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- Title
Subexponential solutions of linear integro-differential equations and transient renewal equations.
- Authors
Appleby, John A. D.; Reynolds, David W.
- 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
- Publication
Proceedings of the Royal Society of Edinburgh: Section A: Mathematics, 2001, Vol 132, Issue 3, p521
- ISSN
0308-2105
- Publication type
Article
- DOI
10.1017/S0308210500001761