Generalise Theorem 1’s qualitative conclusions to linear equations with unbounded delay

Establish that the qualitative conclusions proven in Theorem 1 for the forced pantograph equation z'(t) = a z(qt) + b z(t) + p(t) under b < 0 and |b| > |a|—namely, that p(t) -> 0 implies z(t) -> 0 and that bounded p implies bounded z—extend to linear functional differential equations with unbounded delay, i.e., linear retarded equations whose delay is unbounded in time.

Background

The paper establishes sharp qualitative results for the forced pantograph equation z'(t) = a z(qt) + b z(t) + p(t) with q in (0,1) and stability condition b < 0, |b| > |a|. Theorem 1 shows: (a) if the forcing p(t) tends to zero then z(t) tends to zero; and (b) if the forcing p is bounded then z is bounded. These are the key qualitative stability and boundedness conclusions for the pantograph case.

Immediately after presenting these results, the authors conjecture that the same qualitative properties should hold for a broader class of linear functional differential equations with unbounded delay, suggesting that comparison methods from prior work could be adapted. While Theorem 4 provides related bounds via a direct Dini-derivative approach for the pantograph setting, a full extension to general unbounded-delay equations is left as a conjecture.