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Boundedness results of solutions to the equation $x'''+ax''+g(x)x'+h(x)=p(t)$ without the hypothesis h(x)sgnx$\ge 0$ for $\vert x\vert >R$. (English) Zbl 0722.34027
Using appropriate Lyapunov functions, the author proves that the equation cited in the title admits a bounded solution if the following assumptions are satisfied: $g(x)\equiv b>0;\quad \limsup\sb{t\to \infty}\vert p(t)\vert <\infty,\quad \vert \int\sp{\infty}\sb{0}p(t)dt\vert <\infty;\limsup\sb{\vert x\vert \to \infty}\vert h(x)\vert <\infty,\quad \limsup\sb{\vert x\vert \to \infty}h(x)sgn x<-\vert p(0)\vert.$ This result is of special interest as a supplement to a paper of {\it J. Voráček} [Czech. Math. J. 20(95), 207-219 (1970; Zbl 0201.116)] who proved that the same equation under the same assumptions admits at least one solution x(t) with $\vert x(t)\vert \to \infty$ as $t\to \infty$.
Reviewer: W.Müller (Berlin)

34C11Qualitative theory of solutions of ODE: growth, boundedness