Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras
Abstract In this paper, we prove Hyers–Ulam–Rassias stability of C∗ $C^{*}$-algebra homomorphisms for the following generalized Cauchy–Jensen equation: αμf(x+yα+z)=f(μx)+f(μy)+αf(μz), $$ \alpha\mu f \biggl(\frac{x+y}{\alpha}+z \biggr) = f(\mu x) + f(\mu y) +\alpha f( \mu z), $$ for all μ∈S:={λ∈C∣|λ|...
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Online Access: | http://link.springer.com/article/10.1186/s13660-018-1824-6 |
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doaj-ae76051c5a444235b4ebf705063f3a112020-11-25T02:07:16ZengSpringerOpenJournal of Inequalities and Applications1029-242X2018-09-012018111910.1186/s13660-018-1824-6Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebrasProndanai Kaskasem0Chakkrid Klin-eam1Department of Mathematics, Faculty of Science, Naresuan UniversityDepartment of Mathematics, Faculty of Science, Naresuan UniversityAbstract In this paper, we prove Hyers–Ulam–Rassias stability of C∗ $C^{*}$-algebra homomorphisms for the following generalized Cauchy–Jensen equation: αμf(x+yα+z)=f(μx)+f(μy)+αf(μz), $$ \alpha\mu f \biggl(\frac{x+y}{\alpha}+z \biggr) = f(\mu x) + f(\mu y) +\alpha f( \mu z), $$ for all μ∈S:={λ∈C∣|λ|=1} $\mu\in\mathbb{S}:= \{ \lambda\in\mathbb{C} \mid|\lambda| =1\}$ and for any fixed positive integer α≥2 $\alpha\geq2$, which was introduced by Gao et al. [J. Math. Inequal. 3:63–77, 2009], on C∗ $C^{*}$-algebras by using fixed poind alternative theorem. Moreover, we introduce and investigate Hyers–Ulam–Rassias stability of generalized θ-derivation for such functional equations on C∗ $C^{*}$-algebras by the same method.http://link.springer.com/article/10.1186/s13660-018-1824-6Cauchy–Jensen functional equationsHyers–Ulam–Rassias stabilityC ∗ $C^{*}$ -algebrasFixed point theorem |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Prondanai Kaskasem Chakkrid Klin-eam |
spellingShingle |
Prondanai Kaskasem Chakkrid Klin-eam Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras Journal of Inequalities and Applications Cauchy–Jensen functional equations Hyers–Ulam–Rassias stability C ∗ $C^{*}$ -algebras Fixed point theorem |
author_facet |
Prondanai Kaskasem Chakkrid Klin-eam |
author_sort |
Prondanai Kaskasem |
title |
Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras |
title_short |
Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras |
title_full |
Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras |
title_fullStr |
Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras |
title_full_unstemmed |
Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras |
title_sort |
approximation of the generalized cauchy–jensen functional equation in c∗ $c^{*}$-algebras |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1029-242X |
publishDate |
2018-09-01 |
description |
Abstract In this paper, we prove Hyers–Ulam–Rassias stability of C∗ $C^{*}$-algebra homomorphisms for the following generalized Cauchy–Jensen equation: αμf(x+yα+z)=f(μx)+f(μy)+αf(μz), $$ \alpha\mu f \biggl(\frac{x+y}{\alpha}+z \biggr) = f(\mu x) + f(\mu y) +\alpha f( \mu z), $$ for all μ∈S:={λ∈C∣|λ|=1} $\mu\in\mathbb{S}:= \{ \lambda\in\mathbb{C} \mid|\lambda| =1\}$ and for any fixed positive integer α≥2 $\alpha\geq2$, which was introduced by Gao et al. [J. Math. Inequal. 3:63–77, 2009], on C∗ $C^{*}$-algebras by using fixed poind alternative theorem. Moreover, we introduce and investigate Hyers–Ulam–Rassias stability of generalized θ-derivation for such functional equations on C∗ $C^{*}$-algebras by the same method. |
topic |
Cauchy–Jensen functional equations Hyers–Ulam–Rassias stability C ∗ $C^{*}$ -algebras Fixed point theorem |
url |
http://link.springer.com/article/10.1186/s13660-018-1824-6 |
work_keys_str_mv |
AT prondanaikaskasem approximationofthegeneralizedcauchyjensenfunctionalequationinccalgebras AT chakkridklineam approximationofthegeneralizedcauchyjensenfunctionalequationinccalgebras |
_version_ |
1724930430274109440 |