Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay

In this paper, we consider a second-order abstract linear equation with infinite memory and time delay terms. Under appropriate assumptions on the convolution kernel and on the weight of the delay, we prove the well-posedness and the exponential stability of the system. Our stability estimate proves that the unique dissipation given by the memory term is strong enough to stabilize exponentially the system in presence of delay. Some applications are also given.