The asymptotic behavior of a Volterra-renewal equation
HTML articles powered by AMS MathViewer
- 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 $O({t^{ - 2}})$.References
- Krishna B. Athreya and Peter E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972. MR 0373040, DOI 10.1007/978-3-642-65371-1
- Fred Brauer, On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal. 6 (1975), 312–317. MR 361694, DOI 10.1137/0506031
- J. Chover and P. Ney, The non-linear renewal equation, J. Analyse Math. 21 (1968), 381–413. MR 246386, DOI 10.1007/BF02787676
- William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- J. J. Levin, On a nonlinear Volterra equation, J. Math. Anal. Appl. 39 (1972), 458–476. MR 304994, DOI 10.1016/0022-247X(72)90217-X
- Stig-Olof Londen, On a nonlinear Volterra integral equation, J. Differential Equations 14 (1973), 106–120. MR 340995, DOI 10.1016/0022-0396(73)90080-6
- Charles Stone, On absolutely continuous components and renewal theory, Ann. Math. Statist. 37 (1966), 271–275. MR 196795, DOI 10.1214/aoms/1177699617
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