Oscillation of third-order half-linear neutral difference equations. (English) Zbl 1274.39023

Thandapani, E.; Selvarangam, S.

Math. Bohem. 138, No. 1, 87-104 (2013).

The authors give oscillation criteria for the nonlinear neutral difference equations of the third order, \[ \Delta \big ( a_n\,(\Delta ^2(x_n\pm b_nx_{n-\delta }))\big )^{\alpha }+q_n\,x_{n+1-\tau }^{\alpha }=0. \] These criteria present sufficient conditions for the oscillation of every (nontrivial) solution, or the limit of the solution as \(n\to \infty \) being zero. The proofs involve a Riccati-type technique. On the other hand, the existence of solutions of the above equation is not discussed at all, while the presented criteria have the existence of a solution as an assumption.

Reviewer: Roman Šimon Hilscher (Brno)

MSC:

39A21	Oscillation theory for difference equations
39A10	Additive difference equations

Keywords:

third-order neutral difference equation; oscillation; nonoscillation