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.


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