On a class of second order quasilinear ordinary differential equations
type: Item
instance of: scholarly article
title: On a class of second order quasilinear ordinary differential equations
zbMATH Open document ID: 0835.34034
author: Motohiko Kitano
author: Kusano Takaŝi
published in: Hiroshima Mathematical Journal
publication date: 1996-04-15T00:00:00Z
review text: Consider an equation (A) \((|y' |^{\alpha - 1} y')' + q(t) |y |^{\beta - 1} y = 0\), where \(\alpha\) and \(\beta\) are positive constants, and \(q : [a, \infty) \to \mathbb{R}\) is a continuous function. If \(\alpha = 1\) then (A) reduces to the Emden-Fowler type equation (C) \(y'' + q(t) |y |^{\beta - 1} y = 0\). In the first part of the paper the authors discuss the question of global existence and uniqueness of the initial value problem (A), (B) \(y(t_0) = y_0\), \(y'(t_0) = y_1\). Particularly, the following results is proved: If \(q(t)\) is positive on \([a, \infty)\) and \(q(t)\) has either finite positive variation of finite negative variation on any compact subinterval of \([a, \infty)\), then for any \(y_0, y_1\), there exists a unique solution of the initial value problem (A), (B) defined on the entire interval \([a, \infty)\). In the second part of the paper the authors study the problem of oscillation and nonoscillation of solutions of (A). They are concentrated on the case where the function \(q(t)\) may change its sign infinitely often on \([a, \infty)\). It is shown that all proper solutions of (A) are oscillatory if \[ \int^\infty_a q(t)dt = \lim_{T \to \infty} \int^T_a q(t)dt = \infty. \] This result generalizes \textit{P. Waltman}'s oscillation theorem [J. Math. Anal. Appl. 10, 439-441 (1965; Zbl 0131.089)] for the Emden-Fowler equation (C). In the case of conditional integrability of \(q(t)\), i.e. if \[ \int^\infty_a q(t)dt = \lim_{T \to \infty} \int^T_a q(t)dt \quad \text{exists and is finite}, \tag{*} \] the authors suppose \[ Q(t) : = \int^\infty_t q(s)ds \geq 0 \tag{**} \] and obtain a partial generalization of a result of \textit{M. Naito} [Proc. Am. Math. Soc. 109, 769-774 (1990; Zbl 0701.34044)]. A necessary and sufficient condition for (A) to have a minimal (or maximal) nonoscillatory solution are given in the case \((*)\), which generalizes a theorem of \textit{M. Naito} [Trans. Am. Math. Soc. 282, 577-588 (1984; Zbl 0556.34055)]. Final…
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 811375
zbMATH Keywords: quasilinear differential equations
zbMATH Keywords: Emden-Fowler type equation
zbMATH Keywords: global existence
zbMATH Keywords: uniqueness
zbMATH Keywords: initial value problem
zbMATH Keywords: oscillation
zbMATH Keywords: nonoscillation
reviewed by: J. Kalas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206127714
OpenAlex ID: W2136702328
Wikidata QID: Q128871536
DOI: 10.32917/HMJ/1206127714
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description: scientific article; zbMATH DE number 811375