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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\(\geq 0\) for \(| x| >R\). (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: \(g(x)\equiv b>0;\quad \limsup_{t\to \infty}| p(t)| <\infty,\quad | \int^{\infty}_{0}p(t)dt| <\infty;\limsup_{| x| \to \infty}| h(x)| <\infty,\quad \limsup_{| x| \to \infty}h(x)sgn x<-| p(0)|.\) This result is of special interest as a supplement to a paper of 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 \(| x(t)| \to \infty\) as \(t\to \infty\).
Reviewer: W.Müller (Berlin)

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations

Citations:

Zbl 0201.116
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