On the asymptotic behavior of solutions of certain second-order differential equations

In this paper, the second order non-linear differential equation x¨+a(t)f(x,x˙)x˙+b(t)g(x)=p(t,x,x˙)is considered, and Lyapunov's second method is used to show that uniform boundedness and convergence to zero of all solutions of this equation together with their derivatives of the first order.