## Convergence of solutions of some third order systems of non-linear ordinary differential equations.(English)Zbl 1199.34247

Summary: We consider the convergence of solutions of equations of the form $\dddot X + A\ddot X +G(\dot X ) + H(X) = P(t, X, \dot X , \ddot X ),$ in which $$X \in {\mathbb R }^n$$, $$P:{\mathbb R }\times {\mathbb R }^n \times {\mathbb R }^n \times {\mathbb R }^n \to {\mathbb R}^n$$, $$A$$ is a $$n \times n$$ constant matrix. We assume that the functions $$G$$ and $$H$$ are of class $$C({\mathbb R}^n )$$, and satisfy for any $$X_1 , X_2 , Y_1 , Y_2$$ in $${\mathbb R}^n$$ $G(Y_2) = G(Y_1) + B_g (Y_1, Y_2) (Y_2 - Y_1),$
$H(X_2) = H(X_1) + C_h (X_1, X_2) (X_2 - X_1),$ where $$B_g (Y_1, Y_2), C_h (X_1, X_2)$$ are $$n \times n$$ real continuous operators, having positive eigenvalues.
Under different conditions on $$P$$, we give sufficient conditions to establish the convergence of the solutions.

### MSC:

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