New oscillation theorems for second order quasi-linear difference equations with sub-linear neutral term

Abstract

Abstract In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the second order neutral difference equation Δ(an(Δzn)β)+qnxn−ℓγ=0,n≥n0, $$\Delta \bigl( a_{n} ( \Delta z_{n} )^{\beta} \bigr) + q_{n}x_{n - \ell}^{\gamma} = 0,\quad n \ge n_{0}, $$ where zn=xn+pnxn−kα $z_{n} = x_{n} + p_{n}x_{n - k}^{\alpha} $. The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the importance of the main results.

Keywords

• Oscillation
• Quasi-linear difference equations
• Sub-linear neutral term