The asymptotic behavior of a Volterrarenewal equation
Author:
Peter Ney
Journal:
Trans. Amer. Math. Soc. 228 (1977), 147155
MSC:
Primary 45D05; Secondary 60K05
MathSciNet review:
0440317
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Abstract: Theorem. Assume that the functions satisfy: (i) as bounded, measurable; (ii) Lipschitz continuous for , where I is a closed interval containing the range of ; (iii) G a probability distribution on having nontrivial absolutely continuous component and finite second moment. Let . If , with strict inequality on the left on a set of positive measure, then , where is a constant depending only on h and G. The condition is close to best possible, and cannot, e.g., be replaced by .
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 K. Athreya and P. Ney, Branching processes, SpringerVerlag, Berlin and New York, 1972. MR 0373040 (51:9242)
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 F. Brauer, On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal. 6 (1975), 312317. MR 0361694 (50:14139)
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 J. Chover and P. Ney, The nonlinear renewal equation, J. Analyse Math. 21 (1968), 381413. MR 39 #7690. MR 0246386 (39:7690)
 [4]
 W. Feller, An introduction to probability theory and its applications. Vol. II, 2nd ed., Wiley, New York, 1971. MR 0270403 (42:5292)
 [5]
 J. Levin, On a nonlinear Volterra equation, J. Math. Anal. Appl. 39 (1972), 458476. MR 46 #4124. MR 0304994 (46:4124)
 [6]
 S.O. Londen, On a nonlinear Volterra integral equation, J. Differential Equations 14 (1973), 106120. MR 49 #5745. MR 0340995 (49:5745)
 [7]
 C. Stone, On absolutely continuous components and renewal theory, Ann. Math. Statist. 37 (1966), 271275. MR 33 #4981. MR 0196795 (33:4981)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719770440317X
PII:
S 00029947(1977)0440317X
Keywords:
Nonlinear Volterra equations,
nonlinear renewal theory,
Tauberian theorems,
population growth,
branching processes
Article copyright:
© Copyright 1977
American Mathematical Society
