{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:40:53Z","timestamp":1760240453482,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,6,25]],"date-time":"2019-06-25T00:00:00Z","timestamp":1561420800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We will prove the generalized Hyers\u2013Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, \u2026, xn + yn) = f(x1, x2, \u2026 xn) + f(y1, y2, \u2026, yn). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.<\/jats:p>","DOI":"10.3390\/axioms8020076","type":"journal-article","created":{"date-parts":[[2019,6,25]],"date-time":"2019-06-25T10:52:31Z","timestamp":1561459951000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Generalized Hyers\u2013Ulam Stability of the Additive Functional Equation"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9013-2925","authenticated-orcid":false,"given":"Yang-Hi","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Mathematics Education, Gongju National University of Education, Gongju 32553, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4279-8992","authenticated-orcid":false,"given":"Gwang","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kangnam University, Yongin 16979, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,25]]},"reference":[{"key":"ref_1","unstructured":"Ulam, S.M. (1964). Problems in Modern Mathematics, Wiley."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1073\/pnas.27.4.222","article-title":"On the stability of the linear functional equation","volume":"27","author":"Hyers","year":"1941","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"64","DOI":"10.2969\/jmsj\/00210064","article-title":"On the stability of the linear transformation in Banach algebras","volume":"2","author":"Aoki","year":"1950","journal-title":"J. Math. Soc. Japan"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1090\/S0002-9939-1978-0507327-1","article-title":"On the stability of the linear mapping in Banach spaces","volume":"72","author":"Rassias","year":"1978","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1155\/S016117129100056X","article-title":"On stability of additive mappings","volume":"14","author":"Gajda","year":"1991","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1090\/S0002-9939-1992-1059634-1","article-title":"On the behavior of mappings which do not satisfy Hyers-Ulam stability","volume":"114","author":"Rassias","year":"1992","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1006\/jmaa.1994.1211","article-title":"A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings","volume":"184","year":"1994","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","first-page":"107","article-title":"Hyperstability of a class of linear functional equations","volume":"17","author":"Maksa","year":"2001","journal-title":"Acta Math. Acad. Paedagog. Nyh\u00e1zi"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1215\/S0012-7094-49-01639-7","article-title":"Approximately isometric and multiplicative transformations on continuous function rings","volume":"16","author":"Bourgin","year":"1949","journal-title":"Duke Math. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s00010-012-0168-4","article-title":"Remarks on hyperstability of the the Cauchy equation","volume":"86","year":"2013","journal-title":"Aequationes Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1017\/S0004972713000683","article-title":"A hyperstability result for the Cauchy equation","volume":"89","year":"2014","journal-title":"Bull. Aust. Math. Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/8\/2\/76\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:01:08Z","timestamp":1760187668000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/8\/2\/76"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,25]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["axioms8020076"],"URL":"https:\/\/doi.org\/10.3390\/axioms8020076","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2019,6,25]]}}}