On Lyapunov inequality in stability theory for Hill's equation on time scales
type: Item
instance of: scholarly article
title: On Lyapunov inequality in stability theory for Hill's equation on time scales
zbMATH Open document ID: 0971.39005
DOI: 10.1155/S1025583400000333
author: Ferhan M. Atıcı
author: Billûr Kaymakçalan
author: Gusein Sh. Guseinov
published in: Journal of Inequalities and Applications
publication date: 2001-10-30T00:00:00Z
full work available at URL: https://eudml.org/doc/121338
review text: Let \(T\) be a closed subset of \(\mathbb{R}\). If there exists a positive number \(\Lambda\) such that \(t+n\Lambda\in T\) for all \(t\in T\) and all integers \(n\), then \(T\) is called a periodic time scale with period \(\Lambda\). Consider the linear \(\Delta\)-differential equation \[ \bigl[p(t) y^\Delta(t) \bigr]^\Delta +q(t)y\bigl( \sigma(t)\bigr) =0,\;t\in T,\tag{1} \] where \(\sigma\) is the forward jump operator and \(p(t)>0\) and \(q(t)\) are real smooth \(\Lambda\)-periodic functions defined on \(T\). It is shown that if the nontrivial \(q=q(t)\leq 0\), then all solutions of (1) are unbounded on \(T\); and if \[ \int^\Lambda_0 q(t) \Delta t\geq 0,\;q\neq 0, \] and \[ \left[\max_{t \in[0,\rho (\Lambda)]} {\sigma(t)-t\over p(t)}+ \int^\Lambda_0 {\Delta t\over p(t)}\right] \int^\Lambda_0 \max \bigl\{ q(t), 0\bigr\}\Delta t\leq 4, \] where \(\rho\) is the backward jump operator, then all solutions of (1) are bounded on \(T\).
reviewed by: Sui Sun Cheng
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1563281
zbMATH Keywords: Lyapunov inequality
zbMATH Keywords: Hill's equation
zbMATH Keywords: time scale
zbMATH Keywords: bounded solution
zbMATH Keywords: unbounded solution
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: linear \(\Delta\)-differential equation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1563281