Approximation of the generalized Cauchy–Jensen functional equation in C∗ $C^{*}$-algebras

• Main Authors: Prondanai Kaskasem, Chakkrid Klin-eam
• Format: Article
• Language: English
• Published: SpringerOpen 2018-09-01
• Series: Journal of Inequalities and Applications
• Subjects: Cauchy–Jensen functional equations; Hyers–Ulam–Rassias stability; C ∗ $C^{*}$ -algebras; Fixed point theorem
• Online Access: http://link.springer.com/article/10.1186/s13660-018-1824-6

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.