Stability and oscillation of linear delay differential equations

The paper proves a sharp Λ(s)-threshold for boundedness and decay of ℓ-rapidly oscillating solutions using a carefully constructed auxiliary function ψs and a detailed contradiction argument, with complete handling of both s∈[1,2) and s=2 and examples showing sharpness. The candidate solution asserts a different Volterra-series/recurrence approach but leaves key steps unjustified: (i) the “parity” restriction that the last delayed evaluation lies in Ik or Ik−1 is generally false without extra assumptions on τ; (ii) the claimed sharp bound A+B≤(√(2s)−1)exp(L−(3/2)−(√(2s)−1)²/2) is only sketched and relies on opaque ‘Young-type’ splitting; and (iii) the generating-function manipulations are dimensionally inconsistent. Hence the model’s proof is incomplete and unreliable, whereas the paper’s result is correct and well supported.