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New result in the ultimate boundedness of solutions of a third-order nonlinear ordinary differential equation. (English) Zbl 1173.34321

The author considers the nonlinear third order ordinary differential equation

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

or its equivalent system

$\stackrel{˙}{x}=y,\phantom{\rule{1.em}{0ex}}\stackrel{˙}{y}=z,$
$\stackrel{˙}{z}=-f\left(x,y,z\right)z-g\left(x,y\right)-h\left(x,y,z\right)+p\left(t,x,y,z\right),$

where it is assumed that $f$, $g$, $h$ and $p$ are continuous functions which depend only on the arguments displayed explicitly, the dots denote differentiation with respect to $t$ and the derivatives $\frac{\partial f\left(x,y,z\right)}{\partial x}\equiv {f}_{x}\left(x,y,z\right),\phantom{\rule{1.em}{0ex}}\frac{\partial f\left(x,y,z\right)}{\partial z}\equiv \phantom{\rule{1.em}{0ex}}{f}_{z}\left(x,y,z\right),\phantom{\rule{1.em}{0ex}}\frac{\partial h\left(x,y,z\right)}{\partial x}\equiv {h}_{x}\left(x,y,z\right),\phantom{\rule{1.em}{0ex}}\frac{\partial h\left(x,y,z\right)}{\partial y}\equiv {h}_{y}\left(x,y,z\right),\frac{\partial h\left(x,y,z\right)}{\partial z}\equiv {h}_{z}\left(x,y,z\right)$ and $\frac{\partial g\left(x,y\right)}{\partial x}\equiv {g}_{x}\left(x,y\right)$ exist and are continuous.

Some sufficient conditions have been established for the ultimate boundedness of all solutions of (1). By this way, the author’s result improves the result obtained by the reviewer [C. Tunç, JIPAM, J. Inequal. Pure Appl. Math. 6, No. 1, Paper No. 3, 6 p., electronic only (2005; Zbl 1082.34514)].

##### MSC:
 34C11 Qualitative theory of solutions of ODE: growth, boundedness
##### Keywords:
differential equation; third order; boundedness