Oscillation of delay differential equations via the hyper4 convergence

The paper’s proof hinges on a step that does not follow from condition (C), namely x'(t) ≤ −b(t)x(τ(t)) for eventually positive solutions; (C) only yields x'(t) ≤ −b(t) inf_{r∈I(t)} x(r) = −b(t)x(t) once monotonicity is established, and this does not imply the claimed inequality. The model’s solution also contains a gap: it derives a uniform bound u'(s) ≤ −b(s)m(t) for all s∈[τ(t),t] by replacing u on I(s) with a function whose infimum is m(t)=inf_{I(t)}u. This fails when τ(s)<τ(t), because the value at r=τ(s) may force inf_{I(s)} below m(t) and cannot be freely altered if τ(s)∈H(s). Both arguments can be repaired: the paper by replacing the faulty inequality with the correct log-derivative estimate x'(t)/x(t) ≤ −b(t), and the model by first proving eventual monotonicity and then using the same log-derivative estimate. The repaired argument actually proves property (P) under the weaker assumption lim inf_{t→∞}∫_{τ(t)}^t b(s)ds>0, so the 1/e threshold is not needed.