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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The asymptotic behavior of a Volterra-renewal equation
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by Peter Ney PDF
Trans. Amer. Math. Soc. 228 (1977), 147-155 Request permission

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
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 228 (1977), 147-155
  • MSC: Primary 45D05; Secondary 60K05
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0440317-X
  • MathSciNet review: 0440317