The existence of positive solutions to neutral differential equations
type: Item
instance of: scholarly article
title: The existence of positive solutions to neutral differential equations
zbMATH Open document ID: 0945.34065
author: Ravi P. Agarwal
author: Zhicheng Wang
author: X. H. Tang
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-10-03T00:00:00Z
review text: From authors' abstract: We consider a class of neutral differential equations of the form \[ \Biggl[x(t)- {r(t)\over r(t-\tau)} x(t- \tau)\Biggr]'+ Q(t) x(t-\sigma)= 0,\quad t\geq t_0,\tag{\(*\)} \] with \(\tau\in (0,\infty)\), \(\sigma\in [0,\infty)\), \(Q(t)\in C([t_0, \infty),\mathbb{R}^+)\), \(r(t)\in C([t_0, \infty),(0, \infty))\) with \(r(t)\) nondecreasing on \([t_0- \tau,\infty)\). We show that all positive solutions to \((*)\) can be classified into four types, \(A\), \(B\), \(C\), and \(D\). Sufficient and necessary conditions are obtained for the existence of \(A\)-type, \(B\)-type, and \(D\)-type positive solutions to \((*)\), respectively. A sufficient condition for the existence of \(C\)-type positive solutions to \((*)\) is also given. Finally, a sharp oscillation result is offered for all solutions to \((*)\). These results generalize and improve those established in \textit{B. Yang} and \textit{B. G. Zhang} [Funkcial. Ekvacioj, Ser. Int. 39, No. 2, 347-362 (1996; Zbl 0865.34054)].
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K11
zbMATH DE Number: 1417575
zbMATH Keywords: neutral differential equations
zbMATH Keywords: positive solutions
reviewed by: Waldyr Muniz Oliva
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6598
OpenAlex ID: W2039751470
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DOI: 10.1006/JMAA.1999.6598
description: scientific article; zbMATH DE number 1417575