'' or ``big'' graininess). In particular, the results simplify considerably when \(\mu(t)\geq1\) for large \(t\). New results are obtained even for special time scales with constant graininess \(\mu(t)\equiv h>0\). This is an interesting paper which enriches the theory of half-linear differential/difference/dynamic equations.
reviewed by: Roman Šimon Hilscher
Mathematics Subject Classification ID: 34N05
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 6152196
zbMATH Keywords: half-linear dynamic equation
zbMATH Keywords: time scale
zbMATH Keywords: comparison theorem
author: Pavel Rehak
MaRDI profile type: Publication
cites work: Q4664430
cites work: Q5191572
cites work: Q3392725
cites work: Q2723195
cites work: Analysis on measure chains - a unified approach to continuous and discrete calculus
cites work: Q3630630
cites work: Q4804717
cites work: On certain comparison theorems for half-linear dynamic equations on time scales
cites work: Q3378965
cites work: Comparison of nonlinearities in oscillation theory of half-linear differential equations
cites work: Q3081876
cites work: Comparison theorems for oscillation of second order half-linear differential equations
OpenAlex ID: W2095188884
description: scientific article; zbMATH DE number 6152196