Subexponential solutions of linear integro-differential equations and transient renewal equations

Abstract

This paper studies the asymptotic behaviour of the solutions of the scalar integro-differential equation The kernel k is assumed to be positive, continuous and integrable.If it is known that all solutions x are integrable and x(t) → 0 as t → ∞, but also that x = 0 cannot be exponentially asymptotically stable unless there is some γ > 0 such that Here, we restrict the kernel to be in a class of subexponential functions in which k(t) → 0 as t → ∞ so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) → 0 as t → ∞ is given by The result is proved by determining the asymptotic behaviour of the solution of the transient renewal equation If the kernel h is subexponential, then