Approximate solutions of homomorphisms and derivations of the generalized Cauchy-Jensen functional equation in $C^*$-ternary algebras

Subject Areas : Difference and functional equations

P.  Kaskasem ¹ (Department of Mathematics, Faculty of Science Naresuan University, Phitsanulok, 65000, Thailand)
C.  Klin-eam ² (Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok, Thailand)

Received: 2019-05-22 Accepted : 2019-10-29 Published : 2020-03-01

Keywords: Hyers-Ulam-Rassias stability, Cauchy-Jensen functional equations, C*-ternary algebras, alternative fixed point theorem,

Abstract :

In this paper, we prove Hyers-Ulam-Rassias stability of $C^*$-ternary algebra homomorphism for the following generalized Cauchy-Jensen equation $$\eta \mu f\left(\frac{x+y}{\eta}+z\right) = f(\mu x) + f(\mu y) +\eta f(\mu z)$$ for all $\mu \in \mathbb{S}:= \{ \lambda \in \mathbb{C} : |\lambda | =1\}$ and for any fixed positive integer $\eta \geq 2$ on $C^*$-ternary algebras by using fixed poind alternative theorem. Moreover, we investigate Hyers-Ulam-Rassias stability of generalized $C^*$-ternary derivation for such function on $C^*$-algebras by the same method.

References:

[1] Abramov, R. Kerner, B. L. Roy, Hypersymmetry: a Z3-graded generalization of supersymmetry, J. Math. Phys. 38 (1997), 1650-1669.
[2] P. Ara, M. Mathieu, Local Multipliers of C*-algebras, Springer, London, 2003.
[3] C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta. Math. Sin. 22 (6) (2006), 1789-1796.
[4] J. A. Baker, The stability of certain functional equations, Proc. Am. Math. Soc. 112 (1991), 729-732.
[5] J. Brzdek, W. Fechner, M. S. Moslehian, J. Silorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal. 9 (2015), 278-326.
[6] J. Brzdek, D. Popa, I. Rasa, B. Xu, Ulam Stability of Operators, Mathematical Analysis and its Applications, Academic Press, Elsevier, Oxford, 2018.
[7] L. Cadariu, V. Radu, Fixed points and the stability of Jensens functional equation, J. Ineq. Pure Appl. Math. 4 (1) (2003), 2003:4.
[8] J. B. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Am. Math. Soc. 74 (2) (1968), 305-309.
[9] I. E. Fassai, Non-Archimedean hyperstability of Cauchy-Jensen functional equations on a restricted domain, J. Appl. Anal. 24 (2) (2019), 155-165.
[10] Z. X. Gao, H. X. Cao, W. T. Zheng, L. Xu, Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations, J. Math. Inequal. 3 (1) (2009), 63-77.
[11] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (4) (1941), 222-224.
[12] R. V. Kasdison, G. K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249-266.
[13] P. Kaskasem, C. Klin-eam, Approximation of the generalized Cauchy-Jensen functional equation in C-algebras, J. Inequal. Appl. 2018, 2018:236.
[14] P. Kaskasem, C. Klin-eam, Hyperstability of the generalized Cauchy-Jensen functional equation in ultrametric spaces. Acta. Math. Sci. 39 B(4) (2019), 1017-1032.
[15] P. Kaskasem, C. Klin-eam, On approximation solutions of the Cauchy-Jensen and the additive quadratic functional equation in paranormed spaces, Int. J. Anal. Appl. 17 (3) (2019), 369-387.
[16] P. Kaskasem, C. Klin-eam, Y. J. Cho, On the stability of the generalized Cauchy-Jensen set-valued functional equations, J. Fixed Point Theory Appl. 20 (2018), 2018:76.
[17] R. Kerner, Ternary algebraic structures and their applications in physics, Preprint: arXiv:math-ph/0011023, 2000.
[18] H. M. Kenari, R. Saadati, C. Park, Homomorphisms and derivations in C*-ternary algebras via fixed point method, Adv. Difference Equ. 2012, 2012:137.
[19] W. A. J. Luxemburg, On the covergence of successive approximations in the theory of ordinary differential equations, II Koninkl, Nederl. Akademie van Wetenschappen, Indag. Math. 20 (1958), 540-546.
[20] C. Park, Homomorphisms between Poisson JC*-algebras, Bull. Braz. Math. Soc. 36 (2005), 79-97
[21] C. Park, Isomorphisms between C*-ternary algebras, J. Math. Phys. 47 (2006), 2006:103512.
[22] C. Park, J. S. An, Stability of the Cauchy-Jensen functional equation in C*-algebras: a xed point approach, Fixed Point Theory Appl. 2008, 2008:872190.
[23] C. Park, M. S. Moslehian, On the stability of *-derivations on JB*-triples, Bull. Braz. Math. Soc. 38 (1) (2007), 115-127.
[24] G. Polya, G. Szego, Aufgaben und Lehrsatze aus der Analysis vol. I, Springer, Berlin, 1925.
[25] V. Radu, The fixed point alternative and the stability of functional equations, Fixed point Theory. 4 (2003), 91-96.
[26] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (2) (1978), 297-300.
[27] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts. Pure. Appl. Math, New York, USA, 1960.
[28] L. Vainerman, R. Kerner, On special classes of n-algebras, J. Math. Phys. 37 (1996), 2553-2565.