Oscillation of second order nonlinear dynamic equations on time scales
type: Item
instance of: scholarly article
title: Oscillation of second order nonlinear dynamic equations on time scales
zbMATH Open document ID: 1075.34028
author: Martin Bohner
author: S. H. Saker
published in: Rocky Mountain Journal of Mathematics
publication date: 2005-03-10T00:00:00Z
review text: The authors consider the nonlinear second order dynamic equation \[ (p(t)x^\Delta)^\Delta+q(t)(f\circ x^\sigma)=0,\tag{1} \] where \(p\) and \(q\) are positive, real-valued continuous functions, and the nonlinearity \(f:\mathbb{R}\to\mathbb{R}\) satisfies the sign condition \(xf(x)>0\) and the superlinearity condition \(f(x)>K x\) for some \(K>0\) and every \(x\neq 0\). Two cases, depending on the convergence of the integral \[ \int _1^\infty\frac 1{p(t)}\Delta t\tag{2} \] are discussed separately. New sufficient conditions involving the integral over the coefficients of equation (1) which guarantee that all solutions are oscillatory (in the case when (2) is divergent) or either oscillatory or convergent to zero (in the case of convergence of the integral (2)) are derived. The sharpness of these criteria is shown on the example of the Euler dynamic equation. The authors' main tool is the Riccati transformation.
reviewed by: Robert Mařík
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 2143224
zbMATH Keywords: dynamic equation
zbMATH Keywords: Riccati transformation
zbMATH Keywords: Riccati equation
zbMATH Keywords: oscillation
zbMATH Keywords: Kamenev-type criteria
zbMATH Keywords: Euler dynamic equation
zbMATH Keywords: measure chain
zbMATH Keywords: time scales
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069797
OpenAlex ID: W2038662746
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DOI: 10.1216/RMJM/1181069797
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description: scientific article; zbMATH DE number 2143224