IJPAM: Volume 28, No. 1 (2006)

ASYMPTOTIC PROPERTY OF THE SOLUTION OF
A REPAIRABLE, STANDBY, HUMAN
AND MACHINE SYSTEM

Geni Gupur
College of Mathematics and System Science
Xinjiang University
Urumqi, 830046, P.R. CHINA
e-mails: geni@xju.edu.cn, genigupur@263.net


Abstract.In this paper, we study asymptotic property of the solution of a repairable, standby, human and machine system. First we consider the expression of the solution of a simple system derived from the original system, thus give the expression of the $C_0$-semigroup corresponding to the simple system. Second we prove this $C_0$-semigroup is a quasi-compact operator. Third by using perturbation theory we obtain that the $C_0$-semigroup corresponding to the original system is a quasi-compact operator. Fourth we prove that $0$ is an eigenvalue of the operator corresponding to the original system with algebraic multiplicity one. Last by using all the above results we deduce that the time-dependent solution of the original system converges strongly to the static solution as time tends to infinite.

Received: February 24, 2005

AMS Subject Classification: 47A10, 47D99

Key Words and Phrases: $C_0$-semigroup, quasi-compact operator, eigenvalue

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 28
Issue: 1