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On the boundedness of solutions of a certain fourth-order nonlinear differential equation. (English) Zbl 0719.34060

The paper deals with the differential equation \((1)\quad x^{IV}(t)+ax'''(t)+bx''(t)+cx'(t)+h[x(t)]=p(t)\) where \(a,b,c\in {\mathbb{R}}\), \(a>0\), \(b>0\), \(ab>c>0\) are given constants and the functions h[x(t)], p(t) have continuous first derivatives and are oscillatory on \({\mathbb{R}}\). Under some boundedness hypothesis about the functions \(h(x)\), \(h'(x)\), \(p(t)\), \(p'(t)\) it is proved that all solutions \(x(t)\) of (1) are bounded on \({\mathbb{R}}\) and to each of them corresponds a root \(\bar x\) of the equation \(h(x)=0\) such that \(x(t)-\bar x\) becomes oscillatory.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations

Keywords:

bounded; oscillatory
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References:

[1] Andres J.: Boundedness of solutions of the third order differential equation with oscillatory restoring and forcing terms. Czech. Math. Journal, V.36 111, No. 1, Praha 1986. · Zbl 0608.34039
[2] Vlček V.: A note to a certain nonlinear differential equation of the third order. ACTA UP Olom., F.R.N. Math. XXVII, Vol.91, 1988. · Zbl 0725.34034
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