Boundedness and periodicity results for a certain system of third order nonlinear differential equations. (English) Zbl 0936.34041

Tiryaki, Aydin

Indian J. Pure Appl. Math. 30, No. 4, 361-372 (1999).

The equation \[ \dddot X+ A\ddot X+ G(\dot X)+ H(X)= P(t, X,\dot X,\ddot X)\tag{1} \] is considered. In (1), \(A\) is a constant real \(n\times n\)-matrix, \(X\in\mathbb{R}_n\), \(G\), \(H\), \(P\), are such that, for any \(X_0,Y_0,Z_0\in \mathbb{R}_n\), there is a uniquely defined solution \(X= X(t,X_0,Y_0,Z_0)\) to (1), continuous in \(t\) (where \(X(t_0)= X_0\), \(\dot X(t_0)= Y_0\), \(\ddot X(t_0)= Z_0\)).
Some sufficient conditions are obtained which ensure that all solutions to (1) are ultimately bounded. Finally, some sufficient conditions are given which guarantee that there exists at least one periodic solution to (1).

Reviewer: Bojidar Cheshankov (Sofia)

MSC:

34D40	Ultimate boundedness (MSC2000)
34C25	Periodic solutions to ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations

Keywords:

boundedness; periodicity; third-order nonlinear differential equation