Exponential decay for nonlinear abstract evolution equations with a countably infinite number of time-dependent time delays

In this paper we analyze a nonlinear abstract evolution equation with an infinite number of time-dependent time delays and a Lipschitz continuous nonlinear term. By using a fixed point argument we prove the existence of a mild solution. This allows us to take into account also nonnegative time delays. Furthermore, by using Gronwall estimates, exponential decay of the solution is also proved under some smallness assumptions on the parameters appearing in the system and on the initial data. Finally some examples are illustrated.