On oscillation of a second order impulsive linear delay differential equation (Q1289053)

The authors consider the scalar delay differential equation \[ \ddot x(t)+\sum_{k=1}^ma_k(t) x(g_k(t))=0, \quad t\geq 0, \tag{1} \] with the impulsive conditions \[ x(\tau_j)=A_jx(\tau_j-0),\quad \dot x(\tau_j)= B_j\dot x(\tau_j-0),\quad j=1,2,\dots \tag{2} \] The main result is the equivalence of the four properties: nonoscillation of the differential equation and a corresponding differential inequality, positiveness of a fundamental function and the existence of a solution to a generalized Riccati inequality. Further, explicit conditions for nonoscillation and oscillation and comparison theorems are presented. The theory of Volterra integral operators is used to prove the existence of a positive fundamental function to (1), (2).