Perturbations de fonctions additives



Abstract

Let α ≥ 0, α ≠ 1 and f: [0,∞[→ℝ be such that f(x+y) = f(x)+f(y)+o(max{xα,yα}) as max{x,y}↓0. We show that f(x) = a(x)+o(xα), where a is an additive function, and we use this result in studying a second order generalized derivative for real functions.


1. J. Aczél, Vorlesungen über Funktionalgleichungen und ihre Anwendungen, Birkhäuser, Bâle 1961. (Version anglaise: Academic Press, New York 1966).
2. J. Aczél, J. Dhombres, Functional equations in several variables, University Press, Cambridge 1989.
3. A. Dinghas, Zur Theorie der gewöhnlichen Differentialgleichungen, Ann. Acad. Sci. Fennicae, Ser. AI, n° 375 (1966).
4. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
5. M. Kuczma, An introduction to the theory of functional equations and inequalities, Cauchy's equation and Jensen's inequality, Państwowe Wydawnictwo Naukowe, Varsovie 1985.
6. A. Simon, P. Volkmann, Eine Charakterisierung von polynomialen Funktionen mittels der Dinghasschen Intervall-Derivierten, Results Math. 26 (1994), 382-384.
7. P. Volkmann, Die Äquivalenz zweier Ableitungsbegriffe, Thèse, Université Libre de Berlin 1971.
8. Z. Gajda, Local stability of the functional equation characterizing polynomial functions, Ann. Polon. Math. 52 (1990), 119-137.
9. Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), 499-507.
10. F. Skof, Sull'approssimazione delle applicazioni localmente δ-additive, Atti Accad. Sci. Torino, Cl. Sci. Fiz. Mat. Natur. 117 (1983), 377-389.
11. F. Skof, Proprietà locali e approssimazione di operatori, Rendiconti Sem. Mat. Fis. Milano 53 (1983), 113-129 (1986).
12. J. Tabor, J. Tabor, Remark 15, 34th International Symposium on Functional Equations: Aequationes Math. 53 (1997), 192-193.
Download

Published : 1997-09-30


SimonA., & VolkmannP. (1997). Perturbations de fonctions additives. Annales Mathematicae Silesianae, 11, 21-27. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/14180

Alice Simon 
Départment de Mathématiques, Université d'Orléans, France  France
Peter Volkmann 
Mathematisches Institut I, Universität Karlsruhe, Allemagne  Germany



The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.