Improvement of the stability of solutions of an inhomogeneous perturbed renewal equation on the semiaxis

Kartashov, M.

doi:10.1090/S0094-9000-2012-00854-5

Improvement of the stability of solutions of an inhomogeneous perturbed renewal equation on the semiaxis

Author: M. V. Kartashov
Translated by: O. I. Klesov
Journal: Theor. Probability and Math. Statist. 84 (2012), 65-78
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
DOI: https://doi.org/10.1090/S0094-9000-2012-00854-5
Published electronically: July 31, 2012
MathSciNet review: 2857417
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Abstract:

We consider a generalized inhomogeneous continuous-time renewal equation on the semiaxis that reduces to the Volterra integral equation with a nonnegative bounded (or substochastic) kernel. It is assumed that the kernel can be approximated in a large time scale by a convolution kernel generated by a stochastic distribution. Under some asymptotic assumptions imposed on the perturbation we find an improved condition for the boundedness; the latter condition is used to prove the existence of the limit of a solution of the perturbed equation and to establish estimates for the deviation from a solution of a nonperturbed equation.

Some examples are discussed.

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