Oscillation of sublinear difference equations with positive neutral term
type: Item
instance of: scholarly article
title: Oscillation of sublinear difference equations with positive neutral term
zbMATH Open document ID: 1087.39011
DOI: 10.1007/BF02831940
author: N. E. Zubov
published in: Journal of Applied Mathematics and Computing
publication date: 2006-02-20T00:00:00Z
review text: The authors consider the oscillatory behavior of the solutions for the following equation \[ \Delta(x(n)+p(n)x(\tau(n))) + f(n,x(g_1(n)),\ldots,x(g_m(n)))=0 \] where \(\Delta x(n)=x(n+1)-x(n)\); \(p(n)\) is a sequence of real numbers, \(1<p_1\leq p(n)\leq p_2\), \(\tau(n)<n\) is a sequence of strictly increasing integers, \(\lim_{n\rightarrow\infty}\tau(n)=\infty\); \(g_i(n)\) is a sequence of positive integers, \(\lim_{n\rightarrow\infty}g_i(n)=\infty\), \(f\) is continuous in \(x_1\), \(\ldots\), \(x_m\), \(g(n)=\max_{1\leq i\leq m}\{g_i(n)\}\leq\tau(n)\). Let i) \(f\in C([n_o,\infty)\times \mathbb R^m,\mathbb R)\) be nondecreasing in each \(y_i\) and \(y_1f(n,y_1,\ldots,y_m)>0\) for \(y_1y_i>0\); ii) \(f\) be strongly sublinear; iii) \(\sum_n^\infty f(s,a,\ldots,a)=\infty\), \(\forall a>0\). Then every solution of the above equation is oscillatory. The equation has a bounded non-oscillatory solution which is bounded away from 0 if and only if \(\sum_n^\infty | f(s,d,\ldots,d)| <\infty\) for some \(d\neq 0\) provided the assumptions i)-iii) hold. Under the same conditions i)-iii) and if \(g(n)\leq\tau(n)\) then the solutions of the equations are oscillatory if and only if \(\sum_n^\infty | f(s,a,\ldots,a)| =\infty\) for every \(a\neq 0\).
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 5007513
zbMATH Keywords: Sublinear difference equations
zbMATH Keywords: Oscillatory behavior
zbMATH Keywords: bounded solution
reviewed by: Vladimir Răsvan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5007513