On the stability, boundedness and asymptotic behaviour of solutions of differential equations of third-order

Abstract

In this paper, we construct a complete Lyapunov function which serves as a tool in providing sufficient conditions that ensure uniform asymptotic stability of the zero solution when p(t, x, y, z) ≡ 0; uniform ultimate boundedness and asymptotic behaviour of all solutions when p(t, x, y, z) ̸= 0 of a certain third order non-linear ordinary differential equation. The results in this paper are new and in some ways generalize and improve on some existing results in literature. Finally, we provide an example to justify the correctness of the theorems.