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.
References
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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.
SimonA., & VolkmannP. (1997). Perturbations de fonctions additives. Annales Mathematicae Silesianae, 11, 21-27. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14180
Alice Simon
Départment de Mathématiques, Université d'Orléans, France France
Départment de Mathématiques, Université d'Orléans, France France
Peter Volkmann
Mathematisches Institut I, Universität Karlsruhe, Allemagne Germany
Mathematisches Institut I, Universität Karlsruhe, Allemagne Germany
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