Asymptotic dichotomy in a class of third-order nonlinear differential equations with impulses (Q963184)

The authors consider a third-order nonlinear differential equation with impulses of the form \[ \begin{aligned} & (r(t)x''(t))'+f(t,x)=0,\quad t\geq t_0,\;t\neq t_k,\\ & x^{(i)}(t^+_k)=g_k^{[i]}(x^{(i)}(t_k)), \quad i=0,1,2; \;k=1,2,\dots\\ & x^{(i)}(t^+_0)=x_0^{[i]},\quad i=0,1,2,\dots\end{aligned}\tag{1} \] The authors call a solution \(x(t)\) of (1) nonoscillatory if it is eventually positive or eventually negative. Otherwise it is said to be oscillatory. The authors find sufficiently conditions for the assertion: every solution of (1) either oscillates or tends to zero.