Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations
type: Item
instance of: scholarly article
title: Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations
zbMATH Open document ID: 1082.34030
author: Devrim Çakmak
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-01-11T00:00:00Z
review text: The paper studies oscillation of the nonlinear differential equation \[ (r(t)\Psi (y(t))| y^{\prime }(t)| ^{\alpha -2}y^{\prime }(t))^{\prime }+q(t)f(y(t))g(y^{\prime }(t))=0,\;\alpha>1. \] by averaging of certain functional quantities on a sequence of intervals in the nonnegative half-axis. It is assumed that \(\Psi (y)\), \(yf(y)>0\) for all nonzero \(y\), and \(g(y)\geq K>0\) for a certain constant \(K\). By introducing the Philos-like kernel \(H(t,s)\), a thorough analysis of the functional quantities \(A(h;s,t,\rho )=\int_s^t\rho (v)H(t,v)h(v)\,dv\), \(t\geq s\), and \(B(h;t,s,\rho )=\int_t^s\rho (v)H(t,v)h(v)\,dv\), \(t\leq s\), is performed in a series of technical lemmas. For instance, if a solution \(y(t)\) is positive on \([c,b)\) then the quotient \(u(t)=[r(t)\Psi (y(t))| y^{\prime }(t)| ^{\alpha -2}y^{\prime }(t)]/f(y(t))\) satisfies the inequality \[ A(Kq;c,b,\rho )\leq \rho (c)H(b,c)u(c)+A((r/M^{\alpha -1})(\alpha ^{-1}| \lambda _2H^{-1/\alpha }-(\rho ^{\prime }/\rho )| )^{(\alpha )};c,b,\rho ). \] Here, \(f\) is continuously differentiable, \(f^{\prime }(y)[\Psi (y)| f(y)| ^{\alpha -2}]^{1/(1-\alpha )}\geq M>0\) and \(\rho \) is positive and continuously differentiable. The proofs are based on an integration by parts technique and the maximization of certain functions. Theorem 2.1 reads as: Assume that the above conditions hold and that for some \(c\in (a,b)\), \(H\) in the class of Philos-like kernels and \(\rho >0\), one has \[ \begin{multlined} H(c,a)^{-1}B(Kq;a,c,\rho )+H(b,c)^{-1}A(Kq;c,b,\rho )>\\ M^{1-\alpha }\alpha ^{-\alpha }[H(c,a)^{-1}B(r| \lambda _1H^{-1/\alpha }+(\rho ^{\prime }/\rho )| )^{(\alpha )};a,c,\rho )\\+H(b,c)^{-1}A((r| \lambda _2H^{-1/\alpha }-(\rho ^{\prime }/\rho )| )^{(\alpha )};c,b,\rho )].\end{multlined} \] Then, every solution of the differential equation has a zero in \((a,b)\). This result is further used in connection with a condition for the existence of an infinity of such intervals \((a,b)\) thus ensuring the oscill…
reviewed by: Octavian Mustafa
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 2126374
zbMATH Keywords: interval oscillation criteria
zbMATH Keywords: nonlinear differential equation
zbMATH Keywords: integral averaging technique
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.06.046
OpenAlex ID: W1968686939
cites work: Q3149679
cites work: Stability properties of \((r(t)u')'+a(t)f(u)g(u') = 0\)
cites work: Oscillation criteria for certain forced second-order nonlinear differential equations.
cites work: An Oscillation Criterion for a Forced Second-Order Linear Differential Equation
cites work: An oscillation criterion for certain second order strongly sublinear differential equations
cites work: Q4106500
cites work: Boundedness and convergence to zero of solutions of a forced second-order nonlinear differential equation
cites work: Oscillation and nonoscillation of second order linear differential equations
cites work: Q4061271
cites work: Interval criteria for oscillation of second-order linear ordinary differential equations
cites work: Q4225208
cites work: Oscillation of second order matrix differential systems
cites work: Integral inequalities and second order linear oscillation
cites work: On boundedness of solutions of certain second-order differential equations
cites work: Oscillation criteria for second order linear differential equations
cites work: Oscillation of certain second-order nonlinear differential equations
cites work: Interval oscillation criteria for second-order nonlinear differential equations with damping
cites work: Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations
cites work: Interval oscillation criteria for second-order nonlinear differential equations
cites work: Interval oscillation criteria for a forced second order nonlinear ordinary differential equation
cites work: An oscillation criterion for nonhomogeneous half-linear differential equations
cites work: Interval oscillation criteria for nonlinear second-order differential equations
cites work: Oscillation theorems for linear differential equations of second order
cites work: Oscillation criteria for certain nonlinear differential equations
cites work: On the oscillation of certain second-order nonlinear differential equations
cites work: Some properties of solutions of \(u''(t)+a(t)f(u)g(u') = 0\). II
cites work: Oscillation criteria for a forced second-order linear differential equation
cites work: On Kamenev-type oscillation theorems for second-order differential equations with damping
cites work: Note on Wong's paper
DOI: 10.1016/J.JMAA.2004.06.046
description: scientific article; zbMATH DE number 2126374