AIMS Mathematics, 2020, 5(5): 4482-4493. doi: 10.3934/math.2020288

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Homomorphism-derivation functional inequalities in C*-algebras

1 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
2 Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, Shaanxi, P. R. China

In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1} && \left\|g\left(x+y\right) -g(x) -g(y)\right\| + \left\|2 h\left(\frac{x+y}{2}\right) - h(x) - h(y) \right\| \\ && \le \left\| s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \|t ( h(x+y)-h(x)-h(y))\|, \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $ |t| <1$. Furthermore, we investigate homomorphisms and derivations in complex Banach algebras and unital $C^*$-algebras, associated to the additive-additive $(s,t)$-functional inequality (0.1) under some extra condition.
Moreover, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1t} \nonumber && \|g\left(x+y+z\right) -g(x) -g(y)-g(z)\| +\left\|3h\left(\frac{x+y+z}{3}\right)+ h(x-2y+z) + h(x+y-2z)-3 h(x) \right\| \\ && \le \left\|s\left( 3 g\left(\frac{x+y+z}{3}\right)-g(x)-g(y)-g(z)\right)\right\| \\ && + \left\|t \left( h(x+y+z) + h(x-2y+z) + h(x+y-2z)-3 h(x) \right) \right\|, \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $ |t| <1$. Furthermore, we investigate $C^*$-ternary derivations and $C^*$-ternary homomorphisms in $C^*$-ternary algebras, associated to the additive-additive $(s,t)$-functional inequality (0.1) under some extra condition.
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References

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

2. J. Bae, I. Chang, Some additive mappings on Banach *-algebras with derivation, J. Nonlinear Sci. Appl., 11 (2018), 335-341.

3. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.

4. Y. Ding, Ulam-Hyers stability of fractional impulsive differential equations, J. Nonlinear Sci. Appl., 11 (2018), 953-959.

5. N. Eghbali, J. M. Rassias, M. Taheri, On the stability of a k-cubic functional equation in intuitionistic fuzzy n-normed spaces, Results Math., 70 (2016), 233-248.

6. G. Z. Eskandani, P. Găvruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl., 5 (2012), 459-465.

7. P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.    

8. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224.    

9. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, Academic Press, New York, 1983.

10. C. Park, Homomorphisms between Poisson JC*-algebras, B. Braz. Math. Soc., 36 (2005), 79-97.    

11. C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal., 9 (2015), 17-26.

12. C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal., 9 (2015), 397-407.

13. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96.

14. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300.    

15. F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129.    

16. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publication, New York, 1960.

17. Z. Wang, Stability of two types of cubic fuzzy set-valued functional equations, Results Math., 70 (2016), 1-14.

18. H. Zettl, A characterization of ternary rings of operators, Adv. Math., 48 (1983), 117-143.    

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