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On oscillations of solutions and their derivatives in a nonlinear third- order differential equation. (English) Zbl 0711.34040

The author studies the differential equation \[ (p(t)(p(t)x')')'+4p(t)q(t)x'+2(p(t)q(t))'x=f(t,x,x',(p(t)x')'), \] where \(p,q\in C^ 0(J)\), \(pq\in C^ 1(J)\), \(f\in C^ 0(D)\), \(p(t)>0\) for \(t\in J\), \(J=[t_ 0,\infty)\), \(D=J\times {\mathbb{R}}^ 3.\)
He finds sufficient conditions for the oscillation of (right-maximal) solutions, or for the oscillation of the derivative of solutions, or for the oscillation of the second quasi derivative, \((px')'\), of solutions on an interval \([t_ x,\infty)\), respectively.
Reviewer: A.D.Osborne

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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References:

[1] Staněk S.: Bounds for solutions of a nonlinear differential equations of the third order. Acta Univ. Palackianae Olomucensis, Fac. rerum nat., Math. XXVI, Vol. 88, 1987.
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