{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:41:20Z","timestamp":1760175680153,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T00:00:00Z","timestamp":1585699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Romanian Ministery of Research and Innovation","award":["10PFE\/16.10.2018, PERFORM-TECH-UPT"],"award-info":[{"award-number":["10PFE\/16.10.2018, PERFORM-TECH-UPT"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) . Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B. Kleiner and B. Leeb, using the Hyers-Ulam stability of a Cauchy equation. In this paper, some results on the Ulam\u2019s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm).<\/jats:p>","DOI":"10.3390\/sym12040502","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"502","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ulam\u2019s Type Stability and Generalized Norms"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3110-9805","authenticated-orcid":false,"given":"Laura","family":"Manolescu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Politehnica University of Timi\u015foara, Pia\u0163a Victoriei 2, 300006 Timi\u015foara, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,1]]},"reference":[{"key":"ref_1","unstructured":"Ulam, S.M. (1960). A Collection of Mathematical Problems, Interscience Publ."},{"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. Soc. USA"},{"key":"ref_3","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_4","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_5","first-page":"257","article-title":"On the stability of functional equations with square-symmetric operation","volume":"4","author":"Kim","year":"2001","journal-title":"Math. Ineq. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3","DOI":"10.5644\/SJM.06.1.01","article-title":"An instructive treatment of a generalization of G\u0103vru\u0163a\u2019s stability theorem","volume":"6","author":"Gselmann","year":"2010","journal-title":"Sarajevo J. Math."},{"key":"ref_7","unstructured":"Brzd\u0229k, J., Popa, D., Ra\u015fa, I., and Xu, B. (2018). Ulam Stability of Operators, Academic Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cho, Y.-J., Park, C., Rassias, T.M., and Saddati, R. (2015). Stability of Functional Equations in Banach Algebras, Springer International Publishing.","DOI":"10.1007\/978-3-319-18708-2"},{"key":"ref_9","unstructured":"Czerwik, S. (2003). Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Inc."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hyers, D.H., Isac, G., and Rassias, T.H. (1998). Stability of Functional Equations in Several Variables, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-1790-9"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Jung, S.-M. (2011). Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Science+Business Media.","DOI":"10.1007\/978-1-4419-9637-4"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF02698902","article-title":"Rigidity of quasi-isometries for symmetric spaces Euclidean buildings","volume":"86","author":"Kleiner","year":"1997","journal-title":"Inst. Hautes Etudes Sci. Publ. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"712743","DOI":"10.1155\/2012\/712743","article-title":"Fixed points and generalized Hyers-Ulam stability","volume":"2012","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_14","first-page":"829419","article-title":"Fixed point theory and the Ulam stability","volume":"2014","author":"Cieplinski","year":"2014","journal-title":"J. Funct. Spaces"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/978-3-319-31281-1_12","article-title":"Approximation of functions by additive and by quadratic mappings","volume":"Volume 111","author":"Rassias","year":"2016","journal-title":"Mathematical Analysis, Approximation Theory and Their Applications, Book Series: Springer Optimization and Its Application"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Brzdek, J., Popa, D., and Rassias, T.M. (2019). Approximation by cubic mappings. Ulam Type Stability, Springer International Publishing.","DOI":"10.1007\/978-3-030-28972-0"},{"key":"ref_17","first-page":"381","article-title":"Nonconvex minimization theorems and fixed point theorems in complete metric spaces","volume":"44","author":"Kada","year":"1996","journal-title":"Math. Japonica"},{"key":"ref_18","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. Math. Math. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/S0022-0000(03)00004-7","article-title":"Approximate testing with error relative to imput size","volume":"66","author":"Kiwi","year":"2003","journal-title":"J. Comput. Syst. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1090\/S0002-9939-03-07252-6","article-title":"On an approximate authomorphism on a C*-algebra","volume":"132","author":"Park","year":"2004","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/502\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:14:03Z","timestamp":1760174043000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/502"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,1]]},"references-count":20,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040502"],"URL":"https:\/\/doi.org\/10.3390\/sym12040502","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,4,1]]}}}