Oscillations of second-order nonlinear impulsive ordinary differential equations

Consider the second-order impulsive ordinary differential equation(r(t)(x′(t))σ)′+f(t,x(t))=0,t⩾t0, t≠tk, k=1,2,…,x(tk+)=gk(x(tk)),x′(tk+)=hk(x′(tk)), k=1,2,…, (E)where 0⩽t0<t1<⋯<tk<⋯ with limk→+∞ tk=+∞, σ is any quotient of positive odd integers. We obtain some sufficient conditions ensuring that all solutions of (E) oscillate.