Oscillation tests for delay equations (Q1293289)

The authors study the oscillatory behavior of delay equations of the form \[ x'(t)+ p(t)x(\tau(t))= 0,\quad t\geq T,\tag{1} \] with \(p,\tau\in C([T,\infty),[0,\infty))\), \(\tau(t)\) is decreasing, \(\tau(t)<t\) for \(t\geq T\), \(\lim_{t\to\infty} \tau(t)= \infty\). Let \[ k= \liminf_{t\to\infty} \int^t_{\tau(t)} p(s) ds,\quad L= \limsup_{t\to\infty} \int^t_{\tau(t)} p(s) ds. \] It is proved that when \(L<1\) and \(0< k\leq 1/e\), all solutions to equation (1) oscillate if \[ L> {\ln\lambda_1+ 1\over \lambda_1}- {1-k- \sqrt{1- 2k- k^2}\over 2}, \] where \(\lambda_1\) is the smaller root of the equation \(\lambda= e^{k\lambda}\).