Oscillatory properties of the solutions of nonlinear delay hyperbolic differential equations of neutral type.
type: Item
instance of: scholarly article
title: Oscillatory properties of the solutions of nonlinear delay hyperbolic differential equations of neutral type.
zbMATH Open document ID: 1078.35129
DOI: 10.1007/BF02437652
author: Mengxing He
author: Anping Liu
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 2002-01-01T00:00:00Z
review text: The authors study oscillatory properties of the solutions of nonlinear delay hyperbolic differential equations of neutral type with multi-delays under two type boundary conditions, \[ {\partial^2 \over \partial t^2}[u(t,x)+p(t)u(t-\mu (t),x )]=a(t)h(u)\triangle u+\sum_{i=1}^{m}a_i(t)h_i(u(t-\tau_i (t),x ))\triangle u(t-\tau_i(t),x)- \] \[ -\sum_{j=1}^{n}b_j(t,x)f_j(u(t-\sigma_j (t),x )), \;\;(t,x)\in \mathbb R_+\times \Omega \equiv G. \] Two different cases of boundary problems are considered. The first case is the problem under nonlinear boundary condition, \(\partial u/\partial n =g (t, x, u)\), and the second is \(\partial u/\partial n + cu=0\), on \( \mathbb R_+\times \partial\Omega \), where \(\Omega \) is a bounded domain in \(\mathbb R^N\) with a smooth boundary \(\partial \Omega \), \(\mathbb R_+=[0,\infty )\), \(\triangle \) the Laplacian operator in \(\mathbb R^N\) and \(n\) is the unit outward normal vector on \(\partial\Omega \). It is assumed that \(a,a_i,b_j,g,h^{\prime }(u),h_i^{\prime }(u),f_j(u),\tau_i,\sigma_j,\mu \) are continuous functions in some domains, \(0<\tau_i(t)<\tau \), \(0<\sigma_j(t)<\sigma \), \(0\leq \mu (t)\leq \mu \) (\(c>0,\tau,\sigma,\mu \) are constants, \(i=1,\dots ,m\), \(j=1,\dots ,n\)), \(-1\leq p(t)<0\) is two times differentiable with respect to \(t\) in \(\mathbb R_+\), \(uh(u)g(t,x,u)<0\), \(uh_i(u)g(t,x,u)<0\), \(uh_i^{\prime }(u)\geq 0\), \(uh^{\prime }(u)\geq 0\), \(f_j(u)/u\geq C_j= \text{const} >0\) (\(u\neq 0\)). The main result is that all solutions of the problems under consideration are oscillating in \(G\). Several modified statements for existence of oscillations follow from the main result. The results fully indicate that the oscillations are caused by delay and hence reveal the difference between these equations and those equations without delay. Two interesting examples illustrate the theory.
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1880780
zbMATH Keywords: nonlinear delay
zbMATH Keywords: existence of oscillations
reviewed by: Drumi Bainov
MaRDI profile type: Publication
cites work: Oscillation of hyperbolic functional differential equations with deviating arguments
cites work: Q3836942
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full work available at URL: https://doi.org/10.1007/bf02437652
OpenAlex ID: W1974408252
description: scientific article; zbMATH DE number 1880780