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

Afuwape, Anthony Uyi

An. Științ. Univ. Al. I. Cuza Iași, Ser. Nouă, Mat. 55, No. 1, 11-20 (2009).

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

Keywords:

convergence; Lyapunov functions; nonlinear third order system