Approximation of the generalized Cauchy-Jensen functional equation in C ∗ -algebras.

In this paper, we prove Hyers-Ulam-Rassias stability of C ∗ -algebra homomorphisms for the following generalized Cauchy-Jensen equation: α μ f ( x + y α + z ) = f ( μ x ) + f ( μ y ) + α f ( μ z ) , for all μ ∈ S : = { λ ∈ C ∣ | λ | = 1 } and for any fixed positive integer α ≥ 2 , which was introduced by Gao et al. [J. Math. Inequal. 3:63-77, 2009], on 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 ∗ -algebras by the same method.

Introduction and preliminaries

Throughout this paper, let , and 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: or for all , 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 -algebra homomorphisms and generalized derivations on -algebras by using alternative of fixed point theorem for the Cauchy–Jensen functional equation , 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.

Definition 1.1

([1])

Let G be an n-divisible abelian group where (i.e. is a surjection) and X be a normed space with norm . For a mapping , the equation for all and for any fixed positive integer is said to be a generalized Cauchy–Jensen equation (GCJE, shortly).

In particular, when , it is called a Cauchy–Jensen equation. Moreover, they gave the following useful properties.

Corollary 1.2

([1])

For a mapping , the following statements are equivalent.

f is additive.

, for all .

, for all .

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, and are -algebras with norm and , 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].

Definition 1.3

([11])

Let X be a set. A function is called a generalized metric on X if d satisfies the following conditions:

if and only if ,

, for all ,

, for all .

The following fixed point theorem will play important roles in proving our main results.

Theorem 1.4

([9])

Let be a complete generalized metric space and be a strictly contractive mapping, that is, for all and for some Lipschitz . Then, for each given element , either for all nonnegative integer n or there exists a positive integer such that

for all ,

the sequence converges to a fixed point of T,

is the unique fixed point of T in the set ,

, for all .

The following lemma is useful for proving our main results.

Lemma 1.5

([12])

Let be an additive mapping such that for all and all . Then the mapping f is -linear.

Stability of -algebra homomorphisms

Let f be a mapping of into . We define for all , for all and for any fixed positive integer .

We prove the Hyers–Ulam–Rassias stability of -algebra homomorphisms for the functional equation .

Theorem 2.1

Let be a function such that there exists a satisfying for all . Let f be a mapping of into satisfying for all and for all . Then there exists a unique -algebra homomorphism such that for all .

Proof

Consider the set and introduce the generalized metric on X as follows: It is easy to show that is complete.

Now, we consider the linear mapping such that for all . Next, we will show that T is a strictly contractive self-mapping of X with the Lipschitz constant k. For any , let for some . Then we have By (2.2), we obtain Hence, we obtain Letting and in (2.1), we get for all . By (2.3), we have which implies that for all , that is, for all . It follows from (2.7) that we have By Theorem 1.4, there exists a mapping such that the following conditions hold. From (2.2), for any , we have for all . Since , we obtain for all .

F is a fixed point of T, that is, for all . Then we have for all . Moreover, the mapping F is a unique fixed point of T in the set From (2.7), there exists satisfying for all .

The sequence converges to F. This implies that we have the equality for all .

We obtain , which implies that Therefore, inequality (2.6) holds.

It follows from (2.3), (2.8) and (2.10) that for all . Hence, we have for all . From Corollary 1.2 and (2.11), we see that F is additive, that is, for all . Next, we can show that is -linear. Firstly, we will show that, for any , for all . For each , substituting in (2.1) by , we obtain for all . By (2.3), we have for all . From (2.13), in the case , we obtain the fact that for all . It follows from (2.3), (2.13) and (2.14) that for all . This implies that for all . By (2.10), we have which implies that for all . It follows from (2.12), (2.15) and Lemma 1.5 that is -linear. Next, we will show that F is a -algebra homomorphism. It follows from (2.4) that for all . Hence for all .

Finally, it follows from (2.5) that for all , which implies that for all . Therefore, is a -algebra homomorphism. □

Corollary 2.2

Let , and f be a mapping of into such that for all and for all . Then there exists a unique -algebra homomorphism such that for all .

The proof follows from Theorem 2.1 by taking for all . Then and we get the desired results. □

Theorem 2.3

Let be a function such that there exists a such that for all . Let f be a mapping of into satisfying (2.3), (2.4) and (2.5). Then there exists a unique -algebra homomorphism such that for all .

We consider the linear mapping such that for all . By a similar proof to Theorem 2.1, T is a strictly contractive self-mapping of X with the Lipschitz constant k. Letting and substituting in (2.3) by , we have for all . From inequality (2.22) we get for all , that is, for all . Hence, we obtain By Theorem 1.4, there exists a mapping such that the following conditions hold.

F is a fixed point of T, that is, for all . Then we have for all . Moreover, the mapping F is a unique fixed point of T in the set From (2.7), there exists satisfying for all .

The sequence converges to F. This implies that the equality for all .

We obtain , which implies that Therefore, inequality (2.20) holds.

It follows from (2.19) and same argument in Theorem 2.1 that we obtain for all . It follows from (2.3), (2.23), (2.24) that for all . Hence, we have for all . From Corollary 1.2 and the above equation, we see that F is additive for all . Next, we can show that is -linear. Firstly, we will show that, for any , for all . For each , substituting in (2.1) by , we obtain for all . By (2.3), we have for all . From (2.25), in the case , we obtain the fact that for all . It follows from (2.3), (2.25) and (2.26) that for all . This implies that for all . By (2.24), we have which implies that for all . By Lemma 1.5, we see that F is -linear. The fact that and for all can be obtained in a similar method as in the proof of Theorem 2.1. □

Corollary 2.4

Let , and f be a mapping of into satisfying (2.16), (2.17) and (2.18). Then there exists a unique -algebra homomorphism such that for all .

The proof follows from Theorem 2.3 and Corollary 2.2 by taking for all . Then and we get the desired results. □

Remark 2.5

If , then Theorem 2.1, Corollary 2.2 and Theorem 2.3 we recover Theorem 2.1, Corollary 2.2 and Theorem 2.3 in [10], respectively.

Stability of generalized θ-derivations on -algebras

Let f be a mapping of into . We define for all and all and for any fixed positive integer .

Definition 3.1

A generalized θ-derivation is a -linear map satisfying for all , where is a -linear mapping.

We prove the Hyers–Ulam–Rassias stability of generalized θ-derivation on -algebras for the functional equation .

Theorem 3.1

Let be a function such that there exists a satisfying (2.2). Let be mappings of into itself satisfying for all and for all . Then there exist unique -linear mappings such that for all . Moreover, is a generalized θ-derivation on .

Let be the generalized metric space as in the proof of Theorem 2.1. We consider the linear mapping such that for all and for all . Letting and in (3.3), we get for all , so we have for all . Hence, we obtain It follows from the proof of Theorem 2.1 that By the same reasoning as the proof of Theorem 2.1, there exist a unique involutive -linear mapping and a mapping satisfying (3.5) and (3.6), respectively. The mappings δ and θ are given by and for all , respectively. It follows from (3.2) that for all . Hence for all . Next, we can show that is -linear. Firstly, we will show that, for any , for all . For each , substituting in (3.3) by , we obtain for all . For , we also have for all . It follows from (3.7) and (3.8) that for all . This implies that for all . By (2.2), we have for all . That is, for all . By Lemma 1.5, we obtain that θ is a -linear mapping. Thus, is generalized θ-derivation satisfying (3.5). □

Corollary 3.2

Let , and f be a mapping of into itself such that for all and for all . Then there exist unique -linear mappings such that for all . Moreover, is a generalized θ-derivation on .

The proof follows from Theorem 3.1 by taking for all . Then and we get the desired results. □

Theorem 3.3

Let such that there exists a satisfying for all . Let be mappings of into itself satisfying (3.1), (3.2), (3.3) and (3.4). Then there exist unique -linear mappings such that for all . Moreover, is a generalized θ-derivation on .

The proof is similar to the proofs of Theorem 2.3 and Theorem 3.1. □

Corollary 3.4

Let , and f be a mapping of into itself satisfying (3.9), (3.10), (3.11) and (3.12). Then there exist unique -linear mappings such that for all . Moreover, is a generalized θ-derivation .

The proof follows from Theorem 3.3 by taking for all . Then and we get the desired results. □

We recall definition of generalized derivations on -algebra.

Definition 3.2

([13])

A generalized derivation is involutive -linear and satisfies for all .

Remark 3.5

According to Definition 3.1, If , I is identity mapping on , then a generalized θ-derivation is a generalized derivation. If the mapping h is identity mapping and , Then Theorem 3.1 and Theorem 3.3 we recover Theorem 3.2 and Theorem 3.4 in [10], respectively. Moreover, if we set the mapping h is identity mapping, and in Theorem 3.1 where and , then Theorem 3.1 one recovers Corollary 3.3 in [10] with .

Conclusions

In the first section of main results, we prove Hyers–Ulam–Rassias stability of -algebra homomorphisms for the generalized Cauchy–Jensen equation -algebras by using fixed point alternative theorem. In the second section of main results, we introduce and investigate the Hyers–Ulam–Rassias stability of generalized θ-derivation for such function -algebras by the same method. By our main results we recover partial results of Park and An in [10] by Remark 2.5 and Remark 3.5.