**Title:** Hom-derivations in Banach Algebras

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-22-00015; **Volume 4 / Issue 3 / Year 2022 (Special Issue)**, Pages 282-291

**Document Type:** Research Paper

**Author(s):** Siriluk Paokanta ^{a} , Jung Rye Lee ^{b} , Choonkil Park ^{c}

^{a}Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

^{b}Department of Data Science, Daejin University, Kyunggi 11159, Korea

^{c}Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received:4 April 2022, Accepted:25 December 2022, Published:13 February 2023.

**Corresponding Author:** Jung Rye Lee (Email address: jrlee@daejin.ac.kr)

**Full Text:** PDF

**Abstract**

In this paper, we introduce hom-derivations in complex Banach algebras. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of hom-derivations in complex Banach algebras, associated with the bi-additive *s*-functional inequality

*
*

where *s* is a fixed nonzero complex number with |*s*|<1.

**Keywords:** Hom-biderivation, Complex Banach algebra, Hyers-Ulam stability, Fixed point method, Bi-additive *s*-functional inequality

**References:**

- T. Aoki,
*On the stability of the linear transformation in Banach spaces*, J. Math. Soc. Japan**2**, 64–66, 1950. - J. Bae and W. Park,
*Approximate bi-homomorphisms and bi-derivations in*, Bull. Korean Math. Soc.*C*^{*}-ternary algebras**47**, 195–209, 2010. - L. Cădariu and V. Radu,
*Fixed points and the stability of Jensen’s functional equation*, J. Inequal. Pure Appl. Math.**4 (1)**, 2003; Article ID: 4. - L. Cădariu and V. Radu,
*On the stability of the Cauchy functional equation: A fixed point approach*, Grazer Math. Ber.**346**, 43–52, 2004. - L. Cădariu and V. Radu,
*Fixed point methods for the generalized stability of functional equations in a single variable*, Fixed Point Theory Appl.**2008**, 2008; Article ID: 749392. - J. Diaz and B. Margolis,
*A fixed point theorem of the alternative for contractions on a generalized complete metric space*, Bull. Amer. Math. Soc.**74**, 305–309, 1968. - Iz. EL-Fassi,
*Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-*, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM*β*-Banach spaces**113 (2)**, 675–687, 2019. - M. Eshaghi Gordji, M. B. Ghaemi and B. Alizadeh,
*A fixed point method for perturbation of higher ring derivationsin non-Archimedean Banach algebras,*Int. J. Geom. Methods Mod. Phys.**8 (7)**, 1611–1625, 2011. - M. Eshaghi Gordji and N. Ghobadipour,
*Stability of (*Int. J. Geom. Methods Mod. Phys.*α*,*β*,*γ*)-derivations on Lie*C*^{*}-algebras,**7**, 1097–1102, 2010. - W. Fechner,
*Stability of a functional inequalities associated with the Jordan-von Neumann functional equation*, Aequationes Math.**71**, 149–161, 2006. - P. Gǎvruta,
*A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings*, J. Math. Anal. Appl.**184**, 431–436, 1994. - A. Gilányi,
*Eine zur Parallelogrammgleichung äquivalente Ungleichung*, Aequationes Math.**62**, 303–309, 2001. - A. Gilányi,
*On a problem by K. Nikodem*, Math. Inequal. Appl.**5**, 707–710, 2002. - D.H. Hyers,
*On the stability of the linear functional equation*, Proc. Natl. Acad. Sci. USA**27**, 222–224, 1941. - G. Isac and Th. M. Rassias,
*Stability of*, Int. J. Math. Math. Sci.*ψ*-additive mappings: Applications to nonlinear analysis**19**, 219–228, 1996. - S. Jung, D. Popa and M. Th. Rassias,
*On the stability of the linear functional equation in a single variable on complete metric spaces*, J. Global Optim.**59**, 13–16, 2014. - Y. Lee, S. Jung and M. Th. Rassias,
*Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation*, J. Math. Inequal.**12**, 43–61, 2018. - D. Miheţ and V. Radu,
*On the stability of the additive Cauchy functional equation in random normed spaces*, J. Math. Anal. Appl.**343**, 567–572, 2008. - C. Park,
*Additive*, J. Math. Inequal.*ρ*-functional inequalities and equations**9**, 17–26, 2015. - C. Park,
*Additive*, J. Math. Inequal.*ρ*-functional inequalities in non-Archimedean normed spaces**9**, 397–407, 2015. - C. Park,
*Fixed point method for set-valued functional equations*, J. Fixed Point Theory Appl.**19**, 2297–2308, 2017. - C. Park,
*Set-valued additive*, J. Fixed Point Theory Appl.*ρ*-functional inequalities**20**, 2018; Article ID: 70. - C. Park, Y. Jin and X. Zhang,
*Bi-additive*, Rocky Mountain J. Math.*s*-functional inequalities and quasi-multipliers on Banach algebras**49**, 593–607, 2019. - Th. M. Rassias,
*On the stability of the linear mapping in Banach spaces*, Proc. Amer. Math. Soc.**72**, 297–300, 1978. - S. M. Ulam,
*A collection of the mathematical problems*, Interscience Publ., New York, 1960.