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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The asymptotic behavior of a Volterra-renewal equation


Author: Peter Ney
Journal: Trans. Amer. Math. Soc. 228 (1977), 147-155
MSC: Primary 45D05; Secondary 60K05
MathSciNet review: 0440317
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Abstract: Theorem. Assume that the functions $ x( \cdot ),h( \cdot ),G( \cdot )$ satisfy:

(i) $ 0 \leqslant x(t),t \in [0,\infty );x(t) \to 0$ as $ t \to \infty ;x$ bounded, measurable;

(ii) $ 0 \leqslant h(s);h(s)$ Lipschitz continuous for $ s \in I$, where I is a closed interval containing the range of $ x;h(0) = 0,h'(0 + ) = 1,h''(0 + ) < 0$;

(iii) G a probability distribution on $ (0,\infty )$ having nontrivial absolutely continuous component and finite second moment.

Let $ Hx(t) = \smallint _0^th[x(t - y)]dG(y)$. If $ 0 \leqslant (x - Hx)(t) = o({t^{ - 2}})$, with strict inequality on the left on a set of positive measure, then $ x(t) \sim \gamma /t,t \to \infty $, where $ \gamma $ is a constant depending only on h and G.

The condition $ o({t^{ - 2}})$ is close to best possible, and cannot, e.g., be replaced by $ O({t^{ - 2}})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0440317-X
PII: S 0002-9947(1977)0440317-X
Keywords: Nonlinear Volterra equations, nonlinear renewal theory, Tauberian theorems, population growth, branching processes
Article copyright: © Copyright 1977 American Mathematical Society