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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 .

**[1]**Krishna B. Athreya and Peter E. Ney,*Branching processes*, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196. MR**0373040****[2]**Fred Brauer,*On a nonlinear integral equation for population growth problems*, SIAM J. Math. Anal.**6**(1975), 312–317. MR**0361694****[3]**J. Chover and P. Ney,*The non-linear renewal equation*, J. Analyse Math.**21**(1968), 381–413. MR**0246386****[4]**William Feller,*An introduction to probability theory and its applications. Vol. II.*, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403****[5]**J. J. Levin,*On a nonlinear Volterra equation*, J. Math. Anal. Appl.**39**(1972), 458–476. MR**0304994****[6]**Stig-Olof Londen,*On a nonlinear Volterra integral equation*, J. Differential Equations**14**(1973), 106–120. MR**0340995****[7]**Charles Stone,*On absolutely continuous components and renewal theory*, Ann. Math. Statist.**37**(1966), 271–275. MR**0196795**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
45D05,
60K05

Retrieve articles in all journals with MSC: 45D05, 60K05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0440317-X

Keywords:
Nonlinear Volterra equations,
nonlinear renewal theory,
Tauberian theorems,
population growth,
branching processes

Article copyright:
© Copyright 1977
American Mathematical Society