Hostname: page-component-7bb8b95d7b-nptnm Total loading time: 0 Render date: 2024-09-26T21:35:25.016Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularly varying solutions of a linear functional equation

Published online by Cambridge University Press:  09 April 2009

Marek Kuczma
Marek Kuczma
Affiliation:
Mathematics Department, Silesian University Katowice, Poland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We are concerned with the problem of the existence and uniqueness of regularly varying (in Karamata's sense) solutions ϕ of the linear functional equation in a right neighbourhood of x = 0. Under suitable conditions on the given functions f and h, the uniqueness of solutions depends essentially on whether the series Σh ∘ f1 converges or diverges; here fi denotes the i-th functional iterate of f. The existence of solutions may be proved under further assumptions.

The case of the more general linear functional equation may be reduced to that of equation (*).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 2 , September 1976 , pp. 135 - 143
Copyright
Copyright © Australian Mathematical Society 1976

References

Coifman, R. R. and Kuczma, M. (1969), ‘On asymptotically regular solutions of a linear functional equation’, Aequationes Math. 2, 332–336.CrossRefGoogle Scholar
Feller, W. (1966), ‘An Introduction to Probability Theory and its Applications’, Vol. 2, John Wiley & Sons, New York.Google Scholar
Karamata, J. (1930), ‘Sur une mode de croissance regulière’, Mathematica Cluj 4, 38–53.Google Scholar
Korevaar, J., van Ardenne-Ehrenfest, T. and De Bruijn, N. G. (1949), ‘A note on slowly oscillating functions’, Nieuw Archief voor Wiskunde 23, 77–86.Google Scholar
Kuczma, M. (1968), ‘Functional Equations in a Single Variable’, Monografie Mat. 46, Polish Scientific Publishers, Warszawa.Google Scholar
Kuczma, M. (to appear), ‘On some properties of solutions of a functional equation’.Google Scholar
Seneta, E. (1971), ‘On invariant measures for simple branching process’, J. Appl. Probability 8, 43–51.CrossRefGoogle Scholar