Abstract
We show that unlikely to the single-valued case, the set-valued orthogonally additive equation is unstable. After presenting an example showing this phenomenon, we provide some special cases where a set-valued approximately orthogonally additive function can be approximated by the one which satisfies the equation of orthogonal additivity exactly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aczél, J., Dhombres, J.: Functional equations in several variables. In: Encyclopedia Math. Appl., vol. 31. Cambridge University Press, Cambridge (1989)
Baron K., Volkmann P.: On orthogonally additive functions. Publ. Math. Debr. 52, 291–297 (1998)
Beer, G.: Topologies on closed and closed convex sets. In: Mathematics and its Applications, vol. 268. Kluwer Academic Publishers Group, Dordrecht (1993)
Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions. Springer, Berlin (1977)
Fechner W., Sikorska J.: On the stability of orthogonal additivity. Bull. Pol. Acad. Sci. Math. 58, 23–30 (2010)
Ger R., Sikorska J.: Stability of the orthogonal additivity. Bull. Pol. Acad. Sci. Math. 43, 143–151 (1995)
Gudder S., Strawther D.: Orthogonality and nonlinear functionals. Bull. Am. Math. Soc. 80, 946–950 (1974)
Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)
Heijnen B., Goovaerts M.J.: Additivity and premium calculation principles. Blätter der Deutschen Gesellschaft für Versich. Math. 17, 217–223 (1986)
Mirmostafaee A.K., Mahdavi M.: Approximately orthogonal additive set-valued mappings. Kyungpook Math. J. 53, 639–646 (2013)
Rådström H.: An embedding theorem for spaces of convex set. Proc. Am. Math. Soc. 3, 165–169 (1952)
Rätz J.: On orthogonally additive mappings. Aequationes Math. 28, 35–49 (1985)
Rätz J.: Cauchy functional equation problems concerning orthogonality. Aequationes Math. 62, 1–10 (2001)
Sikorska, J.: Generalized orthogonal stability of some functional equations. J. Inequal. Appl. 2006, Art. ID 12404, 23 pp. (2006)
Sikorska J.: On a direct method for proving the Hyers–Ulam stability of functional equations. J. Math. Anal. Appl. 372, 99–109 (2010)
Sikorska, J.: Orthogonalities and functional equations. Aequationes Math. 89, 215–277 (2015)
Sikorska J.: Set-valued orthogonal additivity. Set-Valued Var. Anal. (2015). doi:10.1007/s11228-015-0321-z
Truesdell C., Muncaster R.G.: Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas. Academic Press, New York (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Zygfryd Kominek on the occasion of his 70th birthday
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Sikorska, J. A Singular Behaviour of a Set-Valued Approximate Orthogonal Additivity. Results. Math. 70, 163–172 (2016). https://doi.org/10.1007/s00025-015-0468-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-015-0468-9