where a is a suitable kernel and x is an element of X. Using a matrix operator, we obtain some spectral conditions on A that ensure the existence of constants C, omega > 0 such that parallel to u( t)parallel to <= Ce (-omega t) parallel to x parallel to, for each x is an element of D ( A) and all t >= 0. With these results, we prove the existence of a uniformly exponential stable resolvent family to an integro-differential equation with in finite delay. Finally, sufficient conditions are established for the existence and uniqueness of bounded mild solutions to this equation.