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Sufficient conditions are established for ultimate boundedness of solutions of certain nonlinear vector differential equations of third-order. Our result improves on Tunc’s [C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third order].

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 $$\stackrel{\text{t}o8pt...}{Xwidth0ptheight5.46pt}+F\left(\ddot{X}\right)+G\left(\dot{X}\right)+H\left(X\right)=P(t,X,\dot{X},\ddot{X})$$ where $X,F\left(\ddot{X}\right)$, $G\left(\dot{X}\right)$, $H\left(X\right)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:{ℝ}^n\to {ℝ}^n$ and $P:ℝ\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ continuous in their respective arguments. We do not necessarily require that $F\left(\ddot{X}\right),G\left(\dot{X}\right)$ and $H\left(X\right)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized.