On the Qualitative Behaviors of a Functional Differential Equation of Second Order

Abstract

The aim of this paper is first to investigate the stability of the zero solution to a new Lienard type equation with multiple variable delays by two different methods. The methods to be used in the proofs involve the Lyapunov-Krasovskii functional approach and the fixed point technique under an exponentially weighted metric, respectively. We make a comparison between the applications of these methods with the established conditions on the same stability problems. Then, we obtain three new results for uniformly stability and boundedness/uniformly boundedness of the solutions to the considered equation by the Lyapunov-Krasovskii functional approach. An example is given to verify the results obtained by the Lyapunov-Krasovskii functional approach. Our results complement and improve some recent ones in the literature.