Convergence results for solutions of certain third-order nonlinear vector differential equations. (English) Zbl 1291.34087

The author gives some new convergence results for systems of the form \[ \dddot{X} + \Psi(X,\dot{X},\ddot{X}) + \Phi(\dot{X}) + c X = P(t,X,\dot{X}, \ddot{X}), \] where \(t\in \mathbb R^+, X\in \mathbb R^n, c \) is a positive constant, \(\Phi\) is a continuous vector function and \(\Psi\) is an \(n\times n\)-continuous symmetric positive definite matrix function, \(P: \mathbb R^+\times \mathbb R^n \times \mathbb R^n \times \mathbb R^n \to \mathbb R^n.\) Under well selected conditions on \(\Phi, \Psi \) and \( P,\) the convergence of solutions is proved, using a suitable Lyapunov function.
The results are particular generalizations of some results in the literature (for example [A. U. Afuwape, Simon Stevin 57, 255–271 (1983; Zbl 0552.34043); A. U. Afuwape and M. O. Omeike, Ann. Differ. Equations 21, No. 4, 533–540 (2005; Zbl 1103.34036)]).


34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations