Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations

summary:In this paper, we shall give sufficient conditions for the ultimate boundedness of solutions for some system of third order non-linear ordinary differential equations of the form $${\ensuremath{\mathop{\smash{X}\vrule width0ptheight5.46pt}\limits^{\hbox to 8pt{\hss \footnotesize \kern1pt.\kern-0.065em.\kern-0.065em.\hss}}}}+F(\ddot{X})+G(\dot{X})+H(X)= P(t,X,\dot{X},\ddot{X})$$ where $X,F(\ddot{X})$, $G(\dot{X})$, $H(X)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:\mathbb{R}^n\rightarrow\mathbb{R}^n$ and $P:\mathbb{R}\times \mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ continuous in their respective arguments. We do not necessarily require that $F(\ddot{X}),G(\dot{X})$ and $H(X)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized