New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function ki, namely, ki '(t)<=-xi i(t)psi i(ki(t)),i=1,2. We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when k(i)(s)=s(p) and p covers the full admissible range [1,2).