Optimal decay for second-order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with infinite memory, time delay, and a kernel functionh:Double-struck capital R+-> Double-struck capital R+satisfying, for allt >= 0,h(')(t) <= -zeta(t)G(h(t))where zeta andGare functions satisfying some specific properties. For this much larger class of kernel functions and under a suitable conditions, we prove well-posedness of solution by using semi-group theory. Then, we establish an explicit and general decay results of the energy solution by introducing a suitable Lyapunov functional and some properties of the convex functions. Finally, some applications are given. This work improves the previous results with finite memory to infinite memory and without time delay term to those with delay.