Oscillation criteria for certain hyperbolic functional differential equations with Robin boundary condition
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.
scientific article; zbMATH DE number 1807638