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

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).

### MSC:

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