Discrete Riccati and Comparison Approaches to Oscillation of Second-Order Advanced Difference Equations with Multiple Deviating Arguments

Abstract

In this work, we develop, from first principles, a discrete oscillation theory for second-order nonlinear advanced difference equations with several deviating arguments, Δr(n)(Δy(n))α+∑i=1mqi(n)Fy(σi(n))=0,σi(n)≥n, studied under the canonical condition ∑s=n0∞r−1/α(s)=∞. This equation models discrete systems governed by anticipatory rather than delayed dynamics, since the second-order difference term depends on a future value y(σi(n)) rather than a past one. We prove an iterative monotonicity lemma, a classification lemma for eventually positive solutions, three oscillation results, a comparison theorem, and a Riccati-type theorem. Two features that are unique to the discrete setting, with no counterpart in the continuous theory, are highlighted. First, the partial sum η(n)=∑s=n∞r−1/α(s) typically lacks a closed form and must be approximated asymptotically. Second, the discrete Riccati step requires an additional power-rule inequality, which introduces a correction term absent from the continuous case. The theoretical thresholds obtained are illustrated by a fully worked example, verified both numerically and symbolically and supported by figures.