Oscillation theorems for certain nonlinear differential equations of second order
type: Item
instance of: scholarly article
title: Oscillation theorems for certain nonlinear differential equations of second order
zbMATH Open document ID: 1057.34019
DOI: 10.1016/S0898-1221(04)90013-2
author: Aydin Tiryaki
author: Bülent Ayanlar
publication date: 2004-06-15T00:00:00Z
review text: The authors are concerned with the general second-order nonlinear differential equation \[ [ a(t)\Psi(x(t))k(x^{\prime}(t))] ^{\prime} +q(t)f(x(t))=r(t),\tag{1} \] with \(q,r\in C[ [t_{0},\infty),\mathbb{R}] ,\) \(a\in C[ [t_{0},\infty),\mathbb{R}^{+}] ,\) \(\Psi\in C[ \mathbb{R} ,\mathbb{R}^{+}] ,\) \(k\in C[ \mathbb{R},\mathbb{R}] ,\) and \(f\in C^{1}[ \mathbb{R},\mathbb{R}] .\) The research follows closely the recent paper by \textit{J. V. Manojlovic} [Acta Sci. Math. 65, No. 3--4, 515--527 (1999; Zbl 0957.34037)], where under certain assumptions on the coefficients it has been shown that solutions of the nonlinear differential equation \[ [ p(t)g(x(t))x^{\prime}(t)] ^{\prime}+q(t)f(x(t))=r(t)\tag{2} \] are either oscillatory or satisfy \[ \liminf_{t\to\infty}| x(t)| =0. \] The main goal of the paper under review, according to the authors, is to present new oscillation criteria and to show that some of Manojlovic's results contain the superfluous condition. Results of the paper are proved using the so-called integral averaging technique similar to that developed by \textit{H. J. Li} [J. Math. Anal. Appl. 194, No. 1, 217--234 (1995; Zbl 0836.34033)]. However, it has been pointed out by O. Došlý in his review on the paper by \textit{J. V. Manojlovic} [Acta Sci. Math. 65, No. 3--4, 515--527 (1999; Zbl 0957.34037)] that a certain assumption in the oscillation criteria of Li is not necessary as observed by \textit{Yu. V. Rogovchenko} [J. Math. Anal. Appl. 203, No. 2, 560--563 (1996; Zbl 0862.34024)]. The criteria of the reviewed paper contain a similar assumption and it seems that this assumption can be removed without affecting the results of the paper. The authors just repeat these comments on several occasions referring to the assumption \[ \int_{t_{0}}^{t}a(s)h^{2}(t,s)\,ds<\infty\qquad\text{for }t\geq t_{0}. \] It is not clear whether the study of equation (1) has been motivated by any real-world applications since all examples in the paper are artificial, while …
reviewed by: Svitlana P. Rogovchenko
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 2075448
zbMATH Keywords: oscillation
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: nonlinear differential equation
zbMATH Keywords: integral averaging technique
MaRDI profile type: Publication
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full work available at URL: s0898-1221(04)90013-2
OpenAlex ID: W2076793670
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2075448