On oscillations of nonautonomous linear impulsive differential equations with general piecewise constant deviating arguments

The paper’s main oscillation criterion (Theorem 7) and its proof repeatedly use integrals with reversed limits (e.g., ∫_{ζ_k}^{t_k} and ∫_{t_{k+1}}^{ζ_k}) and compare them against thresholds 1 and −1 in a way that conflicts with the correct variation-of-constants representation, where the natural “zero-creation” thresholds arise from forward integrals ∫_{t_k}^{ζ_k} and ∫_{ζ_k}^{t_{k+1}}. The fundamental solution derived earlier in the paper shows z(t) = φ(t,ζ_k)[1 + ∫_{ζ_k}^{t} φ(ζ_k,s)b(s)ds]z(ζ_k) (hence the + sign with forward limits) , but the proof of Theorem 7 subsequently replaces this by identities with the upper/lower bounds swapped, e.g., z(t_{k+1}^-)e^{∫_{ζ_k}^{t_{k+1}}a} = z(ζ_k)(1 + ∫_{t_{k+1}}^{ζ_k} e^{∫_{ζ_k}^{s}a}b(s)ds) , leading to incorrect inequalities. The main statement itself employs reversed limits (lim sup ∫_{ζ_k}^{t_k}…>1, lim inf ∫_{t_{k+1}}^{ζ_k}…<−1, etc.) , which do not correspond to the actual zero-creation thresholds identified by the normalized profile y_k(t)=1+∫_{ζ_k}^{t}E(ζ_k,s)b(s)ds (left threshold ≥1 on ∫_{t_k}^{ζ_k}, right threshold ≤−1 on ∫_{ζ_k}^{t_{k+1}}). The paper itself acknowledges the correct zero condition earlier (1+∫_{α_j}^{ζ_j}E(ζ_j,s)b(s)ds=0 for some α_j∈I_j) and provides the right positivity conditions for nonoscillation (Theorem 6(c), equation (6.2)) in forward orientation . Reconciling these shows the proof of Theorem 7 hinges on sign/limit inversions that invalidate the stated criterion. The model’s solution corrects the orientation: it derives the proper transfer identities and the zero-creation lemma with forward limits and shows how oscillation follows under the corrected Aftabizadeh–Wiener-type conditions.