# Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras

## Abstract

Eskandani and Vaezi proved the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras associated with the following Pexiderized Jensen type functional equation

$kf\left(\frac{x+y}{k}\right)={f}_{0}\left(x\right)+{f}_{1}\left(y\right)$

by using direct method. Using fixed point method, we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras. Moreover, we investigate the Pexiderized Jensen type functional inequality in proper Jordan CQ*-algebras.

Mathematics Subject Classification 2010: Primary, 17B40; 39B52; 47N50; 47L60; 46B03; 47H10.

## 1. Introduction and preliminaries

In 1940, Ulam  asked the first question on the stability problem. In 1941, Hyers  solved the problem of Ulam. This result was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. In 1994, a generalization of Rassias' Theorem was obtained by Găvruta . Since then, several stability problems for various functional equations have been investigated by numerous mathematicians (see , M Eshaghi Gordji, unpublished work).

The Jensen equation is $2f\left(\frac{x+y}{2}\right)=f\left(x\right)+f\left(y\right)$, where f is a mapping between linear spaces. It is easy to see that a mapping f : XY between linear spaces with f(0) = 0 satisfies the Jensen equation if and only if it is additive . Stability of the Jensen equation has been studied at first by Kominek .

We recall some basic facts concerning quasi *-algebras.

Definition 1.1. Let A be a linear space and let A0 be a *-algebra contained in A as a subspace. We say that A is a quasi *-algebra over A0 if

(i) the right and left multiplications of an element of A and an element of A0 are defined and linear;

(ii) x1(x2a) = (x1x2)a, (ax 1)x 2 = a(x1x2) and x1(ax2) = (x1a)x2 for all x1, x2 A0 and all a A;

(iii) an involution *, which extends the involution of A0, is defined in A with the property (ab)* = b*a* whenever the multiplication is defined.

Quasi *-algebras [28, 29] arise in natural way as completions of locally convex *-algebras whose multiplication is not jointly continuous; in this case one has to deal with topological quasi *-algebras.

A quasi *-algebra (A, A o ) is said to be a locally convex quasi *-algebra if in A a locally convex topology τ is defined such that

(i) the involution is continuous and the multiplications are separately continuous;

(ii) A o is dense in A[τ].

Throughout this article, we suppose that a locally convex quasi *-algebra (A, A0) is complete. For an overview on partial *-algebra and related topics we refer to .

In a series of articles  many authors have considered a special class of quasi * algebras, called proper CQ*-algebras, which arise as completions of C*-algebras. They can be introduced in the following way:

Let A be a Banach module over the C*-algebra A0 with involution * and C*-norm || . ||0 such that A0 A. We say that (A, A0) is a proper CQ*-algebra if

(i) A0 is dense in A with respect to its norm || · ||;

(ii) (ab)* = b*a* whenever the multiplication is defined;

(iii) || y ||0= supa A, || a ||≤1|| ay || for all y A0.

Definition 1.2. A proper CQ*-algebra (A, A0), endowed with the Jordan product

$z\circ x=\frac{zx+xz}{2}$

for all x A and all z A0, is called a proper Jordan CQ*-algebra.

Definition 1.3. Let (A, A0) be proper Jordan CQ*-algebras.

A -linear mapping δ: A0A is called a Jordan derivation if

$\delta \left(x\circ y\right)=x\circ \delta \left(y\right)+\delta \left(x\right)\circ y$

for all x, y A0.

Park and Rassias  investigated homomorphisms and derivations on proper JCQ*- triples.

Eskandani and Vaezi  proved the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras associated with the following Pexiderized Jensen type functional equation

$kf\left(\frac{x+y}{k}\right)={f}_{0}\left(x\right)+{f}_{1}\left(y\right)$

by using direct method.

In this article, using fixed point method, we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras.

Moreover, we investigate the Pexiderized Jensen type functional inequality in proper Jordan CQ*-algebras.

## 2. Derivations on proper Jordan CQ*-algebras

Throughout this section, assume that (A, A0) is a proper Jordan CQ*-algebra with C*-norm || · ||A0 and norm || · || A .

Theorem 2.1. Let φ : A0 × A0 → [0, + ∞) be a function such that

$\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}\phi \left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)=0$
(2.1)

for all x, y A0 . Suppose that f, f0, f1 : A0A are mappings with f(0) = 0 and

$||\mu f\left(x\right)-{f}_{0}\left(y\right)-{f}_{1}\left(z\right)|{|}_{A}\le {∥kf\left(\frac{\mu x+y+z}{k}\right)∥}_{A}$
(2.2)
$||f\left(x\circ y\right)+x\circ {f}_{1}\left(y\right)+{f}_{0}\left(x\right)\circ y|{|}_{A}\le \phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)$
(2.3)

for all $\mu \in {T}^{1}:\left\{\mu \in ℂ:\phantom{\rule{2.77695pt}{0ex}}|\mu |=1\right\}$ and all x, y, z A0. Then the mapping f : A0A is a Jordan derivation. Moreover,

$f\left(x\right)={f}_{0}\left(0\right)-{f}_{0}\left(x\right)={f}_{1}\left(0\right)-{f}_{1}\left(x\right)$

for all x A0.

Proof. Letting x = yz = 0 in (2.2), we get f0(0) + f1(0) = 0.

Letting µ = 1, y = -x and z = 0 in (2.2), we get

$f\mathsf{\text{(}}x\mathsf{\text{)}}={f}_{\mathsf{\text{0}}}\left(-x\right)+{f}_{\mathsf{\text{1}}}\mathsf{\text{(0)}}={f}_{\mathsf{\text{0}}}\left(-x\right)-{f}_{\mathsf{\text{0}}}\mathsf{\text{(0)}}$
(2.4)

for all x A0. Similarly, we have

$f\mathsf{\text{(}}x\mathsf{\text{)}}={f}_{\mathsf{\text{1}}}\mathsf{\text{(-}}x\mathsf{\text{)}}+{f}_{\mathsf{\text{0}}}\mathsf{\text{(0)}}={f}_{\mathsf{\text{1}}}\mathsf{\text{(-}}x\mathsf{\text{)}}-{f}_{\mathsf{\text{1}}}\mathsf{\text{(0)}}$
(2.5)

for all x A0. By (2.2), we have

$\begin{array}{ll}\hfill ||f\left(x+y\right)-f\left(x\right)-f\left(y\right)|{|}_{A}& =||f\left(x+y\right)-\left({f}_{0}\left(-x\right)+{f}_{1}\left(0\right)\right)-\left({f}_{1}\left(-y\right)+{f}_{0}\left(0\right)\right)|{|}_{A}\phantom{\rule{2em}{0ex}}\\ =||f\left(x+y\right)-{f}_{0}\left(-x\right)-{f}_{1}\left(-y\right)|{|}_{A}=0\phantom{\rule{2em}{0ex}}\end{array}$

for all x, y A0. So the mapping f : A0A is additive. Letting y = -µx and z = 0 in (2.2), we get

$\mu f\left(x\right)={f}_{0}\left(-\mu x\right)+{f}_{1}\left(0\right)=f\left(\mu x\right)$

for all x A0. By the same reasoning as in the proof of [, Theorem 2.1], the mapping f : A0A is -linear. By (2.1) and (2.3), we have

$\begin{array}{l}||f\left(x\circ y\right)-x\circ f\left(y\right)-f\left(x\right)\circ y|{|}_{A}\\ =\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}||f\left({2}^{n}x\circ {2}^{n}y\right)-{2}^{n}x\circ \left({f}_{1}\left(-{2}^{n}y\right)-{f}_{1}\left(0\right)\right)-\left({f}_{0}\left(-{2}^{n}x\right)-{f}_{0}\left(0\right)\right)\circ {2}^{n}y|{|}_{A}\\ =\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}||f\left({2}^{n}x\circ {2}^{n}y\right)-{2}^{n}x\circ {f}_{1}\left(-{2}^{n}y\right)-{f}_{0}\left(-{2}^{n}x\right)\circ {2}^{n}y|{|}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}\frac{\phi \left(-{2}^{n}x,-{2}^{n}y\right)}{{4}^{n}}=0\end{array}$

for all x, y A0. So

$f\left(x\circ y\right)=x\circ f\left(y\right)+f\left(x\right)\circ y$

for all x, y A0. Therefore, the mapping f : A0A is a Jordan derivation.

Since f(-x) = -f(x) for all x A0, it follows from (2.4) that

$f\mathsf{\text{(}}x\mathsf{\text{)}}=-f\left(-x\right)=-\mathsf{\text{(}}{f}_{\mathsf{\text{0}}}\mathsf{\text{(}}x\mathsf{\text{)}}-{f}_{\mathsf{\text{0}}}\mathsf{\text{(0))}}={f}_{\mathsf{\text{0}}}\mathsf{\text{(0)}}-{f}_{\mathsf{\text{0}}}\mathsf{\text{(}}x\mathsf{\text{)}}$

for all x A0. It follows from (2.5) that

$f\mathsf{\text{(}}x\mathsf{\text{)}}=-f\mathsf{\text{(}}-x\mathsf{\text{)}}=-\mathsf{\text{(}}{f}_{\mathsf{\text{1}}}\mathsf{\text{(}}x\mathsf{\text{)}}-{f}_{\mathsf{\text{1}}}\mathsf{\text{(0))}}={f}_{\mathsf{\text{1}}}\mathsf{\text{(0)}}-{f}_{\mathsf{\text{1}}}\mathsf{\text{(}}x\mathsf{\text{)}}$

for all x A0. This completes the proof.

□

Corollary 2.2. Let θ, r0, r1 be nonnegative real numbers with r0 + r1 < 2, and let f, f0, f1 : A0A be mappings satisfying f (0) = 0, (2.2) and

$||f\left(x\circ y\right)+x\circ {f}_{1}\left(y\right)+{f}_{0}\left(x\right)\circ y|{|}_{A}\le \theta ||x|{|}_{{A}_{0}}^{{r}_{0}}||y|{|}_{{A}_{0}}^{{r}_{1}}$

for all x, y A0. Then the mapping f : A0A is a Jordan derivation. Moreover,

$f\left(x\right)={f}_{0}\left(0\right)-{f}_{0}\left(x\right)={f}_{1}\left(0\right)-{f}_{1}\left(x\right)$

for all x A0.

Proof. The proof follows from Theorem 2.1.

□

Corollary 2.3. Let θ, r0, r1 be nonnegative real numbers with r < 2 and let f, f0, f1 : A0A be mappings satisfying f(0) = 0, (2.2) and

$||f\left(x\circ y\right)+x\circ {f}_{1}\left(y\right)+{f}_{0}\left(x\right)\circ y|{|}_{A}\le \theta \left(||x|{|}_{{A}_{0}}^{r}+||y|{|}_{{A}_{0}}^{r}\right)$

for all x, y A0. Then the mapping f : A0A is a Jordan derivation. Moreover,

$f\left(x\right)={f}_{0}\left(0\right)-{f}_{0}\left(x\right)={f}_{1}\left(0\right)-{f}_{1}\left(x\right)$

for all x A.

## 3. Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras

We now introduce one of fundamental results of fixed point theory. For the proof, refer to [39, 40]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. .

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 : XX satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that

$d\left(Tx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)\le Ld\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)$

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 .

Theorem 3.1. 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

$d\left({T}^{m}x,\phantom{\rule{2.77695pt}{0ex}}{T}^{m+1}x\right)=\infty \phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}m\ge 0,$

or other exists a natural number m 0 such that

d(Tmx, Tm+1x) <for all mm0;

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

y* is the unique fixed point of T in

$\mathrm{\Lambda }=\left\{y\in \Omega :d\left({T}^{{m}_{0}}x,\phantom{\rule{2.77695pt}{0ex}}y\right)<\infty \right\};$

$d\left(y,{y}^{*}\right)\le \frac{1}{1-L}d\left(y,Ty\right)$ for all y Λ.

Now we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras by using fixed point method.

Theorem 3.2. Let f, f0, f1 : A0A be mappings with f(0) = 0 for which there exists a function $\phi :{A}_{0}^{2}\to \left[0,\phantom{\rule{0.3em}{0ex}}\infty \right)$ with φ(0, 0) = 0 such that

${∥kf\left(\frac{\mu x+\mu y}{k}\right)-\mu {f}_{0}\left(x\right)-\mu {f}_{1}\left(y\right)∥}_{A}\le \phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right),$
(3.1)
${∥kf\left(\frac{x\circ y}{k}\right)-x\circ {f}_{1}\left(y\right)-{f}_{0}\left(x\right)\circ y∥}_{A}\le \phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)$
(3.2)

for all $\mu \in {T}^{1}$ and all x, y A0. If there exists an L < 1 such that $\phi \left(x,y\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2}\right)$ for all x, y A0, then there exists a unique Jordan derivation δ : A0A such that

$\begin{array}{c}||f\left(x\right)-\delta \left(x\right)|{|}_{A}\le \frac{1}{2k-2kL}\phi \left(kx,\phantom{\rule{2.77695pt}{0ex}}kx\right),\\ ||{f}_{0}\left(x\right)-{f}_{0}\left(0\right)-\delta \left(x\right)|{|}_{A}\le \frac{1}{2-2L}\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}x\right)\end{array}$
(3.3)

for all x A0. Moreover, f0(x) - f0(0) = f1(x) - f1(0) for all x A0.

Proof. Letting x = y = 0 and µ = 1 in (3.1), we get f0(0) + f1(0) = 0.

Letting y = 0 and µ = 1 in (3.1), we get

$kf\left(\frac{x}{k}\right)={f}_{0}\left(x\right)+{f}_{1}\left(0\right)={f}_{0}\left(x\right)-{f}_{0}\left(0\right)$
(3.4)

for all x A0. Similarly, we get

$kf\left(\frac{y}{k}\right)={f}_{1}\left(y\right)+{f}_{0}\left(0\right)={f}_{1}\left(y\right)-{f}_{1}\left(0\right)$
(3.5)

for all y A0. Using (3.4) and (3.5), we get

${f}_{0}\left(x\right)-{f}_{0}\left(0\right)={f}_{1}\left(x\right)-{f}_{1}\left(0\right)$

for all x A0.

Let H : A 0 → A be a mapping defined by

$H\left(x\right):={f}_{0}\left(x\right)-{f}_{0}\left(0\right)={f}_{1}\left(x\right)-{f}_{1}\left(0\right)=kf\left(\frac{x}{k}\right)$

for all x A0. Then we have

$||H\left(\mu x+\mu y\right)-\mu H\left(x\right)-\mu H\left(y\right)|{|}_{A}\le \phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)$
(3.6)

for all $\mu \in {T}^{1}$ and x, y A0.

Consider the set

$X:=\left\{g:{A}_{0}\to A\right\}$

and introduce the generalized metric on X:

$d\left(g,\phantom{\rule{2.77695pt}{0ex}}h\right)=\mathsf{\text{inf}}\left\{C\in {ℝ}_{+}:||g\left(x\right)-h\left(x\right)|{|}_{A}\le C\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}x\right),\phantom{\rule{2.77695pt}{0ex}}\forall x\in {A}_{0}\right\}.$

It is easy to show that (X, d) is complete (see [, Lemma 2.1]).

Now we consider the linear mapping J : XX such that

$Jg\left(x\right):=\frac{1}{2}g\left(2x\right)$

for all x A.

By [, Theorem 3.1],

$d\left(Jg,\phantom{\rule{2.77695pt}{0ex}}Jh\right)\le Ld\left(g,\phantom{\rule{2.77695pt}{0ex}}h\right)$

for all g, h X.

Letting µ = 1 and y = x in (3.6), we get

$||H\left(2x\right)-2H\left(x\right)||\le \phi \left(x,\phantom{\rule{2.77695pt}{0ex}}x\right)$
(3.7)

and so

$||H\left(x\right)-\frac{1}{2}H\left(2x\right)||\le \frac{1}{2}\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}x\right)$

for all x A0. Hence $d\left(H,JH\right)\le \frac{1}{2}$.

By Theorem 3.1, there exists a mapping δ : A0A such that

1. (1)

δ is a fixed point of J, i.e.,

$\delta \left(2x\right)=2\delta \left(x\right)$
(3.8)

for all x A0. The mapping δ is a unique fixed point of J in the set

$Y=\left\{g\in X:d\left(f,\phantom{\rule{2.77695pt}{0ex}}g\right)<\infty \right\}.$

This implies that δ is a unique mapping satisfying (3.8) such that there exists C (0, ∞) satisfying

$||H\left(x\right)-\delta \left(x\right)|{|}_{A}\le C\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}x\right)$

for all x A0.

1. (2)

d(JnH, δ) → 0 as n → ∞. This implies the equality

$\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{H\left({2}^{n}x\right)}{{2}^{n}}=\delta \left(x\right)$
(3.9)

for all x A0.

1. (3)

$d\left(H,\phantom{\rule{0.3em}{0ex}}\delta \right)\le \frac{1}{1-L}d\left(H,JH\right)$, which implies the inequality

$d\left(H,\phantom{\rule{2.77695pt}{0ex}}\delta \right)\le \frac{1}{2-2L}.$

This implies that the inequality (3.3) holds.

It follows from (3.6) and (3.9) that

$\begin{array}{l}||\delta \left(\mu x+\mu y\right)-\mu \delta \left(x\right)-\mu \delta \left(y\right)|{|}_{A}\\ =\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{2}^{n}}||H\left({2}^{n}\mu x+{2}^{n}\mu y\right)-\mu H\left({2}^{n}x\right)-\mu H\left({2}^{n}y\right)|{|}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{2}^{n}}\phi \left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)=0\end{array}$

for $\mu \in {T}^{1}$ and all x, y A0. So

$\delta \left(\mu x+\mu y\right)=\mu \delta \left(x\right)+\mu \delta \left(y\right)$

for $\mu \in {T}^{1}$ and all x, y A0. By the same reasoning as in the proof of [, Theorem 2.1], the mapping δ : A0A is -linear.

It follows from $\phi \left(x,y\right)\le 2L\phi \left(\frac{x}{2},\frac{y}{2}\right)$ that

$\underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}\phi \left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)\le \frac{1}{{2}^{n}}\phi \left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)=0$
(3.10)

for all x, y A0.

It follows from (3.2) and (3.10) that

$\begin{array}{l}||\delta \left(x\circ y\right)-x\circ \delta \left(y\right)-\delta \left(x\right)\circ y|{|}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}{∥kf\left({4}^{n}\left(\frac{x\circ y}{k}\right)\right)-{2}^{n}x\circ \left({f}_{1}\left({2}^{n}y\right)+{f}_{0}\left(0\right)\right)-\left({f}_{0}\left({2}^{n}x\right)+{f}_{1}\left(0\right)\right)\circ {2}^{n}y∥}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}{∥kf\left({4}^{n}\left(\frac{x\circ y}{k}\right)\right)-{2}^{n}x\circ {f}_{1}\left({2}^{n}y\right)-{f}_{0}\left({2}^{n}x\right)\circ {2}^{n}y∥}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}\frac{1}{{4}^{n}}\phi \left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)=0\end{array}$

for all x, y A0. Hence

$\delta \left(x\circ y\right)=x\circ \delta \left(y\right)+\delta \left(x\right)\circ y$

for all x, y A0. So δ : A0A is a Jordan derivation, as desired.

□

Corollary 3.3. [, Theorem 3.1] Let be a nonnegative real number and r0, r1 positive real numbers with λ:= r0 + r1 < 1 and let f, f0, f1 : A0A be mappings with f (0) = 0 such that

${∥kf\left(\frac{\mu x+\mu y}{k}\right)-\mu {f}_{0}\left(x\right)-\mu {f}_{1}\left(y\right)∥}_{A}\le \theta ||x|{|}_{{A}_{0}}^{{r}_{0}}||y|{|}_{{A}_{0}}^{{r}_{1}},$
(3.11)
${∥kf\left(\frac{x\circ y}{k}\right)-x\circ {f}_{1}\left(y\right)-{f}_{0}\left(x\right)\circ y∥}_{A}\le \theta ||x|{|}_{{A}_{0}}^{{r}_{0}}||y|{|}_{{A}_{0}}^{{r}_{1}}$
(3.12)

for all $\mu \in {T}^{1}$ and all x, y A0. Then there exists a unique Jordan derivation δ : A0A such that

$\begin{array}{ll}\hfill ||f\left(x\right)-\delta \left(x\right)|{|}_{A}& \le \frac{{k}^{\lambda -1}\theta }{2-{2}^{\lambda }}||x|{|}_{{A}_{0}}^{\lambda },\phantom{\rule{2em}{0ex}}\\ \hfill ||{f}_{0}\left(x\right)-{f}_{0}\left(0\right)-\delta \left(x\right)|{|}_{A}& \le \frac{\theta }{2-{2}^{\lambda }}||x|{|}_{{A}_{0}}^{\lambda }\phantom{\rule{2em}{0ex}}\end{array}$

for all x A0. Moreover, f0(x) - f0(0) = f1(x) - f1(0) for all x A0.

Proof. The proof follows from Theorem 3.2 by taking

$\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right):=\theta ||x|{|}_{{A}_{0}}^{{r}_{0}}||y|{|}_{{A}_{0}}^{{r}_{1}}$

for all x, y A. Letting L = 2λ-1, we get the desired result.

□

Corollary 3.4. [, Theorem 3.4] Let θ, r be a nonnegative real numbers with 0 < r < 1, and let f, f0, f1 : A0A be mappings with f (0) = 0 such that

${∥kf\left(\frac{\mu x+\mu y}{k}\right)-\mu {f}_{0}\left(x\right)-\mu {f}_{1}\left(y\right)∥}_{A}\le \theta \left(||x|{|}_{{A}_{0}}^{r}+||y|{|}_{{A}_{0}}^{r}\right),$
(3.13)
${∥kf\left(\frac{x\phantom{\rule{0.3em}{0ex}}o\phantom{\rule{0.3em}{0ex}}y}{k}\right)-x\phantom{\rule{0.3em}{0ex}}o\phantom{\rule{0.3em}{0ex}}{f}_{1}\left(y\right)-{f}_{0}\left(x\right)\phantom{\rule{0.3em}{0ex}}o\phantom{\rule{0.3em}{0ex}}y∥}_{A}\le \theta \left(||x|{|}_{{A}_{0}}^{r}+||y|{|}_{{A}_{0}}^{r}\right)$
(3.14)

for all $\mu \in {T}^{1}$ and all x, y A0. Then there exists a unique Jordan derivation δ : A0A such that

$\begin{array}{c}||f\left(x\right)-\delta \left(x\right)|{|}_{A}\le \frac{2{k}^{r-1}\theta }{2-{2}^{r}}||x|{|}_{{A}_{0}}^{r},\\ ||{f}_{0}\left(x\right)-{f}_{0}\left(0\right)-\delta \left(x\right)|{|}_{A}\le \frac{2\theta }{2-{2}^{r}}||x|{|}_{{A}_{0}}^{r}\end{array}$

for all x A0. Moreover, f0(x) - f0(0) = f1(x) - f1(0) for all x A0.

Proof. The proof follows from Theorem 3.2 by taking

$\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right):=\theta \left(||x|{|}_{{A}_{0}}^{r}+||y|{|}_{{A}_{0}}^{r}\right)$

for all x, y A. Letting L = 2r- 1, we get the desired result.

□

Theorem 3.5. Let f, f0, f1 : A0A be mappings with f(0) = f0(0) = f1(0) = 0 for which there exists a function $\phi :{A}_{0}^{2}\to \left[0,\infty \right)$ satisfying (3.1) and (3.2). If there exists an L < 1 such that $\phi \left(x,y\right)\le \frac{L}{4}\phi \left(2x,2y\right)$ for all x, y A0, then there exists a unique Jordan derivationδ: A0A such that

$\begin{array}{ll}\hfill ||f\left(x\right)-\delta \left(x\right)|{|}_{A}& \le \frac{L}{4k-4kL}\phi \left(kx,\phantom{\rule{2.77695pt}{0ex}}kx\right),\phantom{\rule{2em}{0ex}}\\ \hfill ||{f}_{0}\left(x\right)-\delta \left(x\right)|{|}_{A}& \le \frac{L}{4-4L}\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}x\right)\phantom{\rule{2em}{0ex}}\end{array}$
(3.15)

for all x A0. Moreover, f0(x) = f1(x) for all x A0.

Proof. Let (X, d) be the generalized metric space defined in the proof of Theorem 3.2.

Now we consider the linear mapping J : XX such that

$Jg\left(x\right):=2g\left(\frac{x}{2}\right)$

for all x X.

Let $H\left(x\right):={f}_{0}\left(x\right)={f}_{1}\left(x\right)=kf\left(\frac{x}{k}\right)$ for all x A0. It follows from (3.7) that

$∥H\left(x\right)-2H\left(\frac{x}{2}\right)∥\le \phi \left(\frac{x}{2},\frac{x}{2}\right)\le \frac{L}{4}\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)$

for all x A0. Thus $d\left(H,JH\right)\le \frac{L}{4}$. One can show that there exists a mapping δ : A0A such that

$d\left(H,\phantom{\rule{2.77695pt}{0ex}}\delta \right)\le \frac{L}{4-4L}.$

Hence we obtain the inequality (3.15).

It follows from $\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)\le \frac{L}{4}\phi \left(2x,\phantom{\rule{2.77695pt}{0ex}}2y\right)$ that

$\underset{n\to \infty }{\mathsf{\text{lim}}}{4}^{n}\phi \left(\frac{x}{{2}^{n}},\phantom{\rule{2.77695pt}{0ex}}\frac{y}{{2}^{n}}\right)=0$

for all x, y A0. So

$\begin{array}{l}||\delta \left(x\circ y\right)-x\circ \delta \left(y\right)-\delta \left(x\right)\circ y|{|}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}{4}^{n}{∥kf\left({4}^{n}\frac{x\circ y}{{4}^{n}k}\right)-\frac{x}{{2}^{n}}\circ {f}_{1}\left(\frac{y}{{2}^{n}}\right)-{f}_{0}\left(\frac{x}{{2}^{n}}\right)\circ \frac{y}{{2}^{n}}∥}_{A}\\ \le \underset{n\to \infty }{\mathsf{\text{lim}}}{4}^{n}\phi \left(\frac{x}{{2}^{n}},\phantom{\rule{2.77695pt}{0ex}}\frac{y}{{2}^{n}}\right)=0\end{array}$

for all x, y A0. Hence

$\delta \left(x\circ y\right)=x\circ \delta \left(y\right)+\delta \left(x\right)\circ y$

for all x, y A0. So δ : A0A is a Jordan derivation, as desired.

The rest of the proof is similar to the proof of Theorem 3.2.

□

Corollary 3.6. [, Theorem 3.2] Let θ be a nonnegative real number and r0, r1 positive real numbers with λ:= r0 + r1 > 2 and let f, f0, f1 : A0A be mappings satisfying f(0) = f0(0) = f1(0) = 0, (3.11) and (3.12). Then there exists a unique Jordan derivation δ: A0A such that

$\begin{array}{c}||f\left(x\right)-\delta \left(x\right)|{|}_{A}\le \frac{{k}^{\lambda -1}\theta }{{2}^{\lambda }-4}||x|{|}_{{A}_{0}}^{\lambda }\\ ||{f}_{i}\left(x\right)-\delta \left(x\right)|{|}_{A}\le \frac{\theta }{{2}^{\lambda }-4}||x|{|}_{{A}_{0}}^{\lambda }\end{array}$

for all x A0. Moreover, f0(x) = f1(x) for all x A0.

Proof. The proof follows from Theorem 3.3 by taking

$\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right):=\theta ||x|{|}_{{A}_{0}}^{{r}_{0}}||y|{|}_{{A}_{0}}^{{r}_{1}}$

for all x, y A. Letting L = 22-λ, we get the desired result.

□

Corollary 3.7. [, Theorem 3.3] Let θ, r be nonnegative real numbers with r > 2, and let f, f0, f1 : A0A be mappings satisfying f (0) = f0(0) = f1(0) = 0, (3.13) and (3.14). Then there exists a unique Jordan derivation δ : A0A such that

$\begin{array}{ll}\hfill ||f\left(x\right)-\delta \left(x\right)|{|}_{A}& \le \frac{2{k}^{r-1}\theta }{{2}^{r}-4}||x|{|}_{{A}_{0}}^{r},\phantom{\rule{2em}{0ex}}\\ \hfill ||{f}_{0}\left(x\right)-\delta \left(x\right)|{|}_{A}& \le \frac{2\theta }{{2}^{r}-4}||x|{|}_{{A}_{0}}^{r}\phantom{\rule{2em}{0ex}}\end{array}$

for all x A0. Moreover, f0(x) = f1(x) for all x A0.

Proof. The proof follows from Theorem 3.3 by taking

$\phi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right):=\theta \left(||x|{|}_{{A}_{0}}^{r}+||y|{|}_{{A}_{0}}^{r}\right)$

for all x, y A. Letting L = 22-r, we get the desired result.

□

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## Acknowledgements

Choonkil Park was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). Dong Yun Shin was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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Correspondence to Dong Yun Shin.

### 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.

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Park, C., Eskandani, G.Z., Vaezi, H. et al. Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras. J Inequal Appl 2012, 114 (2012). https://doi.org/10.1186/1029-242X-2012-114

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• DOI: https://doi.org/10.1186/1029-242X-2012-114

### Keywords

• Hyers-Ulam stability
• proper Jordan CQ*-algebra
• Jordan derivation
• fixed point method 