Oscillatory and asymptotic behavior of fourth order quasilinear difference equations. (English) Zbl 1189.39013

Thandapani, E.; Vijaya, M.

Electron. J. Qual. Theory Differ. Equ. 2009, Paper No. 64, 15 p. (2009).

Summary: The authors consider the fourth order quasilinear difference equation \[ \Delta^{2}\left(p_{n}|\Delta^{2}x_n|^{\alpha-1}\Delta^{2}x_n\right)+q_{n}|x_{n+3}|^{\beta -1}x_{n+3}=0, \]
where \(\alpha\) and \(\beta\) are positive constants, and \({\{p_{n}\}}\) and \({\{q_{n}\}}\) are positive real sequences. They obtain sufficient conditions for oscillation of all solutions when \(\sum_{n=n_{0}}^{\infty}\left(\frac{n}{p_{n}}\right)^\frac{1}{\alpha}<\infty \) and \(\sum_{n=n_{0}}^{\infty}\left(\frac{n}{{p_{n}}^{\frac{1}{\alpha}}}\right)<\infty.\) The results are illustrated with examples.

MSC:

39A21	Oscillation theory for difference equations
39A22	Growth, boundedness, comparison of solutions to difference equations
39A10	Additive difference equations

Keywords:

fourth order difference equation; nonoscillation; oscillation; asymptotic behavior; fourth order quasilinear difference equation