ULTIMATE BOUNDEDNESS OF SOLUTIONS OF CERTAIN THIRD-ORDER NONLINEAR SYSTEM OF DIFFERENTIAL EQUATIONS

ULTIMATE BOUNDEDNESS OF SOLUTIONS

Abstract

This paper studies the ultimate boundedness of solutions of certain third-order non-linear system of differential equation
$$\stackrel{...}{X}+\Psi(\dot{X})\ddot{X}+\Phi(X)\dot{X}+H(X)=P(t,X,\dot{X},\ddot{X})$$
where $\Psi,\Phi$ are positive definite symmetric matrices, $H,P$ are n-vector functions continuous in their respective arguments, $X\in R^n$ and  $t\in R^+=[0, \infty).$ By using the Lyapunov direct (second) method and  considering a complete Lyapunov function, sufficient conditions which guarantee the ultimate boundedness of solutions of the above equation are obtained. Results obtained extend and generalize those in the literature.