  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.
