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
aDepartment of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
bDepartment of Data Science, Daejin University, Kyunggi 11159, Korea
cDepartment 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: email@example.com)
Full Text: PDF
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 inequalityReferences:
- 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 C*-ternary algebras, Bull. Korean Math. Soc. 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-β-Banach spaces, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM 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 (α, β, γ)-derivations on Lie C*-algebras, Int. J. Geom. Methods Mod. Phys. 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 ψ-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci. 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 ρ-functional inequalities and equations, J. Math. Inequal. 9, 17–26, 2015.
- C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 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 ρ-functional inequalities, J. Fixed Point Theory Appl. 20, 2018; Article ID: 70.
- C. Park, Y. Jin and X. Zhang, Bi-additive s-functional inequalities and quasi-multipliers on Banach algebras, Rocky Mountain J. Math. 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.