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.

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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