Oscillations of impulsive delay differential equations (Q5926755)

The authors consider the second-order impulsive delay differential equation \[ (r(t)x'(t))'+P(t)f(x(t-\tau))=0,\quad t\geq t_0,\;t\neq t_k,\;k=1,2,\dots, \] \[ x(t_k^+)=g_k(x(t_k)),\quad x'(t_k^+)=h_k(x'(t_k)),\quad k=1,2,\dots,\tag{1} \] with \(0\leq t_0<t_1<\dots <t_k<\dots\) and \(\lim_{k\to\infty}t_k=\infty\); \(\tau\) is a positive constant, \(r(t)>0\), \(P(t)\geq 0\), \(xf(x)>0\) for all \(x\neq 0\), \(f'(x)\geq 0\); \(g_k,\;h_k\) are positive functions. Sufficient conditions for the oscillation of all solutions to equation (1) are presented in two theorems and three corollaries.