Abstract and Applied Analysis

Volume 2008 (2008), Article ID 410437, 12 pages

http://dx.doi.org/10.1155/2008/410437

## Jordan -Derivations on -Algebras and -Algebras

^{1}Department of Mathematics Education, Pusan National University, Pusan 609-735, South Korea^{2}Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China^{3}Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 19 September 2008; Revised 20 October 2008; Accepted 31 October 2008

Academic Editor: Ferhan Merdivenci Atici

Copyright © 2008 Jong Su An et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate Jordan -derivations on -algebras and Jordan -derivations on -algebras associated with the following functional inequality for some integer greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras and of Jordan -derivations on -algebras associated with the following functional equation for some integer greater than 1.

#### 1. Introduction and preliminaries

The stability problem of functional equations originated from
a question of Ulam [1] concerning the stability of group homomorphisms.
Hyers [2] gave a
first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’
theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by
considering an unbounded Cauchy difference. In 1982–1994, a generalization of
Hyers-Ulam stability result was proved by J. M. Rassias [5–9]. This author assumed that
Cauchy-Găvruţa-Rassias inequalityis
controlled by a product of different powers of norms, where and such that ,
and retained the condition of continuity of in for each fixed .
In 1999, Găvruţa [10] studied the singular case ,
by constructing a nice counterexample to the above pertinent Ulam stability
problem. Also J. M. Rassias [5–9, 11–13] investigated other conditions and still obtained
stability results. In all these cases, the approach to the existence question
was to prove asymptotic type formulas of the formTheorem 1.1 (see [5–7, 9]). *Let be a real normed vector space and a real complete normed vector space. Assume in
addition that is an approximately additive mapping for which
there exist constants and such that and satisfies**for all
Then, there exists a unique additive mapping satisfying**for all
If, in addition, is a mapping such that the transformation is continuous in for each fixed ,
then is an -linear mapping.*Theorem 1.2 (see [4]). *Let be a mapping from a normed vector space into a Banach space subject to the inequality**for all
where and are constants with and
Then, the limit**exists for all
and is the unique additive mapping which
satisfies**for all
Also, if for each the mapping is continuous in ,
then is linear.*

Th. M. Rassias [14], during the 27th International Symposium on
Functional Equations, asked the question whether such a theorem can also be
proved for .
Gajda [15] following
the same approach as in Th. M. Rassias [4] gave an affirmative solution to this question for .
It was shown by Gajda [15]
as well as by Th. M. Rassias and Šemrl [16] that one cannot prove a
Th. M. Rassias’ type theorem when .
The counterexamples of Gajda [15] as well as of Th. M. Rassias and Šemrl [16] have stimulated several
mathematicians to invent new definitions of *approximately additive* or *approximately linear* mappings, compare
Găvruţa [17] and Jung [18], who among others studied
the Hyers-Ulam stability of functional equations. Theorem 1.2 that was
introduced for the first time by Th. M. Rassias [4] provided a lot of influence in the development of a
generalization of the Hyers-Ulam stability concept. This new concept is known
as *generalized Hyers-Ulam stability* of functional equations (cf. the books of Czerwik [19], Hyers et al. [20]).

Găvruţa [17] provided a further generalization of Th. M. Rassias’ theorem. Isac and Th. M. Rassias [21] applied the Hyers-Ulam stability theory to prove fixed point theorems and study some new applications in nonlinear analysis. In [22], Hyers et al. studied the asymptoticity aspect of Hyers-Ulam stability of mappings. Beginning around the year 1980, the topic of approximate homomorphisms and their stability theory in the field of functional equations and inequalities was taken up by several mathematicians (see [3–18, 21–51]).

Gilányi [26] showed that if satisfies the functional inequalitythen satisfies the Jordan-von Neumann functional equationSee also [52]. Fechner [53] and Gilányi [27] proved the generalized Hyers-Ulam stability of the functional inequality (1.8). Park et al. [42] introduced and investigated 3-variable Cauchy-Jensen functional inequalities and proved the generalized Hyers-Ulam stability of the 3-variable Cauchy-Jensen functional inequalities.

*Definition 1.3. *Let be a -algebra. A -linear mapping is called a
Jordan -derivation if for all

A -algebra , endowed with the Jordan product on , is called a -algebra (see [38, 39]).

*Definition 1.4. *Let be a -algebra. A -linear mapping is called a
Jordan -derivation if for all

This paper is organized as follows. In Section 2, we investigate Jordan -derivations on -algebras associated with the functional inequalityand prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras associated with the functional equation

In Section 3, we investigate Jordan -derivations on -algebras associated with the functional inequality (1.12), and prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras associated with the functional equation (1.13).

Throughout this paper, let be an integer greater than 1.

#### 2. Jordan -derivations on -algebras

Throughout this section, assume that is a -algebra with norm .

Lemma 2.1. *Let be a mapping such that **for all
Then, is Cauchy additive, that is, .*

*Proof. *Letting in
(2.1), we getSo .

Letting and in
(2.1), we getfor all
Hence for all

Letting in
(2.1), we getfor all
Thusfor all
Letting in
(2.5), we get for all
Sofor all
as desired.

Theorem 2.2. *Let and be a nonnegative real number, and let be a mapping such that** for all and all .
Then, the mapping is a Jordan -derivation.**Proof. *Let in
(2.7). By Lemma 2.1, the mapping is Cauchy additive. So and for all

Letting and ,
we getfor all and all
Sofor all
and all .
Hence, for all and all
By the same reasoning as in the proof of [39, Theorem 2.1], the mapping is -linear.

It follows from
(2.8) thatfor all
Thusfor all

Hence the mapping is a Jordan -derivation.

Theorem 2.3. *Let and be a nonnegative real number, and let be a mapping satisfying (2.7) and
(2.8). Then, the mapping is a Jordan -derivation.**Proof. *The proof is similar to the proof of
Theorem 2.2.

We prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras.

Theorem 2.4. *Suppose that is a mapping with for which there exists a function such that**for all
and all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *Putting and ,
and replacing by in
(2.15), we get

Using the induction method, we havefor all and all
It follows that for every ,
the sequence is Cauchy, and hence it is convergent since is complete. SetLet and replace and by and ,
respectively, in (2.15), we getTaking the limit as ,
we obtainfor all and all
Letting and in
(2.21), we getfor all
Hence,for all
in particular, is additive. Now similar to the discussion in
[54, Theorem 2.1], we
show that is -linear. Letting in
(2.18), we getTaking the limit as ,
we havefor all
It follows from [55]
that is unique.

Letting , and replacing by in
(2.15), we obtainSofor all
Letting tend to infinity, we havefor all
Hence is a Jordan -derivation.Corollary 2.5. *Suppose that is a mapping with for which there exist constant and such that**for all and all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *Letting in Theorem 2.4, we obtain the result.Theorem 2.6. *Suppose that is a mapping with for which there exists a function satisfying
(2.15) such that**for all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *Letting and in
(2.15), we getOne can apply the induction
method to prove thatfor all and
It follows that for every ,
the sequence is Cauchy, and hence it is convergent since is complete. SetLetting and replacing and by and ,
respectively, in (2.15), we
getfor all
Taking the limit as ,
we obtainfor all
Hence is additive.

The rest of the proof is similar to the proof of
Theorem 2.4.Corollary 2.7. *Suppose that is a mapping with for which there exist constant and such that**for all and all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *Letting in Theorem 2.6, we obtain the result.

#### 3. Jordan -derivations on -algebras

Throughout this section, assume that is a -algebra with norm .

Theorem 3.1. *Let and be a nonnegative real number, and let be a mapping satisfying (2.7) such that**for all
Then, the mapping is a Jordan -derivation.**Proof. *By the same reasoning as in the
proof of Theorem 2.2, the mapping is -linear.

It follows from
(3.1) thatfor all
Thus,for all

Hence, the mapping is a Jordan -derivation.

Theorem 3.2. *Let and be a nonnegative real number, and let be a mapping satisfying (2.7) and
(3.1). Then, the mapping is a Jordan -derivation.**Proof. *The proof is similar to the proofs
of Theorems 2.2 and 3.1.

We prove the generalized Hyers-Ulam stability of Jordan -derivations on -algebras.

Theorem 3.3. *Suppose that is a mapping with for which there exists a function satisfying
(2.13) and (2.14) such
that**for all and all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *By the same reasoning as in the
proof of Theorem 2.4, there exists a unique -linear mapping such thatfor all
The mapping is given by

Letting , and replacing by in
(3.4), we obtainSofor all
Letting tend to infinity, we havefor all
Hence, is a Jordan -derivation.Corollary 3.4. *Suppose that is a mapping with for which there exist constant and such that**for all and all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *Letting in Theorem 3.3, we obtain the result.Theorem 3.5. *Suppose that is a mapping with for which there exists a function satisfying
(2.31) and (3.4). Then, there exists a unique Jordan -derivation such that**for all **Proof. *The rest of the proof is similar to
the proofs of Theorems 2.4 and 3.3.Corollary 3.6. *Suppose that is a mapping with for which there exist constant and such that**for all and all
Then, there exists a unique Jordan -derivation such that**for all **Proof. *Letting in Theorem 3.5, we obtain the result.

#### Acknowledgment

This work was supported by Korea Research Foundation Grant KRF-2007-313-C00033.

#### References

- S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. View at Zentralblatt MATH · View at MathSciNet - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, no. 4, pp. 222–224, 1941. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Bulletin des Sciences Mathématiques. Deuxième Série*, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “Solution of a problem of Ulam,”
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “Solution of a stability problem of Ulam,”
*Discussiones Mathematicae*, vol. 12, pp. 95–103, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,”
*Discussiones Mathematicae*, vol. 14, pp. 101–107, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Găvruţa, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in
*Advances in Equations and Inequalities*, Hadronic Mathematics, pp. 67–71, Hadronic Press, Palm Harbor, Fla, USA, 1999. View at Google Scholar - J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,”
*Chinese Journal of Mathematics*, vol. 20, no. 2, pp. 185–190, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “On the stability of the general Euler-Lagrange functional equation,”
*Demonstratio Mathematica*, vol. 29, no. 4, pp. 755–766, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. J. Rassias and J. M. Rassias, “On the Ulam stability for Euler-Lagrange type quadratic functional equations,”
*The Australian Journal of Mathematical Analysis and Applications*, vol. 2, no. 1, pp. 1–10, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,”
*Aequationes Mathematicae*, vol. 39, pp. 292–293, 1990. View at Google Scholar - Z. Gajda, “On stability of additive mappings,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 14, no. 3, pp. 431–434, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,”
*Proceedings of the American Mathematical Society*, vol. 114, no. 4, pp. 989–993, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431–436, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-M. Jung, “On the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 204, no. 1, pp. 221–226, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Czerwik,
*Functional Equations and Inequalities in Several Variables*, World Scientific, River Edge, NJ, USA, 2002. View at Zentralblatt MATH · View at MathSciNet - D. H. Hyers, G. Isac, and Th. M. Rassias,
*Stability of Functional Equations in Several Variables*, vol. 34 of*Progress in Nonlinear Differential Equations and Their Applications*, Birkhäuser, Boston, Mass, USA, 1998. View at Zentralblatt MATH · View at MathSciNet - G. Isac and Th. M. Rassias, “Stability of $\psi $-additive mappings: applications to nonlinear analysis,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 19, no. 2, pp. 219–228, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - D. H. Hyers, G. Isac, and Th. M. Rassias, “On the asymptoticity aspect of Hyers-Ulam stability of mappings,”
*Proceedings of the American Mathematical Society*, vol. 126, no. 2, pp. 425–430, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. G. Bourgin, “Classes of transformations and bordering transformations,”
*Bulletin of the American Mathematical Society*, vol. 57, pp. 223–237, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,”
*Aequationes Mathematicae*, vol. 50, no. 1-2, pp. 143–190, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gilányi, “On the stability of monomial functional equations,”
*Publicationes Mathematicae Debrecen*, vol. 56, no. 1-2, pp. 201–212, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gilányi, “Eine zur Parallelogrammgleichung äquivalente Ungleichung,”
*Aequationes Mathematicae*, vol. 62, no. 3, pp. 303–309, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gilányi, “On a problem by K. Nikodem,”
*Mathematical Inequalities & Applications*, vol. 5, no. 4, pp. 707–710, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. M. Gruber, “Stability of isometries,”
*Transactions of the American Mathematical Society*, vol. 245, pp. 263–277, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K.-W. Jun and H.-M. Kim, “On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 2, pp. 1335–1350, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,”
*Journal of Difference Equations and Applications*, vol. 13, no. 12, pp. 1139–1153, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,”
*Journal of Mathematical Analysis and Applications*, vol. 222, no. 1, pp. 126–137, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-M. Jung, “On the Hyers-Ulam-Rassias stability of a quadratic functional equation,”
*Journal of Mathematical Analysis and Applications*, vol. 232, no. 2, pp. 384–393, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-M. Kim, J. M. Rassias, and Y.-S. Cho, “Stability problem of Ulam for Euler-Lagrange quadratic mappings,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 10725, 15 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kurepa, “On the quadratic functional,”
*Publications de l'Institut Mathématique de l'Académie Serbe des Sciences et des Arts*, vol. 13, pp. 57–72, 1961. View at Google Scholar · View at MathSciNet - Y.-S. Lee and S.-Y. Chung, “Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,”
*Applied Mathematics Letters*, vol. 21, no. 7, pp. 694–700, 2008. View at Google Scholar · View at MathSciNet - P. Nakmahachalasint, “On the generalized Ulam-Găvruţa-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 63239, 10 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-G. Park, “Lie $\ast $-homomorphisms between Lie ${C}^{\ast}$-algebras and Lie $\ast $-derivations on Lie ${C}^{\ast}$-algebras,”
*Journal of Mathematical Analysis and Applications*, vol. 293, no. 2, pp. 419–434, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-G. Park, “Homomorphisms between Lie $J{C}^{\ast}$-algebras and Cauchy-Rassias stability of Lie $J{C}^{\ast}$-algebra derivations,”
*Journal of Lie Theory*, vol. 15, no. 2, pp. 393–414, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-G. Park, “Homomorphisms between Poisson $J{C}^{\ast}$-algebras,”
*Bulletin of the Brazilian Mathematical Society*, vol. 36, no. 1, pp. 79–97, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-G. Park, “Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between ${C}^{\ast}$-algebras,”
*Bulletin of the Belgian Mathematical Society*, vol. 13, no. 4, pp. 619–632, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park, “Isomorphisms between quasi-Banach algebras,”
*Chinese Annals of Mathematics. Series B*, vol. 28, no. 3, pp. 353–362, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park, Y. S. Cho, and M.-H. Han, “Functional inequalities associated with Jordan-von Neumann-type additive functional equations,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 41820, 13 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-G. Park and J. Hou, “Homomorphisms between ${C}^{\ast}$-algebras associated with the Trif functional equation and linear derivations on ${C}^{\ast}$-algebras,”
*Journal of the Korean Mathematical Society*, vol. 41, no. 3, pp. 461–477, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. G. Park, J. C. Hou, and S. Q. Oh, “Homomorphisms between $J{C}^{\ast}$-algebras and Lie ${C}^{\ast}$-algebras,”
*Acta Mathematica Sinica*, vol. 21, no. 6, pp. 1391–1398, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park, J. Park, and J. Shin, “Hyers-Ulam-Rassias stability of quadratic functional equations in Banach modules over a ${C}^{\ast}$-algebra,”
*Chinese Annals of Mathematics. Series B*, vol. 24, no. 2, pp. 261–266, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,”
*Demonstratio Mathematica*, vol. 39, no. 3, pp. 523–530, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 246, no. 2, pp. 352–378, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of functional equations in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 264–284, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,”
*Acta Applicandae Mathematicae*, vol. 62, no. 1, pp. 23–130, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias,
*Functional Equations, Inequalities and Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at Zentralblatt MATH · View at MathSciNet - K. Ravi and M. Arunkumar, “On the Ulam-Găvruţa-Rassias stability of the orthogonally Euler-Lagrange type functional equation,”
*International Journal of Applied Mathematics & Statistics*, vol. 7, no. Fe07, pp. 143–156, 2007. View at Google Scholar · View at MathSciNet - J. Rätz, “On inequalities associated with the Jordan-von Neumann functional equation,”
*Aequationes Mathematicae*, vol. 66, no. 1-2, pp. 191–200, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Fechner, “Stability of a functional inequality associated with the Jordan-von Neumann functional equation,”
*Aequationes Mathematicae*, vol. 71, no. 1-2, pp. 149–161, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Amyari, F. Rahbarnia, and Gh. Sadeghi, “Some results on stability of extended derivations,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 2, pp. 753–758, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - C. Baak, “Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces,”
*Acta Mathematica Sinica*, vol. 22, no. 6, pp. 1789–1796, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet