Boundedness, limits, and stability of solutions of a perturbation of a nonhomogeneous renewal equation on a semiaxis

Author:
M. V. Kartashov

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **81** (2010).

Journal:
Theor. Probability and Math. Statist. **81** (2010), 71-83

MSC (2010):
Primary 60J45; Secondary 60A05, 60K05

DOI:
https://doi.org/10.1090/S0094-9000-2010-00811-8

Published electronically:
January 18, 2011

MathSciNet review:
2667311

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a time-nonhomogeneous perturbation of the classical renewal equation with continuous time on the semiaxis that can be reduced to the Volterra integral equation with a nonnegative bounded (or with a substochastic) kernel. We assume that this kernel is approximated by a convolution kernel for large time intervals and that the latter is generated by a substochastic distribution. We find necessary and sufficient conditions for the existence of the limit of a solution of the perturbed equation under the assumption that the corresponding perturbation is small in a certain sense. We also obtain estimates for the deviation of the solutions of the perturbed equations from those of the initial equations.

Several applications are described.

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Additional Information

**M. V. Kartashov**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine

Email:
nkartashov@skif.com.ua

DOI:
https://doi.org/10.1090/S0094-9000-2010-00811-8

Keywords:
Volterra equation,
renewal theorem,
transition kernel,
regularity,
minimal solution,
stability

Received by editor(s):
October 12, 2009

Published electronically:
January 18, 2011

Article copyright:
© Copyright 2010
American Mathematical Society