On Liapunov type inequality for certain second order quasi-linear differential equations
type: Item
instance of: scholarly article
title: On Liapunov type inequality for certain second order quasi-linear differential equations
zbMATH Open document ID: 1114.34010
author: Sihu Hu
author: Zhaohui Yuan
author: Lihong Huang
published in: Southeast Asian Bulletin of Mathematics
publication date: 2006-12-06T00:00:00Z
review text: A Lyapunov-type inequality is established for differential equations of the form \[ [|y'(t)|^\sigma \text{sgn }y']' +q(t)f(y(t))=0,\quad t\geq t_0\geq0, \tag{\(*\)} \] where \(\sigma>0\) is a constant, \(q\) is a real-valued continuous function on \([t_0,\infty), t_0\geq0\) and \(f\) is a real-valued continuous function on \((-\infty,\infty)\) with \(0<y f(y)\leq r|y|^{\sigma+1}, y\neq0\). The main result of the paper runs as follows: If \(y(t)\) is a solution of (\(*\)) with \(y(a) = y(b) = 0\) (\(a < b\)) and \(y(t)\neq0\) for \(t\in (a,b)\), then \(\int_a^b q^+(t)dt>\frac{2^{\sigma+1}}{r(b-a)^\sigma}\), where \(q^+(t)=\max\{q(t),0\}\). Some results related to the number of zeros of a solution of (\(*\)) in an interval \([t_0,T]\) and the distance between two consecutive zeros of a solution of (\(*\)) are obtained.
Mathematics Subject Classification ID: 34A40
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 5077535
zbMATH Keywords: Lyapunov-type inequality
zbMATH Keywords: distance between two zeros of a solution
zbMATH Keywords: second order quasi-linear differential equations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5077535