Oscillation of a Certain Class of Third Order Nonlinear Difference Equations

S. H. Saker

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 3

Résumé

In this paper, we are concerned with oscillation of the nonlinear difference equation $\Delta (c_{n}\left[ \Delta (d_{n}\Delta x_{n})\right] ^{\gamma })+q_{n}f(x_{g(n)})=0,\ n\geq n_{0},$ where $\gamma >0$ is the quotient of odd positive integers, $c_{n}$, $d_{n}$ and $q_{n}$ are positive sequences of real numbers, $g(n)$ is a sequence of nonnegative integers and $f\in C(\mathbf{R,R)}$ such that $uf(u)>0$ for $% u\neq 0.$ We establish some new sufficient conditions for oscillation by employing the Riccati substitution and the analysis of the associated Riccati difference inequality. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results.

Détail

Classification : 34K11, 39A10.