Comparison theorems and necessary/sufficient conditions for the existence of nonoscillatory solutions of forced impulsive differential equations with delay (Q352165)

The paper deals with nonlinear differential systems with delay and ``impulsive effects'' \[ (r(t)x'(t))'+F(t,x(g(t)))=e(t),\quad t\in[0,\infty)\setminus \varUpsilon, \tag{1} \] \[ x(t_k^+)=a_kx(t_k), \quad x'(t_k^+)=b_kx'(t_k),\quad\quad k\in\mathbb{N}, \tag{2} \] where \(\varUpsilon=\{t_1,t_2,\cdots\}\) is a set of real numbers such that \(0=t_0<t_1<t_2<\dots\) and \(\lim_{k\to\infty}t_k=\infty\). Moreover, for each \(k\in\mathbb N\), \(a_k>0\) and \(b_k>0\). The authors introduce a comparison principle, compare forced impulsive equations with unforced impulsive equations and thus they offer necessary and sufficient conditions for nonoscillatory solutions to (1), (2).