- Research Article
- Open access
- Published:
Stability of Homomorphisms and Generalized Derivations on Banach Algebras
Journal of Inequalities and Applications volume 2009, Article number: 595439 (2009)
Abstract
We prove the generalized Hyers-Ulam stability of homomorphisms and generalized derivations associated to the following functional equation on Banach algebras.
1. Introduction
The first stability problem concerning group homomorphisms was raised from a question of Ulam [1]. Letbe a group and let
be a metric group with the metric
. Given
, does there exist
such that if a mapping
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ1_HTML.gif)
for all, then there is a homomorphism
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ2_HTML.gif)
for all
Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Aoki [3] and Rassias [4] provided a generalization of the Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also [5]).
Theorem 1.1 (Rassias).
Let be a mapping from a normed vector space
into a Banach space
subject to the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ3_HTML.gif)
for all , where
and
are constants with
and
. Then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ4_HTML.gif)
exists for all and
is the unique additive mapping which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ5_HTML.gif)
for all . If
then inequality (1.3) holds for
and (1.5) for
. Also, if for each
the mapping
is continuous in
, then
is linear.
In 1994, a generalization of the Rassias' theorem was obtained by Gvruţa [6], who replaced the bound
by a general control function
For the stability problems of various functional equations and mappings and their Pexiderized versions, we refer the readers to [7–15]. We also refer readers to the books in [16–19].
Let be a real or complex algebra. A mapping
is said to be a(ring) derivation if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ6_HTML.gif)
for all If, in addition,
for all
and all
then
is called a linear derivation, where
denotes the scalar field of
. Singer and Wermer [20] proved that if
is a commutative Banach algebra and
is a continuous linear derivation, then
They also conjectured that the same result holds even
is a discontinuous linear derivation. Thomas [21] proved the conjecture. As a direct consequence, we see that there are no nonzero linear derivations on a semisimple commutative Banach algebra, which had been proved by Johnson [22]. On the other hand, it is not the case for ring derivations. Hatori and Wada [23] determined a representation of ring derivations on a semi-simple commutative Banach algebra (see also [24]) and they proved that only the zero operator is a ring derivation on a semi-simple commutative Banach algebra with the maximal ideal space without isolated points. The stability of derivations between operator algebras was first obtained by
emrl [25]. Badora [26] and Miura et al. [8] proved the Hyers-Ulam-Rassias stability of ring derivations on Banach algebras. An additive mapping
is called a Jordan derivation in case
is fulfilled for all
Every derivation is a Jordan derivation. The converse is in general not true (see [27, 28]). The concept of generalized derivation has been introduced by M. Brešar [29]. Hvala [30] and Lee [31] introduced a concept of
-derivation (see also [32]). Let
be automorphisms of
An additive mapping
is called a
-derivation in case
holds for all pairs
An additive mapping
is called a
-Jordan derivation in case
holds for all
An additive mapping
is called a generalized
-derivation in case
holds for all pairs
where
is a
-derivation. An additive mapping
is called a generalized
-Jordan derivation in case
holds for all
where
is a
-Jordan derivation. It is clear that every generalized
-derivation is a generalized
-Jordan derivation.
The aim of the present paper is to establish the stability problem of homomorphisms and generalized -derivations by using the fixed point method (see [7, 33–35]).
Let be a set. A function
is called a generalized metric on
if
satisfies
(i) if and only if
;
(ii) for all
;
(iii) for all
We recall the following theorem by Margolis and Diaz.
Theorem 1.2 (See [36]).
Let be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. Then for each given element
, either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ7_HTML.gif)
for all nonnegative integers or there exists a nonnegative integer
such that
(1) for all
;
(2)the sequence converges to a fixed point
of
;
(3) is the unique fixed point of
in the set
;
(4) for all
.
2. Stability of Homomorphisms
Daróczy et al. [37] have studied the functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ8_HTML.gif)
where is a fixed parameter and
is unknown,
is a nonvoid open interval and (2.1) holds for all
They characterized the equivalence of (2.1) and Jensen's functional equation in terms of the algebraic properties of the parameter
For
in (2.1), we get the Jensen's functional equation. In the present paper, we establish the general solution and some stability results concerning the functional equation (2.1) in normed spaces for
This applied to investigate and prove the generalized Hyers-Ulam stability of homomorphisms and generalized derivations in real Banach algebras. In this section, we assume that
is a normed algebra and
is a Banach algebra. For convenience, we use the following abbreviation for a given mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ9_HTML.gif)
for all .
Lemma 2.1.
Let and
be linear spaces. A mapping
with
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ10_HTML.gif)
for all if and only if
is additive.
Proof.
Let satisfy (2.3). Letting
in (2.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ11_HTML.gif)
for all Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ12_HTML.gif)
for all Letting
in (2.3), we get
for all
Therefore by (2.5) we have
for all
This means that
is odd. Letting
in (2.3) and using the oddness of
, we infer that
for all
Hence by (2.4) we have
for all
Therefore it follows from (2.3) that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ13_HTML.gif)
for all Replacing
and
by
and
in (2.6), respectively, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ14_HTML.gif)
for all Replacing
by
in (2.7) and using the oddness of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ15_HTML.gif)
for all Adding (2.6) to (2.8), we get
for all
Using the identity
and replacing
by
in the last identity, we infer that
for all
Hence
is additive. The converse is obvious.
Theorem 2.2.
Let be a mapping with
for which there exist functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ18_HTML.gif)
for all . If there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ19_HTML.gif)
for all , then there exists a unique (ring) homomorphism
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ21_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ22_HTML.gif)
Proof.
By the assumption, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ23_HTML.gif)
for all Letting
in (2.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ24_HTML.gif)
for all Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ25_HTML.gif)
for all Letting
in (2.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ26_HTML.gif)
for all Therefore by (2.18) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ27_HTML.gif)
for all Letting
in (2.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ28_HTML.gif)
for all Now, it follows from (2.20) and (2.21) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ29_HTML.gif)
for all Let
We introduce a generalized metric on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ30_HTML.gif)
It is easy to show that is a generalized complete metric space [34].
Now we consider the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ31_HTML.gif)
Let and let
be an arbitrary constant with
. From the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ32_HTML.gif)
for all . By the assumption and the last inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ33_HTML.gif)
for all . So
for any
. It follows from (2.22) that
. Therefore according to Theorem 1.2, the sequence
converges to a fixed point
of
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ34_HTML.gif)
and for all
. Also
is the unique fixed point of
in the set
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ35_HTML.gif)
that is, inequality (2.13) holds true for all . It follows from the definition of
, (2.10), and (2.16) that
for all
Since
by Lemma 2.1 the mapping
is additive. So it follows from the definition of
, (2.9), and (2.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ36_HTML.gif)
for all So
is homomorphism. Similarly, we have from (2.9) and (2.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ37_HTML.gif)
for all Since
is homomorphism, we get (2.14) from (2.30).
Finally it remains to prove the uniqueness of . Let
another homomorphism satisfying (2.13). Since
and
is additive, we get
and
for all
, that is,
is a fixed point of
. Since
is the unique fixed point of
in
, we get
We need the following lemma in the proof of the next theorem.
Lemma 2.3 (See [38]).
Let and
be linear spaces and
be an additive mapping such that
for all
and all
Then the mapping
is
-linear.
Lemma 2.4.
Let and
be linear spaces. A mapping
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ38_HTML.gif)
for all and all
if and only if
is
-linear.
Proof.
Let satisfy (2.31). Letting
in (2.31), we get
By Lemma 2.1, the mapping
is additive. Letting
in (2.31) and using the additivity of
we get that
for all
and all
So by Lemma 2.4, the mapping
is
-linear. The converse is obvious.
The following theorem is an alternative result of Theorem 2.2 with similar proof.
Theorem 2.5.
Let be a mapping for which there exist functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ39_HTML.gif)
for all and all
. If there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ40_HTML.gif)
for all , then there exists a unique homomorphism
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ41_HTML.gif)
for all , where
is defined as in Theorem 2.2.
Proof.
It follows from the assumptions that and so
The rest of the proof is similar to the proof of Theorem 2.2 and we omit the details.
Corollary 2.6.
Let be non-negative real numbers with
. Suppose that
is a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ42_HTML.gif)
for all and all
. Then there exists a unique homomorphism
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ43_HTML.gif)
for all
Proof.
The proof follows from Theorem 2.2 by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ44_HTML.gif)
for all . Then we can choose
and we get the desired results.
Corollary 2.7.
Let be non-negative real numbers with
and
Suppose that
is a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ45_HTML.gif)
for all and all
. Then there exists a unique homomorphism
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ46_HTML.gif)
for all
Proof.
The proof follows from Theorem 2.5 by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ47_HTML.gif)
for all . Then we can choose
and we get the desired results.
3. Stability of Generalized
-Derivations
In this section, we assume that is a Banach algebra, and
are automorphisms of
For convenience, we use the following abbreviation for given mappings
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ48_HTML.gif)
for all . Now we prove the generalized Hyers-Ulam stability of generalized
-derivations and generalized
-Jordan derivations in Banach algebras.
Theorem 3.1.
Let be mappings with
for which there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ51_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ52_HTML.gif)
for all . If there exists a constants
such
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ53_HTML.gif)
for all , then there exist a unique
-Jordan derivation
and a unique generalized
-Jordan derivation
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ54_HTML.gif)
for all , where
is defined as in Theorem 2.2.
Proof.
It follows from the assumptions that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ55_HTML.gif)
for all By the proof of Theorem 2.5, there exist unique additive mappings
satisfying (3.7) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ56_HTML.gif)
for all . It follows from the definitions of
(3.3), and (3.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ57_HTML.gif)
for all Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ58_HTML.gif)
for all Hence
is a
-Jordan derivation and
is a generalized
-Jordan derivation.
Remark 3.2.
Applying Theorem 3.1 for the case , there exist a unique
-Jordan derivation
and a unique generalized
-Jordan derivation
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ59_HTML.gif)
for all .
The following theorem is an alternative result of Theorem 3.1 with similar proof.
Theorem 3.3.
Let be mappings with
for which there exists a function
satisfying (3.2)–(3.5). If there exists a constant
such
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ60_HTML.gif)
for all , then there exist a unique
-Jordan derivation
and a unique generalized
-Jordan derivation
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ61_HTML.gif)
for all , where
is defined as in Theorem 2.2.
Remark 3.4.
Applying Theorem 3.3 for the case , there exist a unique
-Jordan derivation
and a unique generalized
-Jordan derivation
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F595439/MediaObjects/13660_2009_Article_1978_Equ62_HTML.gif)
for all .
References
Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1
Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7
Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211
Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bulletin of the Brazilian Mathematical Society 2006,37(3):361–376. 10.1007/s00574-006-0016-z
Miura T, Hirasawa G, Takahasi S-E: A perturbation of ring derivations on Banach algebras. Journal of Mathematical Analysis and Applications 2006,319(2):522–530. 10.1016/j.jmaa.2005.06.060
Moslehian MS: Hyers-Ulam-Rassias stability of generalized derivations. International Journal of Mathematics and Mathematical Sciences 2006, 2006:-8.
Najati A: Hyers-Ulam stability of an
-Apollonius type quadratic mapping. Bulletin of the Belgian Mathematical Society. Simon Stevin 2007,14(4):755–774.
Najati A: On the stability of a quartic functional equation. Journal of Mathematical Analysis and Applications 2008,340(1):569–574. 10.1016/j.jmaa.2007.08.048
Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2008,337(1):399–415. 10.1016/j.jmaa.2007.03.104
Najati A, Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. Journal of Mathematical Analysis and Applications 2007,335(2):763–778. 10.1016/j.jmaa.2007.02.009
Najati A, Park C: On the stability of an
-dimensional functional equation originating from quadratic forms. Taiwanese Journal of Mathematics 2008,12(7):1609–1624.
Park C: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002,275(2):711–720. 10.1016/S0022-247X(02)00386-4
Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.
Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.
Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.
Rassias ThM: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.
Singer IM, Wermer J: Derivations on commutative normed algebras. Mathematische Annalen 1955, 129: 260–264. 10.1007/BF01362370
Thomas MP: The image of a derivation is contained in the radical. Annals of Mathematics 1988,128(3):435–460. 10.2307/1971432
Johnson BE: Continuity of derivations on commutative algebras. American Journal of Mathematics 1969, 91: 1–10. 10.2307/2373262
Hatori O, Wada J: Ring derivations on semi-simple commutative Banach algebras. Tokyo Journal of Mathematics 1992,15(1):223–229. 10.3836/tjm/1270130262
Šemrl P: On ring derivations and quadratic functionals. Aequationes Mathematicae 1991,42(1):80–84. 10.1007/BF01818480
Šemrl P: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equations and Operator Theory 1994,18(1):118–122. 10.1007/BF01225216
Badora R: On approximate derivations. Mathematical Inequalities & Applications 2006,9(1):167–173.
Cusack JM: Jordan derivations on rings. Proceedings of the American Mathematical Society 1975,53(2):321–324. 10.1090/S0002-9939-1975-0399182-5
Herstein IN: Jordan derivations of prime rings. Proceedings of the American Mathematical Society 1957, 8: 1104–1110. 10.1090/S0002-9939-1957-0095864-2
Brešar M: Jordan derivations on semiprime rings. Proceedings of the American Mathematical Society 1988,104(4):1003–1006.
Hvala B: Generalized derivations in rings. Communications in Algebra 1998,26(4):1147–1166. 10.1080/00927879808826190
Lee T-K: Generalized derivations of left faithful rings. Communications in Algebra 1999,27(8):4057–4073. 10.1080/00927879908826682
Liu C-K, Shiue W-K: Generalized Jordan triple
-derivations on semiprime rings. Taiwanese Journal of Mathematics 2007,11(5):1397–1406.
Amyari M, Moslehian MS: Hyers-Ulam-Rassias stability of derivations on Hilbert
-modules. In Topological Algebras and Applications, Contemporary Mathematics. Volume 427. American Mathematical Society, Providence, RI, USA; 2007:31–39.
Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory, Grazer Mathematische Berichte. Volume 346. Karl-Franzens-Universitaet Graz, Graz, Austria; 2004:43–52.
Jung S-M, Kim T-S: A fixed point approach to the stability of the cubic functional equation. BoletÃn de la Sociedad Matemática Mexicana 2006,12(1):51–57.
Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0
Daróczy Z, Lajkó K, Lovas RL, Maksa Gy, Páles Zs: Functional equations involving means. Acta Mathematica Hungarica 2007,116(1–2):79–87. 10.1007/s10474-007-5296-2
Park C: Homomorphisms between Poisson
-algebras. Bulletin of the Brazilian Mathematical Society 2005,36(1):79–97. 10.1007/s00574-005-0029-z
Acknowledgment
The second author was supported by Hanyang University in 2009.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Najati, A., Park, C. Stability of Homomorphisms and Generalized Derivations on Banach Algebras. J Inequal Appl 2009, 595439 (2009). https://doi.org/10.1155/2009/595439
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/595439