LAUSR

In this paper, we consider a second‐order abstract viscoelastic equation in Hilbert spaces with infinite memory, time delay, and a kernel function h:ℝ+→ℝ+ satisfying, for all t ≥ 0, h′(t) ≤ −ζ(t)G(h(t)) where ζ and G are 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.