Nonlinear systems with impulse perturbations (Q2784692)

The authors consider the impulsive system NEWLINE\[NEWLINE(\phi_\beta(x'))'+ f(x)= 0,\quad t\neq t_n,\quad x(t_n+ 0)= x(t_n),\;x'(t_n+ 0)= b_n x(t_n),\tag{1}NEWLINE\]NEWLINE with \(0\leq t_1< t_2<\cdots< t_n< t_{n+1}<\cdots\), \(t_n\to \infty\) \((n\to\infty)\), \(0\leq b_n\leq 1\), \(n= 1,2,3,\dots\); \(\phi_\beta(u)= |u|^\beta \text{sgn }u\), \(\beta> 0\), \(f: \mathbb{R}\to \mathbb{R}\) is continuous and \(uf(u)> 0\), \(u\neq 0\). If \(t_n- t_{n-1}\), \(n=2,3,\dots\), is small enough, then sufficient conditions for the existence of nonoscillatory solutions to (1) are derived. If NEWLINE\[NEWLINE\sum^\infty_{n=1} (t_{n+ 1}- t_n)^{1+{1\over\beta}}= \inftyNEWLINE\]NEWLINE then every nonoscillatory solution tends to zero as \(t\to\infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00016].