{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T15:29:41Z","timestamp":1774279781893,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Tianchi Young-Doctors Talent Introduction Program, Xinjiang Uyghur Autonomous Region, China","award":["5105240151q"],"award-info":[{"award-number":["5105240151q"]}]},{"name":"Tianchi Young-Doctors Talent Introduction Program, Xinjiang Uyghur Autonomous Region, China","award":["12061069"],"award-info":[{"award-number":["12061069"]}]},{"name":"Tianchi Young-Doctors Talent Introduction Program, Xinjiang Uyghur Autonomous Region, China","award":["12071491"],"award-info":[{"award-number":["12071491"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["5105240151q"],"award-info":[{"award-number":["5105240151q"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12061069"],"award-info":[{"award-number":["12061069"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071491"],"award-info":[{"award-number":["12071491"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["5105240151q"],"award-info":[{"award-number":["5105240151q"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12061069"],"award-info":[{"award-number":["12061069"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071491"],"award-info":[{"award-number":["12071491"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Over the past two decades, significant advancements have been made in understanding the stability according to Hyers\u2013Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, \u2225\u03b7(gh)\u2225=\u2225\u03b7(g)+\u03b7(h)\u2225 and \u2225\u03b7(gh\u22121)\u2225=\u2225\u03b7(g)\u2212\u03b7(h)\u2225foreveryg,h\u2208G, where (G,\u00b7) denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function \u03b7:G\u2192B is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where p\u22651. This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers\u2013Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form.<\/jats:p>","DOI":"10.3390\/sym16070825","type":"journal-article","created":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T04:23:43Z","timestamp":1719980623000},"page":"825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Isometries to Analyze the Stability of Norm-Based Functional Equations in p-Uniformly Convex Spaces"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4978-9287","authenticated-orcid":false,"given":"Muhammad","family":"Sarfraz","sequence":"first","affiliation":[{"name":"School of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}]},{"given":"Jiang","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6212-3773","authenticated-orcid":false,"given":"Mazhar","family":"Islam","sequence":"additional","affiliation":[{"name":"College of Electronics and Information Engineering, Shenzhen University, Shenzhen 518060, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4322-308X","authenticated-orcid":false,"given":"Yongjin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,1]]},"reference":[{"key":"ref_1","unstructured":"Ulam, S.M. 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