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

• Main Authors: Jelena Manojlović, J. Milosevic
• Format: Article
• Language: English
• Published: University of Szeged 2008-11-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=346

This paper is concerned with the oscillatory behavior of the fourth-order nonlinear differential equation $$ \bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime} +q(t)|x|^{\beta-1}x=0\,,\tag{E} $$ 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 $\beta>\alpha$).