Oscillation criteria for third-order nonlinear differential equations. (English) Zbl 1174.34052

Baculíkova, B.; Elabbasy, E. M.; Saker, S. H.; Džurina, J.

Math. Slovaca 58, No. 2, 201-220 (2008).

The author primarily considers the oscillation properties of the third order nonlinear differential equation \[ \left( b(t)\left( [a(t)x^{'}(t)]'\right)^{\gamma }\right)^{'}+q(t)x^v(t)=0, \qquad t\geq t_{0} \] where \(b\), \(a\) and \(q\) are positive real-valued continuous functions, \(v\) is the quotient of odd positive integers and it holds \[ \int \limits ^{\infty }_{t_0} \left(\frac {1}{b(t)}\right)^\frac {1}{\gamma }\,dt=\infty , \int \limits ^t_{t_0}\frac {1}{a(t)}\,dt=\infty . \] Using the Riccati transformation techniques, they establish some new sufficient conditions which ensure that the solution of the above equation is oscillatory or converges to zero. The obtained results extend the results known in the literature for \(v=1\). They also establish conditions of Kamenev-type and Philos-type for desired asymptotic behavior of the considered equation. Finally, the authors give some interesting examples to illustrate their main results.

Reviewer: P. Das (Haranguer)

MSC:

34K11	Oscillation theory of functional-differential equations
34K25	Asymptotic theory of functional-differential equations

Keywords:

oscillation; third order; integral averaging technique; Riccati substitution

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