Research

# Almost partial generalized Jordan derivations: a fixed point approach

Madjid E Gordji1, Choonkil Park2 and Jung R Lee3*

Author Affiliations

1 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

2 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea

3 Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea

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Journal of Inequalities and Applications 2012, 2012:119 doi:10.1186/1029-242X-2012-119

 Received: 27 October 2011 Accepted: 29 May 2012 Published: 29 May 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Using fixed point method, we investigate the Hyers-Ulam stability and the superstability of partial generalized Jordan derivations on Banach modules related to Jensen type functional equations.

Mathematics Subject Classification 2010: Primary, 39B52; 47H10; 47B47; 13N15; 39B72; 17C50; 39B82; 17C65.

##### Keywords:
Hyers-Ulam stability; superstability; partial Jordan derivation; partial generalized Jordan derivation; Jensen type functional equation

### 1. Introduction and preliminaries

The following question posed by Ulam [1] in 1940: "When is it true that a mapping which approximately satisfies a functional equation must be somehow close to an exact solution of ?". Hyers [2] proved the problem for the Cauchy functional equation. In 1978, Rassias [3] proved the following theorem.

Theorem 1.1. Let f: E E' be a mapping from a normed vector space E into a Banach space E' subject to the inequality

(1.1)

for all x, y E, where ε and p are constants with ε > 0 and p < 1. Then there exists a unique additive mapping T: E E' such that

(1.2)

for all x E. If p < 0 then inequality (1.1) holds for all x, y ≠ 0, and (1.2) for x ≠ 0. Also, if the function t α f(tx) from into E' is continuous in real t for each xE, then T is ℝ-linear.

In 1991, Gajda [4] answered the question for the case p > 1, which was raised by Rassias. In 1994, a generalization of the Rassias' theorem was obtained by Găvruta as follows [5].

Stability of the Jensen functional equation, , where f is a mapping between linear spaces, has been investigated by several mathematicians (see [6,7]). During the last decades several stability problems of functional equations have been investigated by a number of mathematicians. See [8-17] and references therein for more detailed information.

Let A, B be two Banach algebras. A ℂ-linear mapping d: A B is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) δ: A X such that d(a2) = ad(a) + δ(a)a for all a A.

Generalized derivations and generalized Jordan derivations first appeared in the context of operator algebras [18]. Later, these were introduced in the framework of pure algebra [19,20].

Recently, Badora [21] proved the stability of ring derivations (see also [22,23]). More recently, Eshaghi Gordji and Ghobadipour [24] investigated the stability of generalized Jordan derivations on Banach algebras.

Let be normed algebras over the complex field ℂ and let be a Banach algebra over ℂ. A mapping is called a k-th partial derivation if

and there exists a mapping such that

for all and and all γ, μ ∈ ℂ.

Chu et al. [25] established the Hyers-Ulam stability of partial derivations.

Definition 1.2. Let be normed algebras over the complex field ℂ and let be a Banach module over and . Then

(i) A mapping is called a k-th partial Jordan derivation of Jensen type if

and

for all and and all γ ∈ ℂ.

(ii) A mapping is called a k-th partial generalized Jordan derivation of Jensen type if

and there exists a k-th partial Jordan derivation such that

for all and and all γ ∈ ℂ.

We now introduce one of fundamental results of fixed point theory. For the proof, refer to [26,27]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [28].

Let X be a set. A function d: X × X → [0, ] is called a generalized metric on X if and only if d satisfies:

(GM1) d(x, y) = 0 if and only if x = y;

(GM2) d(x, y) = d(y, x) for all x, y X;

(GM3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z X.

Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

Let (X, d) be a generalized metric space. An operator T: X X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that

for all x, y X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator.

We recall the following theorem by Diaz and Margolis [26].

Theorem 1.3. Suppose that we are given a complete generalized metric space (Ω, d) and a strictly contractive function T: Ω → Ω with Lipschitz constant L. Then for each given x ∈ Ω, either

or other exists a natural number m0 such that

d(Tmx, Tm+1x) < ∞ for all m m0;

the sequence {Tmx} is convergent to a fixed point y* of T;

y* is the unique fixed point of T in

.

The equation (ξ) is called superstable if every approximate solution of (ξ) is an exact solution.

We use the fixed point method to investigate the Hyers-Ulam stability and the superstability of partial generalized Jordan derivations of Jensen type.

### 2. Main results

For n0 ∈ ℕ, we define

and we denote by . Also, we suppose that are normed algebras over the complex field ℂ and is a Banach module over and . We denote that 0k, are zero elements of , respectively.

Theorem 2.1. Let be mappings with . Assume that there exist functions , satisfying

(2.1)

(2.2)

for Sk ∈ {Fk, Tk} and for all and all , . If there exists a constant 0 < L < 1 such that φk(ak, bk) ≤ 2k(2-1ak, 2-1bk), Ψk(ak) ≤ 2LΨk(2-1ak) for all , then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to dk) such that

for all .

Proof. It follows from (2.1) that

(2.3)

for Sk ∈{Fk, Tk} and for all and all , .

In the inequality (2.3), put Sk = Fk, bk = 0, λ = 1 and replace ak with 2xk. Then we obtain

(2.4)

for all . Put and define d: Ω × Ω → [0, ] by

It is easy to show that (Ω, d) is a complete generalized metric space. We define the mapping J: Ω → Ω by

for all . Let Gk, Hk ∈ Ω and let α ∈ (0, ) be arbitrary with d(Gk, Hk) ≤ α. From the definition of d, we have

for all . Hence we have

for all . So

for all Gk, Hk∈ Ω. It follows from (2.4) that

By Theorem 1.3, J has a unique fixed point in the set Ω1 := { Hk∈ Ω; d(Fk, Hk) < ∞}. Let dk be the fixed point of J. dk is the unique mapping which satisfies

for all , and there exists α ∈ (0, ) such that

for all .

On the other hand, we have limmd(Jm(Fk), dk) = 0. It follows that

for all . It follows from that that

This means that

for all . By the inequality φk(ak, bk) ≤ 2k(2-1ak, 2-1bk), we conclude that

for all . In the inequality (2.3), replacing ak, bk by 2mak, 2mbk, respectively, we obtain that

Passing the limit m , we obtain

for all and all . Now, we show that dk is ℂ-linear with respect to k-th variable. First suppose that λ belongs to T1. Then λ = efor some 0 ≤ θ ≤ 2π. We set . Then λ1 belongs to and

for all . It is easy to show that dk is additive with respect to k-th variable. Moreover, if λ belongs to nT1 = {nz z T1} then by additivity of dk on k-th variable, we have

for all . If t ∈ (0, ∞), then by Archimedean property of ℂ, there exists an n ∈ ℕ such that the point (t, 0) lies in the interior of circle with center at origin and radius n. Let and . We have . Then

for all . Let λ ∈ ℂ. Then and so

for all . It follows that dk is ℂ-linear with respect to k-th variable.

By the same reasoning as above, we can show that the limit

exists for all and that Dk is ℂ-linear with respect to k-th variable.

By te inequality Ψk(ak) ≤ 2LΨk(2-1ak), we conclude that

for all .

Now, by (2.2), we have

for all , . Replacing ak by 2mak in the above inequality, we obtain that

Then we have

for all . Passing m , we obtain

for all and all . This shows that dk is a partial Jordan derivation. We have to show that Dk is a partial generalized Jordan derivation related to dk. By (2.2), we have

for all , . Replacing ak by 2mak in the last inequality, we get

for all , . Then we have

for all , . Passing m , we obtain that

for all and all . Hence Dk is a partial generalized Jordan derivation related to dk.

Corollary 2.2. Let p ∈ (0, 1) and θ ∈ [0, ) be real numbers. Let be mappings such that and that

for Sk ∈ {Fk, Tk} and for all and all , . Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to dk) such that

for all .

Proof. It follows from Theorem 2.1 by putting Ψk(ak) = θ(||ak||p), φk(ak, bk) = θ(|| ak ||p + ||bk||p) and L = 2p-1. □

Theorem 2.3. Let be mappings with . Assume that there exist functions , satisfying

for Sk ∈ {Fk, Tk} and for all and all , . If there exists a constant 0 < L < 1 such that φk(ak, bk) ≤ 2-1k(2ak, 2bk), Ψk(ak) ≤ 2-1LΨk(2ak) for all , then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to dk) such that

for all .

Proof. The proof is similar to the proof of Theorem 2.1. □

Corollary 2.4. Let p ∈ (1, ) and θ ∈ [0, ) be real numbers. Let be mappings such that and

for Sk ∈ {Fk, Tk} and for all and all , . Then there exist a unique partial Jordan derivation of Jensen type with respect to k-th variable and a unique partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to dk) such that

for all .

Proof. It follows from Theorem 2.3 by putting Ψk (ak)=θ(||ak||p), φk (ak, bk) = θ(||ak||p+ ||bk||p) and L = 21-p for each . □

Moreover, we have the following result for the superstability of partial generalized Jordan derivations of Jensen type.

Corollary 2.5. Let and θ ∈ [0, ) be real numbers. Let be mappings such that and

for Sk ∈ {Fk, Tk}and for all and all , . Then Fk is a partial Jordan derivation of Jensen type with respect to k-th variable and Tk is a partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to Fk).

Proof. It follows from Theorem 2.1 by putting Ψk (ak)=θ(||ak||p), φk (ak, bk) = θ(||ak||p+ ||bk||p), and L = 22p-1. □

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

### Acknowledgements

This work was supported by the Daejin University Research Grants in 2012.

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