Fixed Points and Stability of an Additive Functional Equation of 𝑛-Apollonius Type in 𝐶^∗-Algebras

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of 𝐶^∗-algebra homomorphisms and of generalized derivations on 𝐶^∗-algebras for the following functional equation of Apollonius type _^𝑛_𝑖=1_𝑓(𝑧_𝑥_𝑖)=_(1/𝑛)____{1_𝑖<𝑗_𝑛}_𝑓(𝑥_𝑖+𝑥_𝑗)+𝑛𝑓(𝑧_(1/𝑛²)_^𝑛_𝑖=1_𝑥_𝑖).

1. Introduction and Preliminaries

A classical question in the theory of functional equations is the following: “when is it true that a function, which approximately satisfies a functional equation ℰ, must be close to an exact solution of ℰ?” If the problem accepts a solution, we say that the equation ℰ is stable. Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [2]. It gave rise to the stability theory for functional equations. The result of Hyers was extended by Aoki [3] in 1950 by considering the unbounded Cauchy differences. In 1978, Rassias [4] proved that the additive mapping 𝑇, obtained by Hyers or Aoki, is linear if, in addition, for each 𝑥∈𝐸, the mapping 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ. Găvruţ a [5] generalized the Rassias' result. Following the techniques of the proof of the corollary of Hyers [2], we observed that Hyers introduced (in 1941) the following Hyers continuity condition about the continuity of the mapping for each fixed point and then he proved homogeneity of degree one and, therefore, the famous linearity. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms (see [6]). Beginning around 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [7–21]).

Rassias [22], following the spirit of the innovative approach of Hyers [2], Aoki [3], and Rassias [4] for the unbounded Cauchy difference, proved a similar stability theorem in which he replaced the factor ‖𝑥‖^𝑝+‖𝑦‖^𝑝 by ‖𝑥‖^𝑝⋅‖𝑦‖^𝑞 for 𝑝,𝑞∈ℝ with 𝑝+𝑞≠1 (see also [23, 24] for a number of other new results).

In 2003, Cădariu and Radu applied the fixed-point method to the investigation of the Jensen functional equation [25] (see also [8, 26–30]). They could present a short and a simple proof (different of the “direct method,” initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of Jensen functional equation [25], for Cauchy functional equation [8], and for quadratic functional equation [26].

The following functional equation:𝑄(𝑥+𝑦)+𝑄(𝑥−𝑦)=2𝑄(𝑥)+2𝑄(𝑦)(1.1)is called a quadratic functional equation, and every solution of (1.1) is said to be a quadratic mapping. Skof [31] proved the Hyers-Ulam stability of the quadratic functional equation (1.1) for mappings 𝑓∶𝐸₁→𝐸₂, where 𝐸₁ is a normed space and 𝐸₂ is a Banach space. In [32], Czerwik proved the Hyers-Ulam stability of the quadratic functional equation (1.1). Borelli and Forti [33] generalized the stability result of the quadratic functional equation (1.1). Jun and Lee [34] proved the Hyers-Ulam stability of the Pexiderized quadratic equation𝑓(𝑥+𝑦)+𝑔(𝑥−𝑦)=2ℎ(𝑥)+2𝑘(𝑦)(1.2)for mappings 𝑓,𝑔,ℎ, and 𝑘. The stability problem of the quadratic equation has been extensively investigated by some mathematicians [35].

In an inner product space, the equality‖𝑧−𝑥‖²+‖𝑧−𝑦‖²=12‖𝑥−𝑦‖²+2‖‖‖𝑧−𝑥+𝑦2‖‖‖²(1.3)holds, then it is called the Apollonius' identity. The following functional equation, which was motivated by this equation,𝑄(𝑧−𝑥)+𝑄(𝑧−𝑦)=12𝑄(𝑥−𝑦)+2𝑄𝑧−𝑥+𝑦2,(1.4)holds, then it is called quadratic (see [36]). For this reason, the functional equation (1.4) is called a quadratic functional equation of Apollonius type, and each solution of the functional equation (1.4) is said to be a quadratic mapping of Apollonius type. The quadratic functional equation and several other functional equations are useful to characterize inner product spaces [37].

In [36], Park and Rassias introduced and investigated a functional equation, which is called a generalized Apollonius type quadratic functional equation. In [38], Najati introduced and investigated a functional equation, which is called a quadratic functional equation of 𝑛-Apollonius type. Recently in [39], Park and Rassias introduced and investigated the following functional equation:𝑓(𝑧−𝑥)+𝑓(𝑧−𝑦)=−12𝑓(𝑥+𝑦)+2𝑓𝑧−𝑥+𝑦4(1.5)which is called an Apollonius type additive functional equation, and whose solution is called an Apollonius type additive mapping. In [40], Park introduced and investigated a functional equation, which is called a generalized Apollonius-Jensen type additive functional equation and whose solution is said to be a generalized Apollonius-Jensen type additive mapping.

In this paper, employing the above equality (1.5), for a fixed positive integer 𝑛≥2, we introduce a new functional equation, which is called an additive functional equation of 𝑛-Apollonius type and whose solution is said to be an additive mapping of 𝑛-Apollonius type; ^𝑛_𝑖=1𝑓(𝑧−𝑥_𝑖)=−1𝑛_{1≤𝑖<𝑗≤𝑛}𝑓(𝑥_𝑖+𝑥_𝑗)+𝑛𝑓𝑧−1𝑛^2𝑛_𝑖=1𝑥_𝑖.(1.6)

We will adopt the idea of Cădariu and Radu [8, 25, 28] to prove the generalized Hyers-Ulam stability results of 𝐶^∗-algebra homomorphisms as well as to prove the generalized Ulam-Hyers stability of generalized derivations on 𝐶^∗-algebra for additive functional equation of 𝑛-Apollonius type.

We recall two fundamental results in fixed-point theory.

Theorem 1.1 (see [25]). Let (𝑋,𝑑) be a complete metric space and let 𝐽∶𝑋→𝑋 be strictly contractive, that is, 𝑑(𝐽𝑥,𝐽𝑦)≤𝐿𝑓(𝑥,𝑦),∀𝑥,𝑦∈𝑋(1.7)for some Lipschitz constant 𝐿<1. Then, the following hold:
(1)the mapping 𝐽 has a unique fixed point 𝑥^∗=𝐽𝑥^∗;(2)the fixed point 𝑥^∗ is globally attractive, that is, lim_𝑛→∞𝐽^𝑛𝑥=𝑥^∗(1.8) for any starting point 𝑥∈𝑋;(3)one has the following estimation inequalities: 𝑑(𝐽^𝑛𝑥,𝑥^∗)≤𝐿^𝑛𝑑(𝑥,𝑥^∗),𝑑(𝐽^𝑛𝑥,𝑥^∗)≤11−𝐿𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥),𝑑(𝑥,𝑥^∗)≤11−𝐿𝑑(𝑥,𝐽𝑥)(1.9) for all nonnegative integers 𝑛 and all 𝑥∈𝑋.

Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies the following:

(1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(3)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.

Theorem 1.2 (see [41]). Let (𝑋,𝑑) be a complete generalized metric space and let 𝐽∶𝑋→𝑋 be a strictly contractive mapping with Lipschitz constant 𝐿<1. Then for each given element 𝑥∈𝑋, either 𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)=∞(1.10)for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that the following hold:
(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞ for all 𝑛≥𝑛₀;(2)he sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽;(3)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦)for all 𝑦∈𝑌.

Throughout this paper, assume that 𝐴 is a 𝐶^∗-algebra with norm ‖⋅‖_𝐴 and that 𝐵 is a 𝐶^∗-algebra with norm ‖⋅‖_𝐵.

2. Stability of 𝐶^∗-Algebra Homomorphisms

Lemma 2.1. Let 𝑋 and 𝑌 be real-vector spaces. A mapping 𝑓∶𝑋→𝑌 satisfies (1.6) for all 𝑥₁,…,𝑥_𝑛,𝑧 if and only if the mapping 𝑓 is additive.

Proof. Letting 𝑥₁=⋯=𝑥_𝑛=𝑧=0 in (1.6), we get that 𝑓(0)=0. Let 𝑗 and 𝑘 be fixed integers with 1≤𝑗<𝑘≤𝑛. Setting 𝑥_𝑖=0 for all 1≤𝑖≤𝑛,𝑖≠𝑗,𝑘 in (1.6), we have𝑓(𝑧−𝑥_𝑗)+𝑓(𝑧−𝑥_𝑘)+(𝑛−2)𝑓(𝑧)=−1𝑛𝑓(𝑥_𝑗+𝑥_𝑘)−𝑛−2𝑛(𝑓(𝑥_𝑗)+𝑓(𝑥_𝑘))+𝑛𝑓𝑧−1𝑛²(𝑥_𝑗+𝑥_𝑘)(2.1)for all 𝑥_𝑗,𝑥_𝑘,𝑧∈𝑋. Replacing 𝑥_𝑗 by −𝑥_𝑗 and 𝑥_𝑘 by 𝑥_𝑗 in (2.1), respectively, we get𝑓(𝑧+𝑥_𝑗)+𝑓(𝑧−𝑥_𝑗)=−𝑛−2𝑛(𝑓(−𝑥_𝑗)+𝑓(𝑥_𝑗))+2𝑓(𝑧)(2.2)for all 𝑥_𝑗,𝑧∈𝑋. Putting 𝑧=0 in (2.2), we conclude that 𝑓(−𝑥_𝑗)=−𝑓(𝑥_𝑗) for all 𝑥_𝑗∈𝑋. This means that 𝑓 is an odd function. Letting 𝑥_𝑘=𝑧=0 in (2.1) and using the oddness of 𝑓, we obtain that𝑓1𝑛²𝑥_𝑗=1𝑛²𝑓(𝑥_𝑗),𝑓(𝑛²𝑥_𝑗)=𝑛²𝑓(𝑥_𝑗)(2.3)for all 𝑥_𝑗∈𝑋. Letting 𝑧=0 in (2.1), using the oddness of 𝑓 and (2.3), we have𝑓(𝑥_𝑗+𝑥_𝑘)=𝑓(𝑥_𝑗)+𝑓(𝑥_𝑘)(2.4)for all 𝑥_𝑗,𝑥_𝑘∈𝑋. Therefore, 𝑓∶𝑋→𝑌 is an additive mapping.
The converse is obviously true.

For a given mapping 𝑓∶𝐴→𝐵 and for a fixed positive integer 𝑛≥2, we define𝐶_𝜇𝑓(𝑧,𝑥₁,…,𝑥_𝑛)∶=^𝑛_𝑖=1𝜇𝑓(𝑧−𝑥_𝑖)+1𝑛_{1≤𝑖<𝑗≤𝑛}𝑓(𝜇𝑥_𝑖+𝜇𝑥_𝑗)−𝑛𝑓𝜇𝑧−1𝑛^2𝑛_𝑖=1𝜇𝑥_𝑖(2.5)for all 𝜇∈𝕋¹∶={𝜈∈ℂ∶|𝜈|=1} and all 𝑧,𝑥₁,…,𝑥_𝑛∈𝐴.

We prove the generalized Hyers-Ulam stability of 𝐶^∗-algebra homomorphisms for the functional equation 𝐶_𝜇𝑓(𝑧,𝑥₁,…,𝑥_𝑛)=0.

Theorem 2.2. Let 𝑓∶𝐴→𝐵 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝐴^𝑛+1→[0,∞) such that ^∞_𝑗=0𝑛²𝑛²−1^2𝑗𝜑𝑛²−1𝑛²^𝑗𝑧,𝑛²−1𝑛²^𝑗𝑥₁,…,𝑛²−1𝑛²^𝑗𝑥_𝑛<∞,(2.6)‖𝐶_𝜇𝑓(𝑧,𝑥₁,…,𝑥_𝑛)‖_𝐵≤𝜑(𝑧,𝑥₁,…,𝑥_𝑛),(2.7)‖𝑓(𝑥𝑦)−𝑓(𝑥)𝑓(𝑦)‖_𝐵≤𝜑(𝑥,𝑦,0,…,0_{𝑛−1times}),(2.8)‖𝑓(𝑥^∗)−𝑓(𝑥)^∗‖_𝐵≤𝜑(𝑥,…,𝑥_{𝑛+1times})(2.9) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑛∈𝐴. If for some 1≤𝑗≤𝑛 there exists a Lipschitz constant 𝐿<1 such that 𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)≤𝑛²−1𝑛²𝐿𝜑⎛⎜⎜⎜⎝𝑛²𝑛²−1𝑥,0,…,0,𝑛²𝑛²−1_{𝑗t}𝑥_h,0,…,0⎞⎟⎟⎟⎠(2.10) for all 𝑥∈𝐴, then there exists a unique 𝐶^∗-algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤𝑛(𝑛²−1)×(1−𝐿)𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)(2.11)for all 𝑥∈𝐴.

Proof. Consider the set𝑋∶={𝑔∶𝐴⟶𝐵,𝑔(0)=0}(2.12)and introduce the generalized metric on 𝑋:𝑑(𝑔,ℎ)=inf𝐶∈ℝ₊∶‖𝑔(𝑥)−ℎ(𝑥)‖_𝐵≤𝐶𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)∀𝑥∈𝐴.(2.13)It is easy to show that (𝑋,𝑑) is complete.
For convenience, set𝜑_𝑗(𝑥,𝑦)∶=𝜑(𝑥,0,…,0,𝑦_𝑗th,0,…,0)(2.14)for all 𝑥,𝑦∈𝐴 and all 1≤𝑗≤𝑛.
Now we consider the linear mapping 𝐽∶𝑋→𝑋 such that𝐽𝑔(𝑥)∶=𝑛𝛼𝑔𝛼𝑛𝑥(2.15)for all 𝑥∈𝐴, where 𝛼=(𝑛²−1)/𝑛.
For any 𝑔,ℎ∈𝑋, we have𝑑(𝑔,ℎ)<𝐶⟹‖𝑔(𝑥)−ℎ(𝑥)‖_𝐵≤𝐶𝜑_𝑗(𝑥,𝑥)∀𝑥∈𝐴⟹‖‖‖𝑛𝛼𝑔𝛼𝑛𝑥−𝑛𝛼ℎ𝛼𝑛𝑥‖‖‖_𝐵≤𝑛𝛼𝐶𝜑_𝑗𝛼𝑛𝑥,𝛼𝑛𝑥⟹‖‖‖𝑛𝛼𝑔𝛼𝑛𝑥−𝑛𝛼ℎ𝛼𝑛𝑥‖‖‖_𝐵≤𝐿𝐶𝜑_𝑗(𝑥,𝑥)⟹𝑑(𝐽𝑔,𝐽ℎ)≤𝐿𝐶.(2.16)
Therefore, we see that𝑑(𝐽𝑔,𝐽ℎ)≤𝐿𝑑(𝑔,ℎ),∀𝑔,ℎ∈𝐴.(2.17)This means 𝐽 is a strictly contractive self-mapping of 𝑋, with the Lipschitz constant 𝐿.
Letting 𝜇=1,𝑧=𝑥_𝑗=𝑥, and for each 1≤𝑘≤𝑛 with 𝑘≠𝑗,𝑥_𝑘=0 in (2.7), we get‖‖‖𝛼𝑓(𝑥)−𝑛𝑓𝛼𝑛𝑥‖‖‖_𝐵≤𝜑_𝑗(𝑥,𝑥)(2.18)for all 𝑥∈𝐴. So‖‖‖𝑓(𝑥)−𝑛𝛼𝑓𝛼𝑛𝑥‖‖‖_𝐵≤1𝛼𝜑_𝑗(𝑥,𝑥)(2.19)for all 𝑥∈𝐴. Hence 𝑑(𝑓,𝐽𝑓)≤1/𝛼.
By Theorem 1.2, there exists a mapping 𝐻∶𝐴→𝐵 such that the following hold:
(1) 𝐻 is a fixed point of 𝐽, that is,𝐻𝛼𝑛𝑥=𝛼𝑛𝐻(𝑥)(2.20)for all 𝑥∈𝐴; the mapping 𝐻 is a unique fixed point of 𝐽 in the set𝑌={𝑔∈𝑋∶𝑑(𝑓,𝑔)<∞};(2.21)and this implies that 𝐻 is a unique mapping satisfying (2.20) such that there exists 𝐶∈(0,∞) satisfying‖𝐻(𝑥)−𝑓(𝑥)‖_𝐵≤𝐶𝜑_𝑗(𝑥,𝑥)(2.22)for all 𝑥∈𝐴.
(2) 𝑑(𝐽^𝑚𝑓,𝐻)→0 as 𝑚→∞; and this implies the equalitylim_𝑚→∞𝑛𝛼^𝑚𝑓𝛼𝑛^𝑚𝑥=𝐻(𝑥)(2.23)for all 𝑥∈𝐴;
(3) 𝑑(𝑓,𝐻)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), which implies the inequality𝑑(𝑓,𝐻)≤1𝛼−𝛼𝐿;(2.24)and this implies that the inequality (2.11) holds.
It follows from (2.6), (2.7), and (2.23) that‖‖‖^𝑛_𝑖=1𝐻(𝑧−𝑥_𝑖)+1𝑛_{1≤𝑖<𝑗≤𝑛}𝐻(𝑥_𝑖+𝑥_𝑗)−𝑛𝐻𝑧−1𝑛^2𝑛_𝑖=1𝑥_𝑖‖‖‖_𝐵=lim_𝑚→∞𝑛𝛼^𝑚‖‖‖^𝑛_𝑖=1𝑓𝛼𝑛^𝑚(𝑧−𝑥_𝑖)+1𝑛_{1≤𝑖<𝑗≤𝑛}𝑓𝛼𝑛^𝑚(𝑥_𝑖+𝑥_𝑗)−𝑛𝑓𝛼𝑛^𝑚𝑧−𝛼𝑛^𝑚×1𝑛^2𝑛_𝑖=1𝑥_𝑖‖‖‖_𝐵≤lim_𝑚→∞𝑛𝛼^𝑚𝜑𝛼𝑛^𝑚𝑧,𝛼𝑛^𝑚𝑥₁,…,𝛼𝑛^𝑚𝑥_𝑛≤lim_𝑚→∞𝑛𝛼^2𝑚𝜑𝛼𝑛^𝑚𝑧,𝛼𝑛^𝑚𝑥₁,…,𝛼𝑛^𝑚𝑥_𝑛=0(2.25)for all 𝑥₁,…,𝑥_𝑛,𝑧∈𝐴. So^𝑛_𝑖=1𝐻(𝑧−𝑥_𝑖)=−1𝑛_{1≤𝑖<𝑗≤𝑛}𝐻(𝑥_𝑖+𝑥_𝑗)+𝑛𝐻𝑧−1𝑛^2𝑛_𝑖=1𝑥_𝑖(2.26)for all 𝑥₁,…,𝑥_𝑛,𝑧∈𝐴. By Lemma 2.1, the mapping 𝐻∶𝐴→𝐵 is Cauchy additive, that is, 𝐻(𝑥+𝑦)=𝐻(𝑥)+𝐻(𝑦) for all 𝑥,𝑦∈𝐴.
By a similar method to the proof of [14], one can show that the mapping 𝐻∶𝐴→𝐵 is ℂ-linear.
It follows from (2.8) that‖𝐻(𝑥𝑦)−𝐻(𝑥)𝐻(𝑦)‖_𝐵=lim_𝑚→∞𝑛𝛼^2𝑚‖‖‖𝑓𝛼𝑛^2𝑚𝑥𝑦−𝑓𝛼𝑛^𝑚𝑥𝑓𝛼𝑛^𝑚𝑦‖‖‖_𝐵≤lim_𝑚→∞𝑛𝛼^2𝑚𝜑𝛼𝑛^𝑚𝑥,𝛼𝑛^𝑚𝑦,0,…,0_{𝑛−1times}=0(2.27)for all 𝑥,𝑦∈𝐴. So𝐻(𝑥𝑦)=𝐻(𝑥)𝐻(𝑦)(2.28)for all 𝑥,𝑦∈𝐴.
It follows from (2.9) that‖𝐻(𝑥^∗)−𝐻(𝑥)^∗‖_𝐵=lim_𝑚→∞𝑛𝛼^𝑚‖‖‖𝑓𝛼𝑛^𝑚𝑥^∗−𝑓𝛼𝑛^𝑚𝑥^∗‖‖‖_𝐵≤lim_𝑚→∞𝑛𝛼^𝑚𝜑𝛼𝑛^𝑚𝑥,…,𝛼𝑛^𝑚_{𝑛+1times}𝑥≤lim_𝑚→∞𝑛𝛼^2𝑚𝜑𝛼𝑛^𝑚𝑥,…,𝛼𝑛^𝑚_{𝑛+1times}𝑥=0(2.29)for all 𝑥∈𝐴. So𝐻(𝑥^∗)=𝐻(𝑥)^∗(2.30)for all 𝑥∈𝐴.
Thus 𝐻∶𝐴→𝐵 is a 𝐶^∗-algebra homomorphism satisfying (2.11) as desired.

Corollary 2.3. Let 𝑟>2 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping such that ‖𝐶_𝜇𝑓(𝑧,𝑥₁,…,𝑥_𝑛)‖_𝐵≤𝜃‖𝑧‖^𝑟_𝐴+^𝑛_𝑖=1‖𝑥_𝑖‖^𝑟_𝐴,(2.31)‖𝑓(𝑥𝑦)−𝑓(𝑥)𝑓(𝑦)‖_𝐵≤𝜃(‖𝑥‖^𝑟_𝐴+‖𝑦‖^𝑟_𝐴),(2.32)
‖𝑓(𝑥^∗)−𝑓(𝑥)^∗‖_𝐵≤(𝑛+1)𝜃‖𝑥‖^𝑟_𝐴(2.33) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧∈𝐴. Then there exists a unique 𝐶^∗-algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤2𝑛(𝑛²−1)^−𝑟𝜃(𝑛²−1)^1−𝑟−𝑛^2(1−𝑟)‖𝑥‖^𝑟_𝐴(2.34)for all 𝑥∈𝐴.

Proof. The proof follows from Theorem 2.2 by taking𝜑(𝑧,𝑥₁,…,𝑥_𝑛)∶=𝜃‖𝑧‖^𝑟_𝐴+^𝑛_𝑖=1‖𝑥_𝑖‖^𝑟_𝐴(2.35)for all 𝑥,𝑦,𝑧∈𝐴. It follows from (2.31) that 𝑓(0)=0. We can choose 𝐿=(𝑛²/(𝑛²−1))^1−𝑟 to get the desired result.

Theorem 2.4. Let 𝑓∶𝐴→𝐵 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝐴^𝑛+1→[0,∞) satisfying (2.7), (2.8), and (2.9) such that ^∞_𝑗=0𝑛²−1𝑛²^𝑗𝜑𝑛²𝑛²−1^𝑗𝑧,𝑛²𝑛²−1^𝑗𝑥₁,…,𝑛²𝑛²−1^𝑗𝑥_𝑛<∞(2.36)for all 𝑧,𝑥₁,…,𝑥_𝑛∈𝐴. If for some 1≤𝑗≤𝑛 there exists a Lipschitz constant 𝐿<1 such that 𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)≤𝑛²𝑛²−1𝐿𝜑𝑛²−1𝑛²𝑥,0,…,0,𝑛²−1𝑛²_{𝑗t}𝑥_h,0,…,0(2.37)for all 𝑥∈𝐴, then there exists a unique 𝐶^∗-algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤𝑛𝐿(𝑛²−1)×(1−𝐿)𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)(2.38)for all 𝑥∈𝐴.

Proof. Similar to proof of Theorem (2.2), we consider the linear mapping 𝐽∶𝑋→𝑋 such that𝐽𝑔(𝑥)∶=𝛼𝑛𝑔𝑛𝛼𝑥(2.39)for all 𝑥∈𝐴, where 𝛼=(𝑛²−1)/𝑛. We can conclude that 𝐽 is a strictly contractive self mapping of 𝑋 with the Lipschitz constant 𝐿.
It follows from (2.18) that‖‖‖𝑓(𝑥)−𝛼𝑛𝑓𝑛𝛼𝑥‖‖‖_𝐵≤1𝑛𝜑_𝑗𝑛𝛼𝑥,𝑛𝛼𝑥≤𝐿𝛼𝜑_𝑗(𝑥,𝑥)(2.40)for all 𝑥∈𝐴. Hence, 𝑑(𝑓,𝐽𝑓)≤(𝐿/𝛼).
By Theorem 1.2, there exists a mapping 𝐻∶𝐴→𝐵 such that the following hold:
(1) 𝐻 is a fixed point of 𝐽, that is,𝐻𝑛𝛼𝑥=𝑛𝛼𝐻(𝑥)(2.41)for all 𝑥∈𝐴; the mapping 𝐻 is a unique fixed point of 𝐽 in the set𝑌={𝑔∈𝑋∶𝑑(𝑓,𝑔)<∞};(2.42)and this implies that 𝐻 is a unique mapping satisfying (2.41) such that there exists 𝐶∈(0,∞) satisfying‖𝐻(𝑥)−𝑓(𝑥)‖_𝐵≤𝐶𝜑_𝑗(𝑥,𝑥)(2.43)for all 𝑥∈𝐴;
(2) 𝑑(𝐽^𝑚𝑓,𝐻)→0 as 𝑚→∞; and this implies the equalitylim_𝑚→∞𝛼𝑛^𝑚𝑓𝛼𝑛^𝑚𝑥=𝐻(𝑥)(2.44)for all 𝑥∈𝐴;
(3) 𝑑(𝑓,𝐻)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), which implies the inequality𝑑(𝑓,𝐻)≤𝐿𝛼−𝛼𝐿,(2.45)which implies that the inequality (2.38) holds.
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.5. Let 𝑟<1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping satisfying (2.31), (2.32), and (2.33). Then there exists a unique 𝐶^∗-algebra homomorphism 𝐻∶𝐴→𝐵 such that ‖𝑓(𝑥)−𝐻(𝑥)‖_𝐵≤2𝑛(𝑛²−1)^𝑟−2𝐿𝜃(𝑛²−1)^𝑟−1−𝑛^2(𝑟−1)‖𝑥‖^𝑟_𝐴(2.46)for all 𝑥∈𝐴 and 𝐿=(𝑛²/(𝑛²−1))^𝑟−1.

Proof. The proof follows from Theorem 2.4 by taking𝜑(𝑧,𝑥₁,…,𝑥_𝑛)∶=𝜃‖𝑧‖^𝑟_𝐴+^𝑛_𝑖=1‖𝑥_𝑖‖^𝑟_𝐴(2.47)for all 𝑧,𝑥₁,…,𝑥_𝑛∈𝐴. It follows from (2.31) that 𝑓(0)=0. We can choose 𝐿=(𝑛²/(𝑛²−1))^𝑟−1 to get the desired result.

3. Stability of Generalized Derivations on 𝐶^∗-Algebras

For a given mapping 𝑓∶𝐴→𝐴 and for a fixed positive integer 𝑛≥2, we define𝐶_𝜇𝑓(𝑧,𝑥₁,…,𝑥_𝑛)∶=^𝑛_𝑖=1𝜇𝑓(𝑧−𝑥_𝑖)+1𝑛_{1≤𝑖<𝑗≤𝑛}𝑓(𝜇𝑥_𝑖+𝜇𝑥_𝑗)−𝑛𝑓𝜇𝑧−1𝑛^2𝑛_𝑖=1𝜇𝑥_𝑖(3.1)for all 𝜇∈𝕋¹ and all 𝑧,𝑥₁,…,𝑥_𝑛∈𝐴.

Definition 3.1 (see [42]). A generalized derivation 𝛿∶𝐴→𝐴 is involutive ℂ-linear and fulfills
𝛿(𝑥𝑦𝑧)=𝛿(𝑥𝑦)𝑧−𝑥𝛿(𝑦)𝑧+𝑥𝛿(𝑦𝑧)(1)for all 𝑥,𝑦,𝑧∈𝐴.
We prove the generalized Hyers-Ulam stability of derivations on 𝐶^∗-algebras for the functional equation 𝐶_𝜇𝑓(𝑧,𝑥₁,…,𝑥_𝑛)=0.

Theorem 3.2. Let 𝑓∶𝐴→𝐴 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝐴^𝑛+1→[0,∞) such that ^∞_𝑗=0𝑛²𝑛²−1^3𝑗𝜑𝑛²−1𝑛²^𝑗𝑧,𝑛²−1𝑛²^𝑗𝑥₁,…,𝑛²−1𝑛²^𝑗𝑥_𝑛<∞,(3.2)‖𝐶_𝜇𝑓(𝑥₁,…,𝑥_𝑛,𝑧)‖_𝐴≤𝜑(𝑧,𝑥₁,…,𝑥_𝑛),(3.3)‖𝑓(𝑥𝑦𝑧)−𝑓(𝑥𝑦)𝑧+𝑥𝑓(𝑦)𝑧−𝑥𝑓(𝑦𝑧)‖_𝐴≤𝜑(𝑥,𝑦,𝑧,0,…,0_{𝑛−2times}),(3.4)‖𝑓(𝑥^∗)−𝑓(𝑥)^∗‖_𝐴≤𝜑(𝑥,…,𝑥_{𝑛+1times})(3.5) for all 𝜇∈𝕋¹ and all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑛∈𝐴. If for some 1≤𝑗≤𝑛 there exists a Lipschitz constant 𝐿<1 such that 𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)≤𝑛²−1𝑛²𝐿𝜑𝑛²𝑛²−1𝑥,0,…,0,𝑛²𝑛²−1_{𝑗t}𝑥_h,0,…,0(3.6)for all 𝑥∈𝐴, then there exists a unique generalized derivation 𝛿∶𝐴→𝐴 such that ‖𝑓(𝑥)−𝛿(𝑥)‖_𝐴≤𝑛(𝑛²−1)×(1−𝐿)𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)(3.7)for all 𝑥∈𝐴.

Proof. By the same reasoning as in the proof of Theorem 2.2, there exists a unique involutive ℂ-linear mapping 𝛿∶𝐴→𝐴 satisfying (3.7). The mapping 𝛿∶𝐴→𝐴 is given by𝛿(𝑥)=𝑛𝛼^𝑚𝑓𝑛𝛼^𝑚𝑥(3.8)for all 𝑥∈𝐴.
It follows from (3.4) that‖𝛿(𝑥𝑦𝑧)−𝛿(𝑥𝑦)𝑧+𝑥𝛿(𝑦)𝑧−𝑥𝛿(𝑦𝑧)‖_𝐴=lim_𝑚→∞𝑛𝛼^3𝑚‖‖‖𝑓𝛼𝑛^3𝑚𝑥𝑦𝑧−𝑓𝛼𝑛^2𝑚𝑥𝑦⋅𝛼𝑛^𝑚𝑧+𝛼𝑛^𝑚𝑥𝑓𝛼𝑛^𝑚𝑦⋅𝛼𝑛^𝑚𝑧−𝛼𝑛^𝑚𝑥𝑓𝛼𝑛^2𝑚𝑦𝑧‖‖‖_𝐴≤lim_𝑚→∞𝑛𝛼^3𝑚𝜑𝛼𝑛^𝑚𝑥,𝛼𝑛^𝑚𝑦,𝛼𝑛^𝑚𝑧,0,…,0_{𝑛−2times}=0(3.9)for all 𝑥,𝑦,𝑧∈𝐴. So 𝛿(𝑥𝑦𝑧)=𝛿(𝑥𝑦)𝑧−𝑥𝛿(𝑦)𝑧+𝑥𝛿(𝑦𝑧)(3.10) for all 𝑥,𝑦,𝑧∈𝐴. Thus 𝛿∶𝐴→𝐴 is a generalized derivation satisfying (3.7).

Theorem 3.3. Let 𝑓∶𝐴→𝐴 be a mapping satisfying 𝑓(0)=0 for which there exists a function 𝜑∶𝐴^𝑛+1→[0,∞) satisfying (2.36),(3.3), (3.4) and (3.5) for all 𝑥,𝑦,𝑧,𝑥₁,…,𝑥_𝑛∈𝐴. If for some 1≤𝑗≤𝑛 there exists a Lipschitz constant 𝐿<1 such that 𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)≤𝑛²𝑛²−1𝐿𝜑𝑛²−1𝑛²𝑥,0,…,0,𝑛²−1𝑛²_{𝑗t}𝑥_h,0,…,0(3.11)for all 𝑥∈𝐴, then there exists a unique generalized derivation 𝛿∶𝐴→𝐴 such that ‖𝑓(𝑥)−𝛿(𝑥)‖_𝐵≤𝑛𝐿(𝑛²−1)×(1−𝐿)𝜑(𝑥,0,…,0,𝑥_𝑗th,0,…,0)(3.12)for all 𝑥∈𝐴.

Proof. The proof is similar to the proofs of Theorems 2.4 and 3.2.

Acknowledgments

This paper is based on final report of the research project of the Ph.D. thesis in University of Tabriz and the third author was supported by Grant no. F01-2006-000-10111-0 from the Korea Science and Engineering Foundation. The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.