Journal of Advances in Mathematics and Computer Science

This paper investigates the oscillation of unbounded solutions for the second-order advanced difference equation Δ²ξ(ζ) = ψ(ζ)ξ(ζ+σ), ζ≥ζ₀ where ψ(ζ) is a positive real sequence on N(ζ₀) and σ is an even positive integer. The study addresses the class of eventually positive non-oscillatory solutions for which ξ(ζ) > 0, Δξ(ζ) > 0 and Δ²ξ(ζ) > 0, and derives sufficient conditions that exclude this class. Under a divergence assumption on the coefficient sequence, preliminary results establish the eventual monotonicity of Δξ(ζ), the unbounded growth of its upper limit, the eventual monotonicity of \(\frac{ξ(ζ)}{ζ}\) , and the increasing behaviour of the transformed sequence α(ζ)ξ(ζ+σ). These properties are then used to obtain limsup-type oscillation criteria for all unbounded solutions. Additional estimates involving auxiliary sequences R1(ζ), R₂(ζ) and R3(ζ) are introduced to bound ξ(ζ+σ) in terms of ξ(ζ), and a further condition involving the ratio \(\frac{ψ(ζ+σ+1)}{ψ(ζ+1)}\) provides a refined criterion. The results extend the analysis of advanced difference equations by linking oscillation to explicit coefficient conditions and monotonicity transformations. The findings are formulated as sufficient, rather than necessary, criteria and are intended to support further work on advanced difference equations with broader advance parameters.