Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T19:36:45.640Z Has data issue: false hasContentIssue false
  • English
  • Français

FIXED POINT THEOREM FOR AN INFINITE TOEPLITZ MATRIX

Part of: Combinatorial probability Special matrices Inversion theorems Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  09 November 2020

VYACHESLAV M. ABRAMOV Open the ORCID record for VYACHESLAV M. ABRAMOV [Opens in a new window]
VYACHESLAV M. ABRAMOV*
Affiliation:
24 Sagan Drive, Cranbourne North, Victoria3977, Australia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising in the theory of stochastic processes and other areas.

Keywords

combinatorial probabilityfixed point theoremgenerating functionsTauberian theoremsToeplitz matrix

MSC classification

Primary: 15B05: Toeplitz, Cauchy, and related matrices
Secondary: 40E05: Tauberian theorems, general 40A30: Convergence and divergence of series and sequences of functions 60C99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 108 - 117
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramov, V. M., ‘Optimal control of a large dam with compound Poisson input and costs depending on water levels’, Stochastics 91(3) (2019), 433–483.CrossRefGoogle Scholar
Hardy, G. H., Divergent Series, 2nd edn (AMS Chelsea, Providence, RI, 2000).Google Scholar
Hardy, G. H. and Littlewood, J. E., ‘Tauberian theorems concerning power series and Dirichlet’s series whose coefficients are positive’, Proc. Lond. Math. Soc. 13 (1914), 174–191.CrossRefGoogle Scholar
Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations (SIAM, Philadelphia, 1995).CrossRefGoogle Scholar
Krasnosel’skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Y. B. and Stetsenko, V. Y., Approximate Solutions of Operator Equations (Wolters-Noordhoff, Groningen, 1972).CrossRefGoogle Scholar
Leontief, W., Input–Output Economics, 2nd edn (Oxford University Press, Oxford, 1986).Google Scholar
Takács, L., Combinatorial Methods in the Theory of Stochastic Processes (John Wiley, New York, 1967).Google Scholar
Takács, L., ‘On the busy periods of single-server queues with Poisson input and general service times’, Oper. Res. 24(3) (1976), 564–571.CrossRefGoogle Scholar
Varga, R. S., ‘Iterative methods for solving matrix equations’, Amer. Math. Monthly 72(2) (1965), 67–74.CrossRefGoogle Scholar