Boundedness of Certain System of Second Order Differential Equations

M. O. Omeike, A. A. Adeyanju, D. O. Adams, A. L. Olutimo

This work is concerned with the ultimate boundedness of solutions of the system of vector differential equations \begin{equation*} \dot{X}=H(Y),\quad \dot{Y}=-F(X,Y)Y-G(X)+P(t,X,Y), \end{equation*} where $t\in\mathbb{R}^+,\ X=X(t)$, $Y=Y(t)\in \mathbb{R}^n$, $F:\mathbb{R}^n×\mathbb{R}^n\to\mathbb{R}^{n×n}, \ G,H:\mathbb{R}^n\to\mathbb{R}^n \ \mbox{and} \ P:\mathbb{R}^+×\mathbb{R}^n×\mathbb{R}^n\to\mathbb{R}^n.$ By using a Lyapunov function as a basic technique, we prove that the solutions of the system of equations are ultimately bounded. In addition, result obtained includes and improves some related results in literature.