Oscillations of second-order nonlinear impulsive ordinary differential equations

Abstract

Consider the second-order impulsive ordinary differential equation (r(t)(x′(t)) σ)′+f(t,x(t))=0, t⩾t 0, t≠t k, k=1,2,…, x(t k +)=g k(x(t k)), x′(t k +)=h k(x′(t k)), k=1,2,…, ( E) where 0⩽ t 0< t 1<⋯< t k <⋯ with lim k→+∞ t k=+∞ , σ is any quotient of positive odd integers. We obtain some sufficient conditions ensuring that all solutions of (E) oscillate.