On the Global Stability Properties and Boundedness Results of Solutions of Third-Order Nonlinear Differential Equations

Abstract

We studied the global stability and boundedness results of third-order nonlinear differential equations of the form $\stackrel{⃛}{x}+\psi (x,\dot{x},\ddot{x})\ddot{x}+f(x,\dot{x},\ddot{x})=P(t,x,\dot{x},\ddot{x})$. Particular cases of this equation have been studied by many authors over years. However, this particular form is a generalization of the earlier ones. A Lyapunov function was used for the proofs of the two main theorems: one with $P\equiv 0$ and the other with $P\ne 0$. The results in this paper generalize those of other authors who have studied particular cases of the differential equations. Finally, a concrete example is given to check our results.