On the Stability and Boundedness of Solutions to Certain Second Order Differential Equation

Abstract

In this paper, we investigate by means of second method of Lyapunov, sufficient conditions that guarantee uniform-asymptotic stability of the trivial solution and ultimate boundedness of all solutions to a certain second order differential equation. We construct a complete Lyapunov function in order to discuss the qualitative properties mentioned earlier. The boundedness result in this paper is new and also complement some boundedness results in literature obtained by using an incomplete Lyapunov function together with a signum function. Finally, we demonstrate the correctness of our results with two numerical examples and graphical representation of the trajectories of solutions to the examples using Maple software.