- Research Article
- Open access
- Published:
Superstability of Generalized Derivations
Journal of Inequalities and Applications volume 2010, Article number: 740156 (2010)
Abstract
We investigate the superstability of the functional equation , where
and
are the mappings on Banach algebra
. We have also proved the superstability of generalized derivations associated to the linear functional equation
, where
.
1. Introduction
The well-known problem of stability of functional equations started with a question of Ulam [1] in 1940. In 1941, Ulam's problem was solved by Hyers [2] for Banach spaces. Aoki [3] provided a generalization of Hyers' theorem for approximately additive mappings. In 1978, Rassias [4] generalized Hyers' theorem by obtaining a unique linear mapping near an approximate additive mapping.
Assume that and
are real normed spaces with
complete,
is a mapping such that for each fixed
the mapping
is continuous on
, and there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ1_HTML.gif)
for all . Then there exists a unique linear mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ2_HTML.gif)
for all .
In 1994, Gvruţa [5] provided a generalization of Rassias' theorem in which he replaced the bound
in (1.1) by a general control function
.
Since then several stability problems for various functional equations have been investigated by many mathematicians (see [6–8]).
The various problems of the stability of derivations and generalized derivations have been studied during the last few years (see, e.g., [9–18]). The purpose of this paper is to prove the superstability of generalized (ring) derivations on Banach algebras.
The following result which is called the superstability of ring homomorphisms was proved by Bourgin [19] in 1949.
Suppose that and
are Banach algebras and
is with unit. If
is surjective mapping and there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ3_HTML.gif)
for all , then
is a ring homomorphism, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ4_HTML.gif)
The first superstability result concerning derivations between operator algebras was obtained by emrl in [20]. In [10], Badora proved the superstability of functional equation
where
is a mapping on normed algebra
with unit. In Section 2, we generalize Badora's result [10, Theorem
] for functional equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ5_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ6_HTML.gif)
where and
are mappings on algebra
with an approximate identity.
In [21, 22], the superstability of generalized derivations on Banach algebras associated to the following Jensen type functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ7_HTML.gif)
where is an integer is considered. Several authors have studied the stability of the general linear functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ8_HTML.gif)
where ,
,
, and
are constants in the field and
is a mapping between two Banach spaces (see [23, 24]). In Section 3, we investigate the superstability of generalized (ring) derivations associated to the linear functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ9_HTML.gif)
where . Our results in this section generalize some results of Moslehian's paper [14]. It has been shown by Moslehian [14, Corollary
] that for an approximate generalized derivation
on a Banach algebra
, there exists a unique generalized derivation
near
. We show that the approximate generalized derivation
is a generalized derivation (see Corollary 3.6).
Let be an algebra. An additive map
is said to be ring derivation on
if
for all
. Moreover, if
for all
, then
is a derivation. An additive mapping (resp., linear mapping)
is called a generalized ring derivation (resp., generalized derivation) if there exists a ring derivation (resp., derivation)
such that
for all
.
2. Superstability of (1.5) and (1.6)
Here we show the superstability of the functional equations (1.5) and (1.6). We prove the superstability of (1.6) without any additional conditions on the mapping .
Theorem 2.1.
Let be a normed algebra with a central approximate identity
and
. Suppose that
and
are mappings for which there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ11_HTML.gif)
for all . Then
for all
.
Proof.
Replacing by
in (2.2), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ12_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ13_HTML.gif)
for all and
. By taking the limit as
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ14_HTML.gif)
for all . Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ15_HTML.gif)
for all .
Let and
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ16_HTML.gif)
Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ17_HTML.gif)
By taking the limit as , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ18_HTML.gif)
Therefore, for all
.
Theorem 2.2.
Let be a normed algebra with a left approximate identity and
. Let
and
be the mappings satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ19_HTML.gif)
for all , where
is a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ20_HTML.gif)
for all . Then
for all
.
Proof.
Let . We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ21_HTML.gif)
Replacing by
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ22_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ23_HTML.gif)
By taking the limit as , we have
. Since
has a left approximate identity, we have
.
In the next theorem, we prove the superstability of (1.5) with no additional functional inequality on the mapping .
Theorem 2.3.
Let be a Banach algebra with a two-sided approximate identity and
. Let
and
be mappings such that
exists for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ24_HTML.gif)
for all , where
is a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ25_HTML.gif)
for all . Then
,
, and
.
Proof.
Replacing by
in (2.15), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ26_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ27_HTML.gif)
for all and
. By taking the limit as
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ28_HTML.gif)
for all .
Fix By (2.19), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ29_HTML.gif)
for all . Then
for all
and all
, and so by taking the limit as
, we have
. Now we obtain
, since
has an approximate identity.
Replacing by
in (2.15), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ30_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ31_HTML.gif)
for all and all
. Sending
to infinity, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ32_HTML.gif)
By (2.23), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ33_HTML.gif)
for all . Therefore, we have
.
The following theorem states the conditions on the mapping under which the sequence
converges for all
.
Theorem 2.4.
Let be a Banach space and
. Suppose that
is a mapping for which there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ34_HTML.gif)
for all . Then
exists and
for all
.
Proof.
3. Superstability of the Generalized Derivations
Our purpose is to prove the superstability of generalized ring derivations and generalized derivations. Throughout this section, is a Banach algebra with a two-sided approximate identity.
Theorem 3.1.
Let such that
. Suppose that
is a mapping with
for which there exist a map
and a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ37_HTML.gif)
for all . Then
is a generalized ring derivation and
is a ring derivation. Moreover,
for all
.
Proof.
Put and
in (3.3). We have
, and so
for all
.
Then by (3.2) and applying Theorem 2.4, we have and
for all
.
Put in (3.3). We get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ38_HTML.gif)
for all . It follows from (3.1) and Theorem 2.3 that
,
, and
for all
.
It suffices to show that and
are additive.
Replacing by
and
by
and putting
in (3.3), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ39_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ40_HTML.gif)
for all and
.
By taking the limit as , we get
, and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ41_HTML.gif)
Putting and replacing
by
in (3.7), we have
. Similarly,
.
Replacing by
and
by
in (3.7), we obtain
for all
. Therefore
is an additive mapping.
Since ,
is additive, and
has an approximate identity,
is additive.
Theorem 3.2.
Let such that
. Suppose that
is a mapping with
for which there exist a map
and a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ43_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ44_HTML.gif)
for all . Then
is a generalized ring derivation and
is a ring derivation. Moreover,
for all
.
Proof.
Replacing ,
by
and putting
in (3.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ45_HTML.gif)
for all . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ46_HTML.gif)
it follows from Theorem 2.4 that exists for all
. By (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ47_HTML.gif)
for all . Putting
in (3.10), it follows from Theorem 2.3 that
and
for all
and
for all
.
Replacing by
and
by
, putting
in (3.10), and multiplying both sides of the inequality by
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ48_HTML.gif)
for all and
. By taking the limit as
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ49_HTML.gif)
for all . Hence, by the same reasoning as in the proof of Theorem 3.1,
and
are additive mappings. Therefore,
is a generalized ring derivation and
is a ring derivation.
Remark 3.3.
We note that Theorems 3.1 and 3.2 and all that following results are obtained with no special conditions on the mapping (see [21, Theorems  2.1 and 2.5]).
Corollary 3.4.
Let or
,
, and
with
. Suppose that
is a mapping with
for which there exist a map
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ50_HTML.gif)
for all . Then
is a generalized ring derivation and
is a ring derivation.
Proof.
Let . For
, if
, then
satisfies (3.1), (3.2), and we apply Theorem 3.1, and if
, then we apply Theorem 3.2 since
has conditions (3.8), (3.9) in this case.
For , apply Theorem 3.2 if
and apply Theorem 3.1 if
.
Theorem 3.5.
Let and let
be a function satisfying either (3.1), (3.2) or (3.8), (3.9). Suppose that
is a mapping with
for which there exists a map
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ51_HTML.gif)
for all and all
. Then
is a generalized derivation and
is a derivation.
Proof.
Let in (3.17). We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ52_HTML.gif)
for all .
Suppose that satisfies (3.1), (3.2). By Theorem 3.1,
is a generalized ring derivation and
is a ring derivation. Moreover,
for all
.
Replacing by
and putting
in (3.17), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ53_HTML.gif)
for all ,
, and
. Since
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ54_HTML.gif)
Hence, by taking the limit as , we get
for all
and
.
Let with
. Then
, and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ55_HTML.gif)
for all . Now by [21, Lemma  2.4],
is a linear mapping and hence
is a linear mapping.
The following result generalizes Corollary and Theorem
of [14].
Corollary 3.6.
Let and
with
. Suppose that
is a mapping with
for which there exist a map
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ56_HTML.gif)
for all and all
. Then
is a generalized derivation and
is a derivation.
Proof.
Define and apply Theorem 3.5.
Theorem 3.7.
Let and let
be a function satisfying either (3.1), (3.2) or (3.8), (3.9). Suppose that
is a mapping with
for which there exists a map
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ57_HTML.gif)
for all . If
is continuous in
for each fixed
, then
is a generalized derivation and
is a derivation.
Proof.
Suppose that satisfies (3.1), (3.2). By Theorem 3.1,
is a generalized ring derivation,
is a ring derivation, and
for all
.
Let . The mapping
, defined by
, is continuous in
. Also, the mapping
is additive, since
is additive. Hence
is
-linear, and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ58_HTML.gif)
for all . Therefore,
is
-linear.
Now let . Since
, there exist
such that
. So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ59_HTML.gif)
for all . Therefore, the mapping
is linear and it follows that
is linear.
Corollary 3.8.
Let or
. Suppose that
is a mapping with
for which there exists a map
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ60_HTML.gif)
for all . Suppose that
is continuous in
for each fixed
. Then
is a generalized derivation and
is a derivation.
Proof.
Let , define
, and apply Theorem 3.7.
Theorem 3.9.
Let be a mapping with
for which there exist a map
and a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ61_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ62_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ63_HTML.gif)
for and all
. If
is continuous in
for each fixed
, then
is a generalized derivation and
is a derivation.
Proof.
Let . By Theorem 3.7, it suffices to prove that
satisfies (3.1), (3.2).
Let . We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ64_HTML.gif)
Then , and so
. Hence
satisfies (3.1).
Let . By (3.28), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ65_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F740156/MediaObjects/13660_2010_Article_2234_Equ66_HTML.gif)
and so satisfies (3.2).
The theorems similar to Theorem 3.9 have been proved by the assumption that the relations similar to (3.29) are true for ,
(see, e.g., [9, 14]). We proved Theorem 3.9, under condition that inequality (3.29) is true for
.
References
Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1964:xvii+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
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
Czerwik S (Ed): Stability of Functional Equations of Ulam-Hyers- Rassias Type. Hadronic Press, Palm Harbor, Fla, USA; 2003.
Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston, Mass, USA; 1998:vi+313.
Rassias ThM (Ed): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.
Amyari M, Rahbarnia F, Sadeghi G: Some results on stability of extended derivations. Journal of Mathematical Analysis and Applications 2007, 329(2):753–758. 10.1016/j.jmaa.2006.07.027
Badora R: On approximate derivations. Mathematical Inequalities & Applications 2006, 9(1):167–173.
Jung Y-S: On the generalized Hyers-Ulam stability of module left derivations. Journal of Mathematical Analysis and Applications 2008, 339(1):108–114. 10.1016/j.jmaa.2007.07.003
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: Almost derivations on
-ternary rings. Bulletin of the Belgian Mathematical Society. Simon Stevin 2007, 14(1):135–142.
Moslehian MS: Hyers-Ulam-Rassias stability of generalized derivations. International Journal of Mathematics and Mathematical Sciences 2006, 2006:-8.
Moslehian MS: Superstability of higher derivations in multi-Banach algebras. Tamsui Oxford Journal of Mathematical Sciences 2008, 24(4):417–427.
Moslehian MS: Ternary derivations, stability and physical aspects. Acta Applicandae Mathematicae 2008, 100(2):187–199. 10.1007/s10440-007-9179-x
Park C-G: Homomorphisms between
-algebras,
-derivations on a
-algebra and the Cauchy-Rassias stability. Nonlinear Functional Analysis and Applications 2005, 10(5):751–776.
Park C-G: Linear derivations on Banach algebras. Nonlinear Functional Analysis and Applications 2004, 9(3):359–368.
Bourgin DG: Approximately isometric and multiplicative transformations on continuous function rings. Duke Mathematical Journal 1949, 16: 385–397. 10.1215/S0012-7094-49-01639-7
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
Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module (I). Journal of Inequalities and Applications 2009, 2009:-10.
Kang S-Y, Chang I-S: Approximation of generalized left derivations. Abstract and Applied Analysis 2008, 2008:-8.
Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996, 48(3–4):217–235.
Jung S-M: On modified Hyers-Ulam-Rassias stability of a generalized Cauchy functional equation. Nonlinear Studies 1998, 5(1):59–67.
Forti G-L: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. Journal of Mathematical Analysis and Applications 2004, 295(1):127–133. 10.1016/j.jmaa.2004.03.011
Brzdęk J, Pietrzyk A: A note on stability of the general linear equation. Aequationes Mathematicae 2008, 75(3):267–270. 10.1007/s00010-007-2894-6
Acknowledgment
The authors express their thanks to the referees for their helpful suggestions to improve the paper.
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
Ansari-Piri, E., Anjidani, E. Superstability of Generalized Derivations. J Inequal Appl 2010, 740156 (2010). https://doi.org/10.1155/2010/740156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/740156