**Sharp oscillation criteria for fourth order sub-half-linear and super-half-linear differential equations**

**J. V. Manojlovic**, University of Nis, Nis, Serbia and Montenegro**J. Milosevic**, University of Nis, Nis, Serbia and Montenegro

E. J. Qualitative Theory of Diff. Equ., No. 32. (2008), pp. 1-13.

Communicated by J. R. Graef. | Received on 2008-07-26 Appeared on 2008-11-01 |

**Abstract: **This paper is concerned with the oscillatory behavior of the fourth-order nonlinear differential equation

$$

\bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime}

+q(t)|x|^{\beta-1}x=0\,,\leqno{\rm(E)}

$$

where $\alpha>0$, $\beta>0$ are constants and $p,q:[a,\infty)\to(0,\infty)$ are continuous functions satisfying conditions

$$

\int_a^{\infty}\left( \frac{t}{p(t)}\right)^{\frac{1}{\alpha}}\,dt<\infty,

\int_a^{\infty}\frac{t}{\left(p(t)\right)^{\frac{1}{\alpha}}}\,dt<\infty .

$$

We will establish necessary and sufficient condition for oscillation of all solutions of the sub-half-linear equation (E) (for $\beta<\alpha$) as well as of the super-half-linear equation (E) (for $\beta>\alpha$).

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