{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T17:05:11Z","timestamp":1761930311945,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,16]],"date-time":"2022-08-16T00:00:00Z","timestamp":1660608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Prince Sultan University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability theorem can be applied to numerous functional equations in investigating the stability when approximated in Banach spaces, Banach algebra, and so on. The main focus of this study is to analyse the relationship between functional equations, Hyers\u2013Ulam\u2013Rassias stability, Banach space, quasi-beta normed spaces, and fixed-point theory in depth. The significance of this work is the incorporation of the stability of the generalised additive functional equation in Banach space and quasi-beta normed spaces by employing concrete techniques like direct and fixed-point theory methods. They are powerful tools for narrowing down the mathematical models that describe a wide range of events. Some classes of functional equations, in particular, have lately emerged from a variety of applications, such as Fourier transforms and the Laplace transforms. This study uses linear transformation to explain our functional equations while providing suitable examples.<\/jats:p>","DOI":"10.3390\/sym14081700","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T03:15:27Z","timestamp":1660706127000},"page":"1700","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Direct and Fixed-Point Stability\u2013Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2542-3627","authenticated-orcid":false,"given":"Agilan","family":"Pasupathi","sequence":"first","affiliation":[{"name":"Department of Mathematics, St. Joseph\u2019s College of Engineering, OMR, Chennai 600 119, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3108-9899","authenticated-orcid":false,"given":"Julietraja","family":"Konsalraj","sequence":"additional","affiliation":[{"name":"Department of Mathematics, St. Joseph\u2019s College of Engineering, OMR, Chennai 600 119, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1511-9760","authenticated-orcid":false,"given":"Nahid","family":"Fatima","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6697-4500","authenticated-orcid":false,"given":"Vallinayagam","family":"Velusamy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, St. Joseph\u2019s College of Engineering, OMR, Chennai 600 119, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9150-7426","authenticated-orcid":false,"given":"Nizar","family":"Souayah","sequence":"additional","affiliation":[{"name":"Department of Natural Sciences, Community College Al-Riyadh, King Saud University, Riyadh 11451, Saudi Arabia"},{"name":"Ecole Sup\u00e9rieure des Sciences Economiques et Commerciales de Tunis, Universit\u00e9 de Tunis, Boulevard du 9 Avril, Tunis 1938, Tunisia"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,16]]},"reference":[{"key":"ref_1","first-page":"214","article-title":"Recherches sur la courbe que forme une corde tendue mise en vibration","volume":"Volume 5","year":"1747","journal-title":"Histoire de l\u2019Acad\u00e9mie Royale des Sciences et Belles-Lettres de Berlin HAB Pour L\u2019ann\u00e9e"},{"unstructured":"Science (1964). Problems in Modern Mathematics, Wiley. Chapter VI, Some Questions in Analysis: 1, Stability.","key":"ref_2"},{"key":"ref_3","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_4","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_5","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. Amer. Math. Soc."},{"key":"ref_6","first-page":"95","article-title":"Solution of a stability problem of Ulam","volume":"12","author":"Rassias","year":"1992","journal-title":"Discuss. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1016\/S0022-247X(03)00136-7","article-title":"On the Ulam stability of Jensen and Jensen type mappings on restricted domains","volume":"281","author":"Rassias","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.bulsci.2006.03.011","article-title":"Refined Hyers Ulam approximation of approximately Jensen type mappings","volume":"131","author":"Rassias","year":"2007","journal-title":"Bull. Sci. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1006\/jmaa.2000.7046","article-title":"On the stability of functional equations in Banach spaces","volume":"251","author":"Rassias","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","first-page":"185","article-title":"On the stability of the Euler-Lagrange functional equation","volume":"20","author":"Rassias","year":"1992","journal-title":"Chin. J. Math."},{"key":"ref_11","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."},{"doi-asserted-by":"crossref","unstructured":"Benyamini, Y., and Lindenstrauss, J. (2000). Geometric Nonlinear Functional Analysis, American Mathematical Society. American Mathematical Society Colloquium Publications.","key":"ref_12","DOI":"10.1090\/coll\/048"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s00009-010-0082-8","article-title":"Generalized Hyers-Ulam Stability for a General Mixed Functional Equation in Quasi-\u03b2-normed Spaces","volume":"8","author":"Eskandani","year":"2011","journal-title":"Mediterr. J. Math."},{"key":"ref_14","first-page":"459","article-title":"On the stability problem in quasi-Banach spaces","volume":"5","author":"Eskandani","year":"2012","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/0022-1236(82)90048-9","article-title":"On approximately of approximately linear mappings by linear mappings","volume":"46","author":"Rassias","year":"1982","journal-title":"J. Funct. Anal. USA"},{"key":"ref_17","first-page":"445","article-title":"On approximately of approximately linear mappings by linear mappings","volume":"108","author":"Rassias","year":"1984","journal-title":"Bull. Sci. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.jmaa.2009.03.005","article-title":"Generalized Hyers-Ulam stability for general additive functional equations in quasi-\u03b2-normed spaces","volume":"356","author":"Rassias","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1907","DOI":"10.1007\/s10114-011-0179-4","article-title":"Approximate (m,n)-Cauchy\u2013Jensen Additive Mappings in C*-algebras","volume":"27","author":"Rassias","year":"2011","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/0021-9045(89)90041-5","article-title":"Solution of a problem of Ulam","volume":"57","author":"Rassias","year":"1989","journal-title":"J. Approx. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/j.bulsci.2005.02.001","article-title":"Asymptotic behavior of alternative Jensen and Jensen type functional equations","volume":"129","author":"Rassias","year":"2005","journal-title":"Bull. Sci. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"10","DOI":"10.26637\/mjm101\/002","article-title":"Additive functional equation and inequality are stable in Banach space and its applications","volume":"1","author":"Arunkumar","year":"2013","journal-title":"Malaya J. Mat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"25","DOI":"10.22457\/apam.v15n1a3","article-title":"Solution and Generalized Ulam-Hyers Stability of a n- Dimensional Additive Functional Equation in Banach Space and Banach Algebra: Direct and Fixed Point Methods","volume":"15","author":"Arunkumar","year":"2017","journal-title":"Ann. Pure Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"276","DOI":"10.26637\/MJM0601\/0031","article-title":"Ulam-Hyers Stability of Euler - Lagrange Additive Functional Equation in Intuitionistic Fuzzy Banach Spaces: Direct and Fixed Point Methods","volume":"6","author":"Arunkumar","year":"2018","journal-title":"Malaya J. Mat."},{"key":"ref_25","first-page":"93","article-title":"Solution and Ulam-Hyers Stability of an Dimensional Additive Functional Equation in Banach Space and Banach Algebra: The Direct and Fixed Point Methods","volume":"120","author":"Arunkumar","year":"2018","journal-title":"Int. J. Pure Appl. Math."},{"unstructured":"G\u0103vrut\u0103, P. (1999). An answer to a question of J.M. Rassias concerning the stability of Cauchy functional equation. Adv. Equ. Inequal. Hadron. Math. Ser., 67\u201371.","key":"ref_26"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1006\/jmaa.2001.7539","article-title":"On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings","volume":"261","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","first-page":"36","article-title":"On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation","volume":"3","author":"Ravi","year":"2008","journal-title":"Int. J. Math. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1504\/IJCAET.2020.107441","article-title":"Generalised Ulam-Hyers stability of a n-dimensional additive functional equation in two different methods","volume":"12","author":"Arunkumar","year":"2020","journal-title":"Int. Comput. Aided Eng. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"012011","DOI":"10.1088\/1742-6596\/1377\/1\/012011","article-title":"Generalized Ulam-Hyers Stability of Complex Additive Functional Equation","volume":"1377","author":"Agilan","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_31","first-page":"305","article-title":"A fixed point theorem of the alternative for contractions on a generalized complete metric space","volume":"126","author":"Margoils","year":"1968","journal-title":"Bull. Amer. Math. Soc."},{"doi-asserted-by":"crossref","unstructured":"Debnath, P., Konwar, N., and Radenovic, S. (2021). Metric Fixed Point Theory, Applications in Science, Engineering and Behavioural Sciences. Forum for Interdisciplinary Mathematics, Springer.","key":"ref_32","DOI":"10.1007\/978-981-16-4896-0"},{"doi-asserted-by":"crossref","unstructured":"Todorcevic, V. (2019). Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, Springer Nature.","key":"ref_33","DOI":"10.1007\/978-3-030-22591-9"},{"doi-asserted-by":"crossref","unstructured":"Cho, Y.J., Jleli, M., Mursaleen, M., Samet, B., and Vetro, C. (2021). Advances in Metric Fixed Point Theory and Applications, Springer.","key":"ref_34","DOI":"10.1007\/978-981-33-6647-3"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1186\/s13660-020-02520-8","article-title":"Hyers-Ulam stability of functional inequalities: A fixed point approach","volume":"2020","author":"Batool","year":"2020","journal-title":"J. Inequalities Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"908","DOI":"10.3934\/math.2021054","article-title":"Ulam stability of a functional equation deriving from quadratic and additive mappings in random normed spaces","volume":"6","author":"Tamilvanan","year":"2021","journal-title":"AIMS Math."},{"key":"ref_37","first-page":"8028634","article-title":"Ulam stability and nonstability of additive functional equation in IFN-spaces and 2-Banach spaces by different methods","volume":"2022","author":"Uthirasamy","year":"2022","journal-title":"J. Funct. 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