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Abstract
In this paper, we obtain the general solution and the Hyers-Ulam stability of the system of biadditive functional equations
in complex Banach spaces. Furthermore, we prove the Hyers–Ulam stability of f-biderivations in complex Banach algebras.
Mathematics Subject Classifications:
1. Introduction
In the fall of 1940, Ulam [Citation1] raised the first stability problem. He proposed a question whether there exists an exact homomorphism near an approximate homomorphism. An answer to the problem was given by Hyers [Citation2] in the setting of Banach spaces. Since then the stability problems have been extensively investigated for a variety of functional equations and spaces. In most cases, a functional equation is algebraic in nature whereas the stability is rather metrical. Hence, a normed linear space is a suitable choice to work with the stability of functional equations. We refer to [Citation3–7] for results, references and examples.
Hyers was the first mathematician to present the concequence concerning the stability of functional equations. He answered the question of Ulam for the case of approximate additive mappings under the assumption that and
are Banach spaces (see [Citation2]).
The method provided by Hyers [Citation2] which produces the additive function will be called a direct method. This method is the most important and powerful tool to concerning the stability of different functional equations. That is, the exact solution of the functional equation is explicitly constructed as a limit of a sequence, starting from the given approximate solution [Citation6, Citation8]. The other significant method is fixed point theorem, that is, the exact solution of the functional equation is explicitly created as a fixed point of some certain map [Citation9–12].
Let be a complex Banach algebra. A mapping
is biadditive if f is additive in each variable. Furthermore, f is called a
-bilinear mapping if f is
-linear in each variable. Recently, the Hyers–Ulam stability and the hyperstability of biadditive functional equations were proved in [Citation13]. Moreover, many mathematicians have studied the Hyers–Ulam stability of some derivations in algebras and rings (see [Citation14]).
Lemma 1.1
[Citation15]
Let and
be complex Banach algebras and
be a bi-additive mapping such that
for all
and all
, then f is
-bilinear.
Remark 1.1
In Lemma 1.1, is too big. We can take a smaller set such as a part of the unit circle
or we can take a connected path to obtain the same result as in Lemma 1.1 (see [Citation16]).
Definition 1.2
[Citation17, Citation18]
Let be a ring. A biadditive mapping
is called a symmetric biderivation on
if g satisfies
for all
.
Definition 1.3
[Citation19]
Let be a complex Banach algebra. A
-bilinear mapping
is called a biderivation on
if g satisfies
for all
.
See [Citation11, Citation20–22] for more information on biderivations in several spaces.
In this paper, we introduce f-biderivations in a Banach algebra.
Definition 1.4
Let be a complex Banach algebra and
be a
-bilinear mapping. A
-bilinear mapping
is called an f-biderivation on
if g satisfies
for all
.
Example 1.5
Let be defined by
and
be defined by
for all
. Then g is an f-biderivation.
In this paper, we consider the following system of functional equations (1)
(1)
for all
.
The aim of the present paper is to solve the system of biadditive functional equations (Equation1(1)
(1) ) and prove the Hyers–Ulam stability of f-biderivations in complex Banach algebras by using the fixed point method.
Throughout this paper, assume that is a complex Banach algebra.
2. Stability of the system of biadditive functional equations (1)
We solve and investigate the system of biadditive functional equations (Equation1(1)
(1) ) in complex Banach algebras.
Lemma 2.1
Let be mappings satisfying (Equation1
(1)
(1) ) for all
. Then the mappings
are biadditive.
Proof.
Setting x = y = z = w = 0 in (Equation1(1)
(1) ), we have
Putting y = z = w = 0 in (Equation1
(1)
(1) ), we have
for all
. Hence
for all
. Taking x = y = w = 0 in (Equation1
(1)
(1) ), we obtain
and so
for all
.
Letting y = w = 0 in (Equation1(1)
(1) ), we have
(2)
(2)
for all
. So
for all
. Therefore the mapping
is biadditive and thus by (Equation2
(2)
(2) ) the mapping
is biadditive.
Using the fixed point method, we prove the Hyers–Ulam stability of the system of biadditive functional equations (Equation1(1)
(1) ) in complex Banach algebras.
Theorem 2.2
Suppose that is a function such that there exists an L<1 with
(3)
(3)
for all
. Let
be mappings satisfying
(4)
(4)
for all
. Then there exist unique biadditive mappings
such that
(5)
(5)
for all
.
Proof.
Letting x = y = z = w = 0 in (Equation4(4)
(4) ), we obtain
and hence
.
Letting y = x and w = z in (Equation4(4)
(4) ), we get
for all
. So
(6)
(6)
for all
.
We define
and introduce the generalized metric on H as follows:
by
and we consider
. Then d is a complete generalized metric on H (see [Citation23]).
Now, we define the mapping such that
for all
.
Assume that such that
. Then
for all
. It follows that
. Thus
for all
and all
.
From (Equation6(6)
(6) ), we have
and
.
Using the fixed point alternative (see [Citation24]), we deduce the existence of unique fixed points of , that is, the existence of mappings
, respectively, such that
with the following property: there exist
satisfying
for all
.
Since and
,
for all
.
Next, and
which imply
for all
.
Using (Equation3(3)
(3) ) and (Equation4
(4)
(4) ), we conclude that
and
for all
, since L<1. Hence
for all
. So by Lemma 2.1, the mappings
are biadditive.
Corollary 2.3
Let be nonnegative real numbers with p>2 and
be mappings satisfying
for all
. Then there exist unique biadditive mappings
such that
(7)
(7)
for all
.
Proof.
The proof follows from Theorem 2.2 by taking and
for all
.
Corollary 2.4
Let p, q, r, s be nonnegative real numbers with p + q + r + s>4 and be mappings satisfying
for all
. Then there exist unique biadditive mappings
such that
(8)
(8)
for all
.
Proof.
The proof follows from Theorem 2.2 by taking and
for all
.
3. Stability of f-biderivations in Banach algebras
In this section, by using the fixed point technique, we prove the Hyers–Ulam stability of f-biderivations in complex Banach algebras.
Lemma 3.1
Let be mappings satisfying
(9)
(9)
for all
and all
. Then the mappings
are
-bilinear.
Proof.
If we put in (Equation9
(9)
(9) ), then f and g are biadditive and
(10)
(10)
for all
by Lemma 2.1.
Now, taking y = x and w = z in (Equation9(9)
(9) ), we have
(11)
(11)
for all
and all
. From (Equation10
(10)
(10) ) and (Equation11
(11)
(11) ), we obtain
for all
and all
. Thus by Lemma 1.1 the mappings f and g are
-bilinear.
Theorem 3.2
Suppose that is a function such that there exists an L<1 with
(12)
(12)
for all
. Let
be mappings satisfying
(13)
(13)
and
(14)
(14)
for all
and all
. Then there exist unique
-bilinear mappings
such that G is an F-biderivation and satisfying (Equation5
(5)
(5) ).
Proof.
Let in (Equation13
(13)
(13) ). By the same reasoning as in the proof of Theorem 2.2, there exist unique biadditive mappings
satisfying (Equation5
(5)
(5) ), which are given by
for all
.
From (Equation12(12)
(12) ) and (Equation13
(13)
(13) ), we have
for all
and all
. Similarly, one can show that
for all
and all
. So
for all
and all
. Therefore, by Lemma 3.1, the mappings
are
-bilinear.
Next, from (Equation12(12)
(12) ) and (Equation14
(14)
(14) ) it follows that
and
for all
and all
, since L<1. Thus
for all
. Hence the mapping
is an F-biderivation.
Corollary 3.3
Let be nonnegative real numbers with p>4 and
be mappings satisfying
for all
and all
. Then there exist unique
-bilinear mappings
such that G is an F-biderivation and satisfying (Equation7
(7)
(7) ).
Proof.
The proof follows from Theorem 3.2 and Corollary 2.3.
Corollary 3.4
Let p, q, r, s be nonnegative real numbers with p + q + r + s>16 and be mappings satisfying
for all
. Then there exist unique
-bilinear mappings
such that G is an F-biderivation and satisfying (Equation8
(8)
(8) ).
Proof.
The proof follows from Theorem 3.2 and Corollary 3.4.
Remark 3.1
We can obtain some results on asymptotically generalized Lie bi-derivations corresponding to the results given in [Citation25].
Declarations
Human and animal rights: We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being.
Authors' contributions
The authors equally 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.
Acknowledgments
We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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