Bilateral Bounds for Norms of Solutions and Boundedness/Stability and Instability of Some Nonlinear Systems with Delays and Variable Coefficients

The paper’s Theorem 1 asserts a bilateral comparison inequality (3.13) via the scalar auxiliaries (3.11)–(3.12). However, (i) the lower auxiliary (3.12) contains a +F(t) term, whereas the Dini-derivative lower bound for ||y|| yields a −||F(t)|| contribution in general; keeping a plus sign breaks the lower comparison without extra sign restrictions on F (the paper later clips z(t) at 0, but that ad hoc fix is not part of the differential problem itself) . (ii) The comparison step is justified by “solutions cannot intersect due to uniqueness,” which is not a valid comparison principle for retarded scalar equations; monotonicity in the delayed arguments is needed (the paper mentions this only as an optional remark around Lemma/Corollary) . (iii) The formal proof of Theorem 1 is essentially a pointer back to steps 3.1–3.4 and does not supply the needed first‑crossing argument or a matrix‑measure/Dini‑derivative estimate that would rigorously connect the vector norm dynamics to (3.11)–(3.12) . In contrast, the model’s solution supplies the standard logarithmic‑norm bounds and a correct scalar delay comparison (with L nondecreasing in delayed arguments, locally Lipschitz in the current value, and the necessary −||F|| in the lower comparator), from which z(t) ≤ ||V^{-1}x(t)|| ≤ Z(t) follows. The model also diagnoses the sign error and specifies the exact condition under which the paper’s printed +F can be salvaged (additional positivity structure on F in the chosen basis).