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On boundedness and stability results for the solution of certain fourth-order differential equations via the intrinsic method. (English) Zbl 0880.34051

Summary: We first construct a Lyapunov function for

${x}^{\left(4\right)}+\varphi \left(\stackrel{¨}{x}\right)\stackrel{⃛}{x}+f\left(x,\stackrel{˙}{x}\right)\stackrel{¨}{x}+k\left(x\right)\stackrel{˙}{x}+h\left(x\right)=p\left(t,x,\stackrel{˙}{x},\stackrel{¨}{x},\stackrel{⃛}{x}\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

in which the functions $\varphi$, $f$, $k$, $h$ and $p$ depend only on the arguments displayed and the dots denote differentiation with respect to $t$. The functions $\varphi$, $f$, $k$, $h$ and $p$ are continuous for all values of their respective arguments. Moreover, the derivatives $\frac{\partial f\left(x,\stackrel{˙}{x}\right)}{\partial x}={f}_{x}\left(x,\stackrel{˙}{x}\right)$, $\frac{dh}{dx}={h}^{\text{'}}\left(x\right)$ exist and are continuous.

We show the asymptotic stability in the large of the trivial solution $x=0$ for the case $p\equiv 0$, and give a boundedness result for the solutions of (1) for the case $p\ne 0$. These results improve several well-known results.

##### MSC:
 34D20 Stability of ODE
##### Keywords:
Lyapunov function; asymptotic stability
##### References:
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