Oscillation criteria for certain hyperbolic functional differential equations with Robin boundary condition
type: Item
instance of: scholarly article
title: Oscillation criteria for certain hyperbolic functional differential equations with Robin boundary condition
zbMATH Open document ID: 1010.35110
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2003-01-26T00:00:00Z
review text: The author studies the oscillation of the following hyperbolic functional-differential equations with Robin boundary condition: \[ \partial /\partial t \{p(t)\partial /\partial t(u+\sum_{i=1}^{n}p_i(t)u(x,\tau_i(t)))\}= \] \[ a(t)\triangle u+\sum_{j=1}^{n}a_j(t)\triangle u(x,\rho_j(t))-h(x,t)u-\int_{a}^{b}q(x,t,\zeta)f(u[x,g(t,\zeta)])d\sigma (\zeta). \] A new method of investigation is used which differs from that introduced by Wang and Yu (1999). To investigate the equation with Robin boundary condition, \(\partial u /\partial n +\beta (x)u(x,t) =0\) (\(x\in\partial \Omega \)), one makes use of the Robin eigenvalue problem \(\triangle \Phi +\lambda \Phi =0\) (\(x\in\Omega \)), \(\partial \Phi /\partial n +\beta (x)\Phi =0\) (\(x\in\partial \Omega \)). Here \(\triangle \) is the Laplacian in \(\mathbb{R}^n\), \(\Omega \) is a bounded domain in \(\mathbb{R}^n\) with a piecewise continuous smooth boundary \(\partial \Omega \), \(n\) denotes the unit exterior vector normal to \(\partial \Omega \) and the Stieltjes integral is considered. Several oscillatory criteria are obtained if \(p_i(t)\), \(p(t)\) satisfy certain inequalities.
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 1807638
zbMATH Keywords: nonlinear delay hyperbolic equation
zbMATH Keywords: Robin boundary condition
author: Lihu Deng
reviewed by: Drumi Bainov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1807638