Oscillation properties of first-order difference equations with nonmonotone delays: new insights and criteria

In this paper, we examine the oscillatory behavior of the first-order linear delay difference equation Δy(n) + p(n) y(τ(n)) = 0, n ∈ ℕ0. It is known in the literature that, for any C > 0, the condition lim supn→∞ ∑i=τ(n)n p(i) > C, with τ(n) nonmonotone, is not, in general, sufficient to establish the oscillation of the difference equation. We prove that, for certain classes of difference equations, the aforementioned condition is sufficient to ensure the oscillation property. Specific oscillation results are presented in terms of the limit superior. We obtain several novel results by employing new methods and techniques to analyze the properties of positive solutions of the equation under study. Two illustrative examples are included to demonstrate the effectiveness of our results.