Boundedness and stability properties of solutions of certain fourth order differential equations via the intrinsic method.

*(English)*Zbl 0868.34039Summary: Firstly a Lyapunov function is derived for

$${x}^{\left(4\right)}+\varphi \left(\ddot{x}\right)\stackrel{\u20db}{x}+f(x,\dot{x})\ddot{x}+g\left(\dot{x}\right)+h\left(x\right)=P(t,x,\dot{x},\ddot{x},\stackrel{\u20db}{x})\phantom{\rule{2.em}{0ex}}(*)$$

by applying the intrinsic method introduced by *P. S. M. Chin* [Int. J. Control 48, No. 4, 1561-1567 (1988; Zbl 0653.93038)]. Secondly equation (*) is examined in two cases: (i) $P\equiv 0$, (ii) $P\neg \equiv 0$. For case (i) the asymptotic stability in the large of the trivial solution $x=0$ is investigated and for case (ii) boundedness results are obtained for solutions of equation (*). These results improve and include several well-known results.

##### MSC:

34D20 | Stability of ODE |