Oscillatory Behavior of Linear Nonautonomous Advanced and Delayed Impulsive Differential Equations with Discontinuous Deviating Arguments via Difference Equations

The paper derives the discrete evolution z(n+1) = a_n z(n) + b_n z(n−k) on each unit interval via variation of constants (equations (5.5)–(5.6)) and the impulse at n+1 , then normalizes to obtain the reduced difference equation Δy_n + Q_n y_{n−k} = 0 with Q_n exactly as in (7.5) . Under k ∈ N and Q_n > 0, the paper invokes Erbe–Zhang-type discrete oscillation criteria, here written equivalently as (A) lim sup(…)< −k^k/(k+1)^{k+1} or (B) lim inf ∑(…)< −1 (Lemma 11) , and earlier establishes that oscillation of the discrete samples implies oscillation of the full IDEPCA solution (Lemma 2) . The candidate solution follows the same pipeline—(i) intervalwise variation of constants, (ii) the same normalization producing the identical Q_n, (iii) application of the Erbe–Zhang criteria in the equivalent Q_n form, and (iv) a bridge back to z(t) (via an IVT-based continuity argument that is equivalent in effect to Lemma 2).