Impulsive differential systems and the pulse phenomena (Q1825334)

Consider the impulsive differential system \[ (1)\quad x'=f(t,x),\quad t\neq \tau_ k(x),\quad x(t_ 0)=x_ 0,\quad t\geq 0,\quad \Delta x=I_ k(x),\quad t=\tau_ k(x), \] where \(f\in C[{\mathbb{R}}_+\times \Omega,{\mathbb{R}}^ n]\), \(\Omega \subset {\mathbb{R}}^ n\) be an open set. The solution of this system may hit a given surface, \(t=\tau_ k(x)\), finite or infinite number of times causing rhythmical beating, so called pulse phenomena. The authors give conditions such that every solution of (1) meets any given surface \(t=\tau_ k(x)\) at most once or exactly once, or several times.