Approximation of the generalized Cauchy–Jensen functional equation in C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{*}$\end{document}-algebras

In this paper, we prove Hyers–Ulam–Rassias stability of C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{*}$\end{document}-algebra homomorphisms for the following generalized Cauchy–Jensen equation: αμf(x+yα+z)=f(μx)+f(μy)+αf(μz),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha\mu f \biggl(\frac{x+y}{\alpha}+z \biggr) = f(\mu x) + f(\mu y) +\alpha f( \mu z), $$\end{document} for all μ∈S:={λ∈C∣|λ|=1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu\in\mathbb{S}:= \{ \lambda\in\mathbb{C} \mid|\lambda| =1\}$\end{document} and for any fixed positive integer α≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\geq2$\end{document}, which was introduced by Gao et al. [J. Math. Inequal. 3:63–77, 2009], on C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{*}$\end{document}-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∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{*}$\end{document}-algebras by the same method.


Introduction and preliminaries
Throughout this paper, let N, R and C be the set of natural numbers, the set of real numbers, the set of complex numbers, respectively. The stability problem of functional equations was initiated by Ulam in 1940 [2] arising from concern over the stability of group homomorphisms. This form of asking the question is the object of stability theory. In 1941, Hyers [3] provided a first affirmative partial answer to Ulam's problem for the case of approximately additive mapping in Banach spaces. In 1978, Rassias [4] gave a generalization of Hyers' theorem for linear mapping by considering an unbounded Cauchy difference. A generalization of Rassias' result was developed by Găvruţa [5] in 1994 by replacing the unbounded Cauchy difference by a general control function.
In 2006, Baak [6] investigated the Cauchy-Rassis stability of the following Cauchy-Jensen functional equations: for all x, y, z ∈ X, in Banach spaces. The fixed point method was applied to study the stability of functional equations by Baker in 1991 [7] by using the Banach contraction principle. Next, Radu [8] proved a stability of functional equation by the alternative of fixed point which was introduced by Diaz and Margolis [9]. The fixed point method has provided a lot of influence in the development of stability.
In 2008, Park and An [10] proved the Hyers-Ulam-Rassias stability of C * -algebra homomorphisms and generalized derivations on C * -algebras by using alternative of fixed point theorem for the Cauchy-Jensen functional equation 2f ( x+y 2 + z) = f (x) + f (y) + 2f (z), which was introduced and investigated by Baak [6] The definition of the generalized Cauchy-Jensen equation was given by Gao et al. [1] in 2009 as follows.
G is a surjection) and X be a normed space with norm · X . For a mapping f : G → X, the equation for all x, y, z ∈ G and for any fixed positive integer n ≥ 2 is said to be a generalized Cauchy-Jensen equation (GCJE, shortly).
In particular, when n = 2, it is called a Cauchy-Jensen equation. Moreover, they gave the following useful properties.

Corollary 1.2 ([1])
For a mapping f : G → X, the following statements are equivalent.
(i) f is additive.
It is obvious that a vector space is n-divisible abelian group, so Corollary 1.2 works for a vector space G.
All over this paper, A and B are C * -algebras with norm · A and · B , respectively. We recall a fundamental result in fixed point theory. The following is the definition of a generalized metric space which was introduced by Luxemburg in 1958 [11]. The following fixed point theorem will play important roles in proving our main results. Theorem 1.4 ([9]) Let (X, d) be a complete generalized metric space and T : X → X be a strictly contractive mapping, that is, for all x, y ∈ X and for some Lipschitz k < 1. Then, for each given element x ∈ X, either d T n x, T n+1 x = ∞ for all nonnegative integer n or there exists a positive integer n 0 such that The following lemma is useful for proving our main results.

Stability of C * -algebra homomorphisms
Let f be a mapping of A into B. We define for all μ ∈ S, for all x, y, z ∈ A and for any fixed positive integer α ≥ 2. We prove the Hyers-Ulam-Rassias stability of C * -algebra homomorphisms for the functional equation E μ f (x, y, z) = 0.
for all x, y, z ∈ A. Let f be a mapping of A into B satisfying for all μ ∈ S and for all x, y, z ∈ A. Then there exists a unique C * -algebra homomorphism for all x ∈ A.
Proof Consider the set and introduce the generalized metric on X as follows: It is easy to show that (X, d) is complete. Now, we consider the linear mapping T : X → X such that for all x ∈ A. Next, we will show that T is a strictly contractive self-mapping of X with the Lipschitz constant k. For any g, h ∈ X, let d(g, h) = K for some K ∈ R + . Then we have By (2.2), we obtain Hence, we obtain Letting μ = 1 and x = y = z in (2.1), we get for all x ∈ A. By (2.3), we have which implies that for all x ∈ A, that is, for all x ∈ A. It follows from (2.7) that we have By Theorem 1.4, there exists a mapping F : A → B such that the following conditions hold.
for all x ∈ A. Moreover, the mapping F is a unique fixed point of T in the set for all x ∈ A. (2) The sequence {T n f } converges to F. This implies that we have the equality Therefore, inequality (2.6) holds. From (2.2), for any j ∈ N, we have for all x, y, z ∈ A. Since 0 < k < 1, we obtain for all x, y, z ∈ A. It follows from (2.3), (2.8) and (2.10) that for all x, y, z ∈ A. Hence, we have for all x, y, z ∈ A. From Corollary 1.2 and (2.11), we see that F is additive, that is, for all x, y ∈ A. Next, we can show that F : A → B is C-linear. Firstly, we will show that, for any x ∈ A, F(μx) = μF(x) for all μ ∈ S. For each μ ∈ S, substituting x, y, z in (2.1) by ( 2+α α ) n x, we obtain for all x ∈ A. From (2.13), in the case μ = 1, we obtain the fact that for all x ∈ A. It follows from (2.3), (2.13) and (2.14) that for all x ∈ A. This implies that for all x ∈ A. By (2.10), we have for all x ∈ A. It follows from (2.12), (2.15) and Lemma 1.5 that F : A → B is C-linear. Next, we will show that F is a C * -algebra homomorphism. It follows from (2.4) that for all x, y ∈ A. Hence

F(xy) = F(x)F(y)
for all x, y ∈ A. Finally, it follows from (2.5) that for all x ∈ A, which implies that for all x ∈ A. Therefore, F : A → B is a C * -algebra homomorphism.
for all μ ∈ S and for all x, y, z ∈ A. Then there exists a unique C * -algebra homomorphism Proof The proof follows from Theorem 2.1 by taking for all x, y, z ∈ A. Then k = ( 2+α α ) p-1 and we get the desired results.
be a function such that there exists a k < 1 such that for all x, y, z ∈ A. Let f be a mapping of A into B satisfying (2.3), (2.4) and (2.5). Then there exists a unique C * -algebra homomorphism F : for all x ∈ A.
Proof We consider the linear mapping T : X → X such that for all x ∈ A. By a similar proof to Theorem 2.1, T is a strictly contractive self-mapping of X with the Lipschitz constant k. Letting μ = 1 and substituting x, y, z in (2.3) by α 2+α x, we have for all x ∈ A. From inequality (2.22) we get for all x ∈ A, that is, for all x ∈ A. Hence, we obtain By Theorem 1.4, there exists a mapping F : A → B such that the following conditions hold.
(1) F is a fixed point of T, that is, TF(x) = F(x) for all x ∈ A. Then we have for all x ∈ A. Moreover, the mapping F is a unique fixed point of T in the set for all x ∈ A.
(2) The sequence {T n f } converges to F. This implies that the equality for all x ∈ A.
Therefore, inequality (2.20) holds. It follows from (2.19) and same argument in Theorem 2.1 that we obtain for all x, y, z ∈ A. It follows from (2.3), (2.23), (2.24) that for all x, y, z ∈ A. Hence, we have for all x, y, z ∈ A. From Corollary 1.2 and the above equation, we see that F is additive for all x, y ∈ A. Next, we can show that F : A → B is C-linear. Firstly, we will show that, for any x ∈ A, F(μx) = μF(x) for all μ ∈ S. For each μ ∈ S, substituting x, y, z in (2.1) by ( α 2+α ) n x, we obtain for all x ∈ A. By (2.24), we have for all x ∈ A. By Lemma 1.5, we see that F is C-linear. The fact that F(xy) = F(x)F(y) and F(x * ) = F(x) * for all x, y ∈ A can be obtained in a similar method as in the proof of Theorem 2.1.
for all x ∈ A.

Stability of generalized θ -derivations on C * -algebras
Let f be a mapping of A into A. We define for all μ ∈ S and all x, y, z ∈ A and for any fixed positive integer α ≥ 2.
for all x, y, z ∈ A, where θ : A → A is a C-linear mapping.
We prove the Hyers-Ulam-Rassias stability of generalized θ -derivation on C * -algebras for the functional equation E μ f (x, y, z) = 0.

4)
for all μ ∈ S and for all x, y, z ∈ A. Then there exist unique C-linear mappings δ, θ : A → A such that

6)
for all x ∈ A. Moreover, δ : A → A is a generalized θ -derivation on A.
Proof Let (X, d) be the generalized metric space as in the proof of Theorem 2.1. We consider the linear mapping T : X → X such that Tg(x) := α 2 + α g 2 + α α x for all x ∈ A and for all g ∈ X. Letting μ = 1 and y = x in (3.3), we get for all x ∈ A, so we have for all x ∈ A. Hence, we obtain