Oscillatory Properties of Second Order Half-Linear Delay Difference Equations

Abstract

This study explores, some necessary and sufficient conditions that are established for oscillatory  properties of second order half-linear delay difference equations of the form
   Δ(p(ξ)(Δx(ξ))^r )+q(ξ)x^s (σ(ξ))=0, for ξ≥ξ_0 .    
Under the assumption  ∑_(t=ξ_0)^(ξ-1) 1/(p^(1/r) (t))=∞. Two cases are considered for r<s and r>s, where r and s are the quotients of two positive odd integers. The effectiveness and applicability of the result are illustrated through few examples.