{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:01:22Z","timestamp":1760058082791,"version":"build-2065373602"},"reference-count":75,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T00:00:00Z","timestamp":1741737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Jouf University through the Fast-Track Research Funding Program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We show how to obtain new results on the Ulam stability of the quadratic equation q(a+b)+q(a\u2212b)=2q(a)+2q(b) using the Banach limit and the fixed point theorem obtained quite recently for some function spaces. The equation is modeled on the parallelogram identity used by Jordan and von Neumann to characterize the inner product spaces. Our main results state that the maps, from the Abelian groups into the set of reals, that satisfy the equation approximately (in a certain sense) are close to its solutions. In this way, we generalize several previous similar outcomes, by giving much finer estimations of the distances between such solutions to the equation. We also present a simplified survey of the earlier related outcomes.<\/jats:p>","DOI":"10.3390\/axioms14030206","type":"journal-article","created":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T07:31:42Z","timestamp":1741764702000},"page":"206","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Banach Limit and Fixed Point Approach in the Ulam Stability of the Quadratic Functional Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4955-0842","authenticated-orcid":false,"given":"El-sayed","family":"El-hady","sequence":"first","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6529-4885","authenticated-orcid":false,"given":"Janusz","family":"Brzd\u0119k","sequence":"additional","affiliation":[{"name":"Faculty of Applied Mathematics, AGH University of Krak\u00f3w, Mickiewicza 30, 30-059 Krak\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hyers, D.H., Isac, G., and Rassias, T.M. (1998). Stability of Functional Equations in Several Variables, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-1790-9"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Jung, S.-M. (2011). Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer.","DOI":"10.1007\/978-1-4419-9637-4"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Brzd\u0119k, J., Popa, D., Ra\u015fa, I., and Xu, B. (2018). Ulam Stability of Operators, Academic Press.","DOI":"10.1007\/978-3-030-28972-0"},{"key":"ref_4","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. Nat. Acad. Sci. USA"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"64","DOI":"10.2969\/jmsj\/00210064","article-title":"On the stability of the linear transformation in Banach spaces","volume":"2","author":"Aoki","year":"1950","journal-title":"J. Math. Soc. Jpn."},{"key":"ref_6","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_7","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_8","doi-asserted-by":"crossref","first-page":"397","DOI":"10.4134\/BKMS.2008.45.2.397","article-title":"On the stability of the monomial functional equation","volume":"45","author":"Lee","year":"2008","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Brzd\u0119k, J. (2024). Some remarks on the best Ulam constant. Symmetry, 16.","DOI":"10.3390\/sym16121644"},{"key":"ref_10","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_11","doi-asserted-by":"crossref","unstructured":"Badora, R., Brzd\u0119k, J., and Ciepli\u0144ski, K. (2021). Applications of Banach limit in Ulam stability. Symmetry, 13.","DOI":"10.3390\/sym13050841"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"719","DOI":"10.2307\/1968653","article-title":"On inner product in linear metric spaces","volume":"36","author":"Jordan","year":"1935","journal-title":"Ann. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alsina, C., Sikorska, J., and Tom\u00e1s, M.S. (2009). Norm Derivatives and Characterizations of Inner Product Spaces, World Scientific.","DOI":"10.1142\/9789814287272"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Acz\u00e9l, J., and Dhombres, J. (1989). Functional Equations in Several Variables, Cambridge University Press.","DOI":"10.1017\/CBO9781139086578"},{"key":"ref_15","first-page":"40","article-title":"A note on a generalization of the quadratic functional equation","volume":"15","author":"Fadli","year":"2016","journal-title":"Math-Rech. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/BF02941618","article-title":"On the stability of the quadratic mapping in normed spaces","volume":"62","author":"Czerwik","year":"1992","journal-title":"Abh. Math. Sem. Univ. Hamburg."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF02924890","article-title":"Local properties and approximations of operators","volume":"53","author":"Skof","year":"1983","journal-title":"Rend. Sem. Math. Fis. Milano"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1007\/BF02192660","article-title":"Remarks on the stability of functional equations","volume":"27","author":"Cholewa","year":"1984","journal-title":"Aequat. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1016\/j.jmaa.2003.11.021","article-title":"The stability of the quadratic functional equation on amenable groups","volume":"291","author":"Yang","year":"2004","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","unstructured":"Czerwik, S. (1994). The stability of the quadratic functional equation. Stability of Mappings of Hyers-Ulam Type, Hadronic Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1006\/jmaa.1998.5916","article-title":"On the Hyers-Ulam stability of the functional equations that have the quadratic property","volume":"222","author":"Jung","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1155\/S0161171201004707","article-title":"On the stability of the quadratic mapping in normed spaces","volume":"25","author":"Kim","year":"2001","journal-title":"Int. J. Math. 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E-Notes"},{"key":"ref_27","first-page":"645","article-title":"Hyers-Ulam stability of the quadratic equation of Pexider type","volume":"38","author":"Jung","year":"2001","journal-title":"J. Korean Math. Soc."},{"key":"ref_28","first-page":"247","article-title":"On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution","volume":"12","author":"Bouikhalene","year":"2007","journal-title":"Nonlin. Funct. Anal. Appl."},{"key":"ref_29","first-page":"27","article-title":"Ulam-G\u0103vru\u0163a-Rassias stability of the Pexider functional equation","volume":"7","author":"Bouikhalene","year":"2007","journal-title":"Int. J. Appl. Math. Stat. IJAMAS"},{"key":"ref_30","first-page":"805","article-title":"On the Hyers\u2013Ulam stability of approximately Pexider mappings","volume":"11","author":"Bouikhalene","year":"2008","journal-title":"Math. Ineq. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Elqorachi, E., Manar, Y., and Rassias, T.M. (2011). Hyers-Ulam stability of the quadratic functional equation. Functional Equations in Mathematical Analysis, Springer.","DOI":"10.1007\/978-1-4614-0055-4_8"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1017\/S0004972717001137","article-title":"On the hyperstability of a pexiderized \u03c3-quadratic functional equation on semigroups","volume":"97","year":"2018","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/s10474-016-0602-5","article-title":"On stability of a functional equation of quadratic type","volume":"149","author":"Moslehian","year":"2016","journal-title":"Acta Math. Hungar."},{"key":"ref_34","first-page":"2596","article-title":"Hyperstability results for generalized quadratic functional equations in (2, \u03b1)-Banach spaces","volume":"13","author":"Sintunavarat","year":"2023","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_35","first-page":"5636101","article-title":"On the stability of quadratic functional equations in F-spaces","volume":"2016","author":"Yang","year":"2016","journal-title":"J. Funct. Spaces"},{"key":"ref_36","first-page":"89","article-title":"Hyers-Ulam-Rassias Stability of the K-quadratic functional equation","volume":"8","author":"Sibaha","year":"2007","journal-title":"J. Ineq. Pure Appl. Math."},{"key":"ref_37","first-page":"67","article-title":"Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation","volume":"21","author":"Chahbi","year":"2015","journal-title":"Arab J. Math. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1016\/S0252-9602(17)30103-0","article-title":"On hyperstability of the biadditive functional equation","volume":"37","author":"Chahbi","year":"2017","journal-title":"Acta Math. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3418","DOI":"10.1016\/j.camwa.2011.08.057","article-title":"On the generalized Hyers-Ulam stability of multi-quadratic mappings","volume":"62","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"415053","DOI":"10.1155\/2013\/415053","article-title":"Solution and stability of the multiquadratic functional equation","volume":"2013","author":"Zhao","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"5230","DOI":"10.3934\/math.2020336","article-title":"Almost multi-quadratic mappings in non-Archimedean spaces","volume":"5","author":"Bodaghi","year":"2020","journal-title":"AIMS Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1186\/s13660-021-02682-z","article-title":"Functional inequalities for generalized multi-quadratic mappings","volume":"2021","author":"Bodaghi","year":"2021","journal-title":"J. Inequal. Appl."},{"key":"ref_43","first-page":"39","article-title":"Stability of two multi-quadratic mappings by a fixed point method","volume":"6","author":"Mazdarani","year":"2024","journal-title":"Math. Anal. Cont. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"493751","DOI":"10.1155\/2008\/493751","article-title":"Generalized Hyers-Ulam stability of quadratic functional equations: A fixed point approach","volume":"2008","author":"Park","year":"2008","journal-title":"Fixed Point Theory Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"193035","DOI":"10.1155\/2009\/193035","article-title":"Fixed points and stability of a generalized quadratic functional equation","volume":"2009","author":"Najati","year":"2009","journal-title":"J. Inequal. Appl."},{"key":"ref_46","first-page":"123","article-title":"On the Hyers-Ulam-Rassias stability problem for quadratic functional equations","volume":"16","author":"Gordji","year":"2010","journal-title":"East J. Approx."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1186\/1029-242X-2011-79","article-title":"On the Ulam-Hyers stability of a quadratic functional equation","volume":"2011","author":"Lee","year":"2011","journal-title":"J. Inequal. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1186\/1029-242X-2012-177","article-title":"Stability of quadratic functional equations in tempered distributions","volume":"2012","author":"Lee","year":"2012","journal-title":"J. Inequal. Appl."},{"key":"ref_49","first-page":"283173","article-title":"Hyers-Ulam stability for a class of quadratic functional equations via a typical form","volume":"2013","author":"Kim","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_50","first-page":"61","article-title":"On the stability of \u03c3-quadratic functional equation","volume":"2","author":"Bounader","year":"2013","journal-title":"Res. Comm. Math. Math. Sci."},{"key":"ref_51","first-page":"1","article-title":"Stability of quadratic functional equations in 2-Banach space","volume":"15","author":"Patel","year":"2013","journal-title":"Gen. Math. Notes"},{"key":"ref_52","first-page":"17","article-title":"A fixed point approach to the stability of additive-quadratic-quartic functional equations","volume":"11","author":"Bodaghi","year":"2020","journal-title":"Int. J. Nonlin. Anal. Appl."},{"key":"ref_53","first-page":"9953214","article-title":"Hyers-Ulam stability of functional equation deriving from quadratic mapping in non-Archimedean (n, \u03b2)- normed spaces","volume":"2021","author":"Alessa","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_54","first-page":"1167","article-title":"Stability of a quadratic functional equation","volume":"16","author":"Karthikeyan","year":"2021","journal-title":"Adv. Dyn. Sys. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Tamilvanan, K., Alanazi, A.M., Rassias, J.M., and Alkhaldi, A.H. (2021). Ulam stabilities and instabilities of Euler-Lagrange-Rassias quadratic functional equation in non-Archimedean IFN spaces. Mathematics, 9.","DOI":"10.3390\/math9233063"},{"key":"ref_56","first-page":"3021457","article-title":"Characterization and stability of multi-Euler-Lagrange quadratic functional equations","volume":"2022","author":"Bodaghi","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Tamilvanan, K., Alanazi, A.M., Alshehri, M.G., and Kafle, J. (2021). Hyers-Ulam stability of quadratic functional equation based on fixed point technique in Banach spaces and non-Archimedean Banach spaces. Mathematics, 9.","DOI":"10.3390\/math9202575"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1007\/s00010-014-0317-z","article-title":"On stability of the general linear equation","volume":"89","author":"Bahyrycz","year":"2015","journal-title":"Aequat. Math."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s00010-016-0418-y","article-title":"Hyperstability of general linear functional equation","volume":"90","author":"Bahyrycz","year":"2016","journal-title":"Aequat. Math."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1007\/s00010-020-00703-8","article-title":"Ulam-stability of a generalized linear functional equation, a fixed point approach","volume":"94","author":"Benzarouala","year":"2020","journal-title":"Aequat. Math."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1007\/s00010-018-0619-7","article-title":"The hyperstability of general linear equation via that of Cauchy equation","volume":"93","author":"Phochai","year":"2019","journal-title":"Aequat. Math."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1017\/S0004972720000556","article-title":"Hyperstability of generalised linear functional equations in several variables","volume":"102","author":"Phochai","year":"2020","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1017\/S0004972715000416","article-title":"On Hyperstability of generalised linear functional equations in several variables","volume":"92","author":"Zhang","year":"2015","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/s00010-015-0393-8","article-title":"On Hyers\u2013Ulam stability of generalized linear functional equation and its induced Hyers\u2013Ulam programming problem","volume":"90","author":"Zhang","year":"2016","journal-title":"Aequat. Math."},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Aboutaib, I., Benzarouala, C., Brzd\u0119k, J., Le\u015bniak, Z., and Oubbi, L. (2022). Ulam stability of a general linear functional equation in modular spaces. Symmetry, 14.","DOI":"10.3390\/sym14112468"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1007\/s00025-023-01840-7","article-title":"On Ulam stability of the inhomogeneous version of the general linear functional equation","volume":"78","author":"Benzaruala","year":"2023","journal-title":"Results Math."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s11784-022-01034-8","article-title":"A fixed point theorem and Ulam stability of a general linear functional equation in random normed spaces","volume":"25","author":"Benzaruala","year":"2023","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_68","unstructured":"Banach, S. (1932). Th\u00e9orie des Op\u00e9rations Lin\u00e9aires, Monografje Matematyczne I."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF02393648","article-title":"A contribution to the theory of divergent sequences","volume":"80","author":"Lorentz","year":"1948","journal-title":"Acta Math."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"308","DOI":"10.2307\/2316038","article-title":"Banach limits","volume":"74","author":"Sucheston","year":"1967","journal-title":"Am. Math. Mon."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"461","DOI":"10.4171\/lem\/61-3\/4-8","article-title":"La trace de Dixmier et autres traces","volume":"61","author":"Guichardet","year":"2015","journal-title":"Enseign. Math."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1007\/s41478-019-00184-2","article-title":"Banach limits: Some new thoughts and perspectives","volume":"29","author":"Sofi","year":"2021","journal-title":"J. Anal."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1070\/RM9901","article-title":"Geometry of Banach limits and their applications","volume":"75","author":"Semenov","year":"2020","journal-title":"Russ. Math. Surv."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s13398-022-01219-3","article-title":"Banach limit, fixed points and Ulam stability","volume":"116","year":"2022","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Ser. A Matem\u00e1ticas"},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Ahmad, H., Din, F.U., Younis, M., and Guran, L. (2024). Analyzing the Chaotic Dynamics of a Fractional-Order Dadras\u2013Momeni System Using Relaxed Contractions. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8120699"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/206\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:51:06Z","timestamp":1760028666000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/206"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,12]]},"references-count":75,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030206"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030206","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,3,12]]}}}