The nonoscillatory solutions of delay differential equations with impulses
The author proves that eventually positive (negative) solutions of the scalar delay differential equation \(x'(t)=- \sum^n_{i=1} p_i(t) x(t-r_i(t))\), where \(p_i,r_i\in C([t_0,+\infty)), \mathbb{R}_+)\), \(\lim_{t\to+\infty} (t-r_i(t))= +\infty\), conserve their nonoscillatory properties under impulsive perturbations of the form \(x(t^+_k)- x(t_k)= I_k(x(t_k))\) with \(I_k(-u)= -I_k(u)\), \(uI_k(u)\geq 0\), \(\forall u\in \mathbb{R}\), \(k\in\mathbb{N}\), and \(t_k< t_{k+1}\to+\infty\) as \(k\to+\infty\).
impulsive differential equation with variable delays
scientific article; zbMATH DE number 911908
scientific article