On almost everywhere K-additive set-valued maps
Abstract
Let X be an Abelian group, Y be a commutative monoid, K⊂Y be a submonoid and F:X→2Y\{∅} be a set-valued map. Under some additional assumptions on ideals 𝓘1 in X and 𝓘2 in X^2, we prove that if F is 𝓘2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G:X→2Y\{∅} such that F=G 𝓘1-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].
Keywords
monoid; Abelian group; K-additive set-valued map; ideal; almost everywhere
References
N.G. de Bruijn, On almost additive functions, Colloq. Math. 15 (1966), 59–63.
J. Chmieliński and J. Rätz, Orthogonality equation almost everywhere, Publ. Math. Debrecen 52 (1998), no. 3-4, 317–335.
J.L. Denny, Sufficient conditions for a family of probabilities to be exponential, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 1184–1187.
J.L. Denny, Cauchy’s equation and sufficient statistics on arcwise connected spaces, Ann. Math. Statist. 41 (1970), 401–411.
P. Erdös, P 310, Colloq. Math. 7 (1960), no. 2, p. 311.
R. Ger, Note on almost additive functions, Aequationes Math. 17 (1978), no. 1, 73–76.
R. Ger, Almost additive functions on semigroups and a functional equation, Publ. Math. Debrecen 26 (1979), no. 3-4, 219–228.
E. Jabłońska and W. Jabłoński, Properties of K–additive set–valued maps, Results Math. 78 (2023), no. 6, Paper No. 221, 18 pp.
E. Jabłońska and K. Nikodem, K–subadditive and K–superadditive set–valued functions bounded on "large" sets, Aequationes Math. 95 (2021), no. 6, 1221–1231.
W.B. Jurkat, On Cauchy’s functional equation, Proc. Amer. Math. Soc. 16 (1965), 683–686.
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, PWN–Uniwersytet Śląski, Warszawa–Kraków–Katowice, 1985. 2nd edition: Birkhäuser, Basel–Boston–Berlin, 2009.
K. Nikodem, K-convex and K-concave Set-valued Functions, Zeszyty Nauk. Politechniki Łódzkiej Mat. 559; Rozprawy Mat. 114, Łódź, 1989.
Wydział Matematyki Stosowanej, Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie Poland
https://orcid.org/0000-0002-0347-0214
This work is licensed under a Creative Commons Attribution 4.0 International License.
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.
- 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. - 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. - 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. - 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.
