Volterra integral equations on time scales: stability under constant perturbations via Liapunov direct method (Q891904)

The authors study the following Volterra integral equation on a time scale \(\mathbb T\subset \mathbb R\), \[ x(t)= f(t)+ \int_0^t k(t,s)x(s)\Delta s, \quad t\in [t_0,\infty)\cap \mathbb T, \] where the integral sign stands for the delta-integral and thus includes ordinary integrals, sums or combinations thereof depending on the time scale \(\mathbb T\). The necessary background material on time scales is reviewed and then the authors state and prove an extension to the time scale case of a stability result proved with the aid of a Lyapunov theorem that is well known for continuous and for discrete Volterra equations, thus providing a unified treatment.