In this paper, we study to solve the Cauchy, Jensen and Cauchy-Jensen additive function inequalities with 3k-variables related to Jordan-von Neumann type in the spirit of the Rassias stability approach for approximate homomorphisms in Banach space. These are the main results of this paper.
Cite this paper
An, L. V. (2023). Generalized Stability of Functional Inequalities with 3k-Variables Associated for Jordan-von Neumann-Type Additive Functional Equation. Open Access Library Journal, 10, e9681. doi: http://dx.doi.org/10.4236/oalib.1109681.
Hyers, D.H. (1941) On the Stability of the Functional Equation. Proceedings of the National Academy of the United States of America, 27, 222-224.
https://doi.org/10.1073/pnas.27.4.222
Rassias, T.M. (1978) On the Stability of the Linear Mapping in Banach Spaces. Proceedings of the American Mathematical Society, 72, 297-300.
https://doi.org/10.1090/S0002-9939-1978-0507327-1
Gajda, Z. (1991) On Stability of Additive Mappings. International Journal of Mathematics and Mathematical Sciences, 14, Article ID: 817959.
https://doi.org/10.1155/S016117129100056X
Găvrut, P. (1994) A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings. Journal of Mathematical Analysis and Applications, 184, 431-436. https://doi.org/10.1006/jmaa.1994.1211
Rätz, J. (2003) On Inequalities Associated with the Jordan-von Neumann Functional Equation. Aequationes Mathematicae, 66, 191-200.
https://doi.org/10.1007/s00010-003-2684-8
Rassias, T.M. and Semrl, P. (1992) On the Behavior of Mappings Which Do Not Satisfy Hyers-Ulam Stability. Proceedings of the American Mathematical Society, 114, 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
Hyers, D.H., Isac, G. and Rassias, T.M. (1998) Stability of Functional Equations in Several Variables. In: Brezis, H., Ed., Progress in Nonlinear Differential Equations and Their Applications, Vol. 34, Birkhäuser, Boston.
https://doi.org/10.1007/978-1-4612-1790-9
Rassias, J.M. (1982) On Approximation of Approximately Linear Mappings by Linear Mappings. Journal of Functional Analysis, 46, 126-130.
https://doi.org/10.1016/0022-1236(82)90048-9
Jun, K.-W. and Lee, Y.-H. (2004) A Generalization of the Hyers-Ulam-Rassias Stability of the Pexiderized Quadratic Equations. Journal of Mathematical Analysis and Applications, 297, 70-86. https://doi.org/10.1016/j.jmaa.2004.04.009
Park, C.-G. (2005) Homomorphisms between Poisson JC-Algebras. Bulletin of the Brazilian Mathematical Society, 36, 79-97.
https://doi.org/10.1007/s00574-005-0029-z
Park, C. (2008) Hyers-Ulam-Rassias Stability of Homomorphisms in Quasi-Banach Algebras. Bulletin des Sciences Mathématiques, 132, 87-96.
https://doi.org/10.1016/j.bulsci.2006.07.004
Aoki, T. (1950) On the Stability of the Linear Transformation in Banach Space. Journal of the Mathematical Society of Japan, 2, 64-66.
https://doi.org/10.2969/jmsj/00210064
Bahyrycz, A. and Piszczek, M. (2014) Hyers Stability of the Jensen Function Equation. Acta Mathematica Hungarica, 142, 353-365.
https://doi.org/10.1007/s10474-013-0347-3
Balcerowski, M. (2013) On the Functional Equations Related to a Problem of Z. Boros and Z. Dróczy. Acta Mathematica Hungarica, 138, 329-340.
https://doi.org/10.1007/s10474-012-0278-4
Fechner, W. (2006) Stability of a Functional Inequlities Associated with the Jordan-von Neumann Functional Equation. Aequationes Mathematicae, 71, 149-161.
https://doi.org/10.1007/s00010-005-2775-9
Prager, W. and Schwaiger, J. (2013) A System of Two Inhomogeneous Linear Functional Equations. Acta Mathematica Hungarica, 140, 377-406.
https://doi.org/10.1007/s10474-013-0315-y
Maligranda, L. (2008) Tosio Aoki (1910-1989). In: Kato, M. and Maligranda, L., Eds., International Symposium on Banach and Function Spaces (14/09/2006-17/09/2006), Yokohama Publishers, Yokohama, 1-23.
Najati, A. and Eskandani, G.Z. (2008) Stability of a Mixed Additive and Cubic Functional Equation in Quasi-Banach Spaces. Journal of Mathematical Analysis and Applications, 342, 1318-1331. https://doi.org/10.1016/j.jmaa.2007.12.039
Fechner, W. (2010) On Some Functional Inequalities Related to the Logarithmic Mean. Acta Mathematica Hungarica, 128, 31-45.
https://doi.org/10.1007/s10474-010-9153-3
Park, C. (2014) Additive β—Functional Inequalities. Journal of Nonlinear Sciences and Applications, 7, 296-310. https://doi.org/10.22436/jnsa.007.05.02
An, L.V. (2019) Hyers-Ulam Stability of Functional Inequalities with Three Variables in Banach Spaces and Non-Archimedean Banach Spaces. International Journal of Mathematical Analysis, 13, 519-537. https://doi.org/10.12988/ijma.2019.9954
Park, C. (2015) Functional in Equalities in Non-Archimedean Normed Spaces. Acta Mathematica Sinica, English Series, 31, 353-366
https://doi.org/10.1007/s10114-015-4278-5
Jung Rye Lee, J.R., Park, C. and Shin, D.Y. (2014) Additive and Quadratic Functional in Equalities in Non-Archimedean Normed Spaces. International Journal of Mathematical Analysis, 8, 1233-1247. https://doi.org/10.12988/ijma.2014.44113
Cho, Y.J., Park, C. and Saadati, R. (2010) Functional Inequalities in Non-Archimedean Normed Spaces. Applied Mathematics Letters, 23, 1238-1242.
https://doi.org/10.1016/j.aml.2010.06.005
An, L.V. (2020) Hyers-Ulam Stability Additive β-Functional Inequalities with Three Variables in Non-Archimedean Banach Space and Complex Banach Spaces. International Journal of Mathematical Analysis, 14, 219-239.
https://doi.org/10.12988/ijma.2020.91169
An, L.V. (2021) Generalized Hyers-Ulam Stability of the Additive Functional Inequalities with 2n-Variables in Non-Archimedean Banach Spaces. Bulletin of Mathematics and Statistics Research, 9, 67-73.
http://bomsr.com/9.3.21/67-73%20LY%20VAN%20AN.pdf