Stability and superstability of n-Jordan ∗-homomorphisms in Frechet locally C*-algebras

Document Type : Research Paper


1 Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, P.O. Box 98135-674, Zahedan, Iran

2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea


Using fixed point methods, we prove the Hyers-Ulam stability and the superstability of $n$-Jordan $*$-homomorphisms in Fr'echet locally $C^*$-algebras for the generalized Jensen-type functional equation
$$r f\left(\frac{ a+b}{ r} \right) + r f\left( \frac{a-b}{r}\right) =2f(a),$$
where $r$ is a fixed  real number greater than $1$.


[1] R.P.B. Agarwal, B. Xu and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl. 288 (2003), 852–869.

[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.

[3] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1.

[4] L. C˘adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer. Math. Ber. 346 (2004), 43–52.

[5] S. Czerwik, Function equations and inequalities in several variables, World Scientific, River Edge, NJ, USA, 2002.

[6] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Am. Math. Soc. 74 (1968), 305–309.

[7] M. Eshaghi Gordji, n-Jordan homomorphisms, Bull. Aust. Math. Soc. 80 (2009), 159–164.

[8] M. Eshaghi Gordji, On approximate n-ring homomorphisms and n-ring derivations, Nonlinear Funct. Anal. Appl. 2009 (2009), 1–8.

[9] M. Eshaghi Gordji, Nearly Ring Homomorphisms and Nearly Ring Derivations on Non-Archimedean Banach Algebras, Abstr. Appl. Anal. 2010 (2010), 1–12.

[10] M. Eshaghi Gordji and Z. Alizadeh, Z., Stability and superstability of ring homomorphisms on nonarchimedean banach algebras, Abstr. Appl. Anal. 2011, Article ID 123656, 10 pages, 2011.

[11] M. Eshaghi Gordji, A. Bodaghi and I.A., Alias, On the stability of quadratic double centralizers and quadratic multipliers: a fixed point approach, J. Inequal. Appl. 2011 (2011), Article ID 957541, 12 pages.

[12] M. Eshaghi Gordji, M.B. Ghaemi, S. Kaboli Gharetapeh, S. Shams and A. Ebadian, On the stability of J∗- derivations, J. Geom. Phys. 60 (2010), 454–459.

[13] G.L. Forti, Hyers-Ulam Stability of functional equations in several variables, Aequationes Math. 50 (1995), 143–190.

[14] P. Gavruta, A generalization of the Hyers-Ulam stability of approximately additive mapping, J. Math. Anal. Appl. 184 (1994), 431–436.

[15] S. Ghaffary and K. Ghasemi, Hyers-Ulam-Rassias Stability of n-jordan *-homomorphisms on C*-algebras, Bull. Iran. Math. Soc. 39 (2013), no. 2, 347–353.

[16] S. Hejazian, M. Mirzavaziri and M.S. Moslehian, n-Homomorphisms, Bull. Iran. Math. Soc. 31 (2005), 13–23.

[17] Hyers, D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.

[18] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and Their Applications, Boston, MA, USA, 1998.

[19] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153.
[20] A. Inoue, Locally C∗-algebra, Mem. Fac. Sci. Kyushu Univ. Ser. A. 25 (1971), 197–235.

[21] G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of ψ-additive mapping, J. Approx. Theory 72 (1993), 131–137.

[22] G. Isac and Th. M. Rassias, Stability of ψ-additive mapping, J. Math. Sci. 19 (1996), 219–228.

[23] J. Jamalzadeh, K. Ghasemi and S. Ghaffary, n-Jordan ∗-derivations in Fr´echet locally C∗-algebras, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 555–562.

[24] S. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Harmonic Press, Palm Harbor, FL, USA, 2001.

[25] S. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer, 2011.

[26] Y.-H. Lee, S.-M. Jung and Th. M. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic additive equation, J. Math. Inequal. 12 (2018), no. 1, 43—61.

[27] C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. 132 (2008), 87–96.

[28] C. Park, Homomorphisms between Poisson JC∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97.

[29] C. Park, S. Ghaffary, K. Ghasemi and S.Y. Jang, Fuzzy n-Jordan *-homomorphisms in induced fuzzy C*-algebras, Adv. Differ. Equ. 42 (2012), 1–10.

[30] C. Park, S. Ghaffary and K. Ghasemi, Approximate n-Jordan *-derivations on C*-algebras and JC*-algebras, J. Inequal. Appl. 273 (2012), 1–10.

[31] C. Park and J.M. Rassias, Stability of the Jensen-type functional equation in C∗-algebras: A fixed point approach, Abstr. Appl. Anal. 2009 (2009), Article ID 360432, 17 pages.

[32] C. Park and Th. M. Rassias, Hom-derivations in C*-ternary algebras, Acta Math. Sinica, English Ser., 36 (2020),no. 9, 1025–1038.

[33] C. Park and Th. M. Rassias, Additive functional equations and partial multipliers in C*-algebras, Rev. Real Acad. Ciencias Exactas, Ser. A. Mat. 113 (2019), no. 3, 2261–2275.

[34] C.N. Philips, Inverse limits of C∗-algebras, J. Operator Theory 19 (1988), 159–195.

[35] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

[36] Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mapping, J. Math. Anal. Appl. 246 (2000), 352–378.

[37] S.M. Ulam, Problems in modern mathematics, Chapter VI. Science ed. Wily, New York, 1940.

[38] B. Xu and W. Zhang, Hyers-Ulam stability for a nonlinear iterative equation, Colloq. Math. 93 (2002), 1–9.
Volume 14, Issue 2
February 2023
Pages 23-30
  • Receive Date: 15 October 2020
  • Revise Date: 10 January 2021
  • Accept Date: 04 March 2021