Oscillatory properties of third-order quasilinear difference equations

Abstract

Some new oscillation criteria are obtained for the third-order quasilinear difference equation $\Delta^2\left(p_n\left(\Delta x_n\right)^\alpha\right)-$ $q_n\left(\Delta x_n\right)^\alpha+r_n f\left(x_n\right)=0, n=0,1,2, \ldots$, where $\alpha>0$ is the ratio of odd positive integers. The method uses techniques based on Schwarz's inequality. Example is inserted to illustrate the result.