On the qualitative behaviour of solutions to certain second order nonlinear differential equation with delay
type: Item
instance of: scholarly article
title: On the qualitative behaviour of solutions to certain second order nonlinear differential equation with delay
zbMATH Open document ID: 1387.34096
published in: Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche
publication date: 2018-01-18T00:00:00Z
review text: In this paper, the authors consider the following two nonlinear differential equation of second order without and with constant delay, respectively \[ x''+a(t)f(x,x')+g(x)=p(t,x,x') \] and \[ x''+a(t)f(x,x')+g(x(t-\tau))=p(t,x,x'), \] where \(\tau\) is a constant delay, \(x\in\mathbb{R}\), \(\mathbb{R}=(-\infty,\infty)\), \(t\in\mathbb{R}^+\), \(\mathbb{R}^+=[0,\infty)\), \(a,f,g\) and \(p\) are continuous functions of the arguments shown explicitly such that the derivatives \(a'(t)=\frac{d}{dt}a(t),\) and \(g'(x)=\frac{dg}{dx}\) exist and are continuous. In addition, it is assumed that the functions \(f,g\) and \(p\) satisfy a Lipschitz condition with respect to their arguments, except \(t\). Hence, the existence and uniqueness of the solutions of the equations considered are guaranteed. In this paper, the Lyapunov's direct method is employed to obtain sufficient conditions for globally asymptotically stability of the zero solution of the first equation and the boundedness solutions and their derivatives of first order for the given first and second differential equations. The obtained results are new and complement existing results in the literature. Two good examples are given to illustrate the main results.
Mathematics Subject Classification ID: 34K12
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 6827405
zbMATH Keywords: boundedness
zbMATH Keywords: delay differential equation
zbMATH Keywords: global asymptotic stability
zbMATH Keywords: Lyapunov function
author: Babatunde Sunday Ogundare
author: A. T. Ademola
author: M. O. Ogundiran
author: Olufemi Adeyinka Adesina
reviewed by: Cemil Tunç
MaRDI profile type: Publication
full work available at URL: s11565-016-0262-y
OpenAlex ID: W2529333715
Wikidata QID: Q115377865
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DOI: 10.1007/S11565-016-0262-Y
description: scientific article; zbMATH DE number 6827405