Oscillation criteria of nonlinear hyperbolic equations with functional arguments (Q1586261)

The authors investigate a class of hyperbolic equations with functional arguments \[ \begin{multlined}\left(\partial^2/\partial t^2\left[u+\sum^n_{i=1}\lambda_i(t)u(x,\tau_i(t)\right)\right] = a(t)\Delta u+ \sum^m_{j=1} a_j(t)u\bigl(x,\rho_j(t)\bigr)-p(x,t)u- \\ -\int^b_a q(x,t,\xi) f\biggl(u\bigl[x,g(t,\xi)\bigr]\biggr)d\sigma(\xi),\end{multlined} \] where \((x,t)\in\Omega\times\mathbb{R}_+\). The boundary condition has the form \(\partial u/\partial n + \nu(x,t)u = 0\) and \(u=0\) on \(\partial\Omega\times \mathbb{R}_+\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with a piecewise smooth boundary, \(n\) denotes the unit exterior normal vector to \(\partial\Omega\). The functions \(a(t),a_j(t), \lambda_i(t), \rho_j(t), p(x,t), q(x,t,\xi), f(u)\), \(g(t,\xi), \sigma(\xi)\) \((i = 1,2,\dots,n\); \(j=1,2,\dots,m)\) are continuous with respect to their arguments. Taking into account these assumptions the authors state that there exist oscillations for the solutions of the equation under consideration with two different kinds of boundary condition stated above.