We consider the quasilinear dynamic equation in a Banach space on unbounded above and below time scales T with rd-continuous, regressive right-hand side.We define the corresponding Green-type map. Using the integral functional technique, we find a new simpler, but at the same time, more general sufficient condition for the existence of a bounded solution on the time scales expressed in terms of integrals of the Green-type map. We construct previously unknown linear scalar differential equation, which does not possess exponentially dichotomy, but for which the integral of the corresponding Green-type map is uniformly bounded. The existence of such example allows, on the one hand, to obtain the new sufficient condition for the existence of bounded solution and, on the other hand, to prove Hyers–Ulam stability for a much broader class of linear dynamic equations even in the classical case.