Abstract
We prove, using the fixed point approach, some stability results for the general linear functional equation. Namely we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. Our results generalize a lot of the well known and recent outcomes concerning stability. In some examples we indicate how our method may be used to check if the particular functional equation is stable and we discuss the optimality of obtained bounding constants.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aoki T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)
Brzdęk J.: A note on stability of additive mappings. In: Rassias, Th.M., Tabor, J. (eds.) Stability of Mappings of Hyers- Ulam Type, pp. 19–22. Hadronic Press, Inc., Florida (1994)
Brzdęk J., Chudziak J., Páles Zs.: A fixed point approach to stability of functional equations. Nonlinear Anal. 74, 6728–6732 (2011)
Brzdęk J.: Stability of the equation of the p-Wright affine functions. Aequationes Math. 85, 497–503 (2013)
Brzdęk J.: Ciepliński, hyperstability and superstability. Abstr. Appl. Anal. (2013). doi:10.1155/2013/401756
Forti G.-L.: Elementary remarks on Ulam-Hyers stability of linear functional equations. J. Math. Anal. Appl. 328, 109–118 (2007)
Gajda Z.: On stability of additive mappings. Int. J. Math. Sci. 14, 431–434 (1991)
Gilányi, A., Páles, Zs.: On Dinghas-type derivatives and convex functions of higher order. Real Anal. Exchange, 27, 485–493 (2001/2002)
Hyers D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)
Hyers D.H., Isac G., Rassias Th. M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998)
Jung S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer Optimization and Its Applications, vol. 48, Springer, New York-Dordrecht-Heidelberg-London (2011)
Jung S.-M.: Hyers-Ulam-Rassias stability of Jensen’s equation and its application. Proc. Am. Math. Soc. 126(11), 3137–3143 (1998)
Lajkó K.: On a functional equation of Alsina and García-Roig. Publ. Math. Debrecen 52, 507–515 (1998)
Maksa Gy., Nikodem K., Páles Zs.: Results on t-Wright convexity. C. R. Math. Rep. Acad. Sci. Canada 13, 274–278 (1991)
Paneah B.: A new approach to the stability of linear functional operators. Aequationes Math. 78, 45–61 (2009)
Rassias M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)
Skof F.: Proprieta locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Bahyrycz, A., Olko, J. On stability of the general linear equation. Aequat. Math. 89, 1461–1474 (2015). https://doi.org/10.1007/s00010-014-0317-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-014-0317-z