Subexponential solutions of linear integro-differential equations and transient renewal equations
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 asymptotica...
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Published in | Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 132; no. 3; pp. 521 - 543 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Edinburgh, UK
Royal Society of Edinburgh Scotland Foundation
01.06.2002
Cambridge University Press |
Online Access | Get full text |
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Summary: | 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 |
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Bibliography: | PII:S0308210500001761 istex:1B05133975214A0485368D40615F65F6E8F2CCE6 ArticleID:00176 ark:/67375/6GQ-X5NGBGMF-C ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0308-2105 1473-7124 |
DOI: | 10.1017/S0308210500001761 |