Document Type : Research Paper


Department of Mathematics, Payame Noor University, Tehran, Iran.


In this paper, we introduce the notion of the ternary generalized Jordan ring homomorphism on ternary non-Archimedean Banach algebras. Utilizing  alternative fixed point methods, we establish the generalized Hyers-Ulam stability of ternary generalized Jordan ring homomorphisms on ternary non-Archimedean Banach algebras associated with the generalized additive functions in several variables.


Main Subjects

[1] R. Badora, On approximate ring homomorphisms, J. Math. Anal. Appl., 276 (2002), pp. 589-597.
[2] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), pp. 223-237.
[3] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16 (1949), pp. 385-397.
[4] L. Cadariu and V. Radu, Fixed Point Methods for the Generalized Stability of Functional Equations in a Single Variable, Fixed Point Theory Appl., 2008, Article ID 749392 (2008), 15 pages.
[5] A. Ebadian, I. Nikoufar, Th. M. Rassias and N. Ghobadipour, Stability of generalized derivations on Hilbert $C^*$-modules associated to a Pexiderized Cauchy-Jensen type functional equation, Acta Math. Sci., 32B (2012), pp. 1226-1238.
[6] M. Eshaghi Gordji, $n$-Jordan homomorphisms, Bull. Austral. Math. Soc., 80 (2009), pp. 159-164.
[7] M. Eshaghi Gordji, T. Karimi and S.K. Gharetapeh, Approximately $n$-Jordan homomorphisms on Banach algebras, J. Inequal. Appl., 870843 (2009).
[8] M. Eshaghi Gordji, Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras, Abstr. Appl. Anal., 2010, Article ID 393247 (2010), 12 pages.
[9] M. Eshaghi Gordji and A. Fazeli, Stability and superstability of homomorphisms on $C^*$-ternary algebras, An. t. Univ. Ovidius Constana, 20(1) (2012), pp. 173-188.
[10] M. Eshaghi Gordji, A. Jabbari and E. Karapinar, Automatic continuity of surjective $n$-homomorphisms on Banach algebras, Bull. Iran. Math. Soc., 41 (2015), pp. 1207-1211.
[11] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), pp. 431-436.
[12] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), pp. 222-224.
[13] H. Khodaei and Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl., 1 (2010), pp. 22-41.
[14] H. Khodaei, Asymptotic behavior of $n$-Jordan homomorphisms, Mediterr. J. Math., 17, 143 (2020).
[15] A. Khrennikov, Non-Archimedean Analysis: Quantum paradoxes, dynamical systems and biological models, Kluwer Academic Publishers, Dordrecht, 1997.
[16] Y.-H. Lee, Stability of $n$-Jordan homomorphisms from a normed algebra to a Banach algebra, Abstr. Appl. Anal., 2013, Article ID 691025 (2013), pp. 1-5.
[17] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on the generalized complete metric space, Bull. Amer. Math. Soc., 126 (1968), pp. 305-309.
[18] A. Najati, Cauchy-Rassias stability of homomorphisms associated to a Pexiderized Cauchy-Jensen type functional equation, J. Math. Ineq., 3(2) (2009), pp. 257-265.
[19] L. Narici and E. Beckenstein, Strange terrain-non-Archimedean spaces, Amer. Math. Monthly, 88 (1981), pp. 667-676.
[20] I. Nikoufar, Jordan $(\theta, \phi)$-derivations on Hilbert $C^*$-modules, Indag. Math. (N.S.), 26(2) (2015), pp. 421-430.
[21] I. Nikoufar, Superstability of $m$-additive maps on complete non-Archimedean spaces, Sahand Commun. Math. Anal., 2(1) (2015), pp. 19-25.
[22] I. Nikoufar, Refined almost double derivations and Lie $*$-double derivations, Miskolc Math. Notes, 16 (2015), pp. 1063-1071.
[23] I. Nikoufar, Functions near some $(\alpha_1,\alpha_2)$-double Jordan derivations in $p$-Banach algebras, Boll. Unione Mat. Ital., 10(2) (2017), pp. 191-198.
[24] I. Nikoufar, A correction to approximation of generalized homomorphisms in quasi-Banach algebras, Miskolc Math. Notes, 19(1) (2018), pp. 423-430.
[25] I. Nikoufar, Stability of multi-quadratic functions in Lipschitz spaces, Iran J. Sci. Technol. Trans. Sci., 43 (2019), pp. 621-625. 
[26] T. Miura, S.-E. Takahasi and G. Hirasawa, Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl., 2005(4) (2005), pp. 435-441.
[27] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), pp. 297-300.
[28] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), pp. 126-130.
[29] N. Shilkret, Non-Archimedian Banach algebras, Thesis (Ph.D.)-Polytechnic University. 178 pages, ProQuest LLC, Thesis, 1968.
[30] S.M. Ulam, A collection of the mathematical problems, Interscience Publ., New York, 1960.