1. Introduction and Preliminaries
The stability problem of functional equations originated from a question of Ulam [
1] concerning the stability of group homomorphisms.
The functional equation
is called the
Cauchy equation. In particular, every solution of the Cauchy equation is said to be an
additive mapping. 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 Rassias [
4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [
5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.
Gilányi [
6] showed that if
f satisfies the functional inequality
then
f satisfies the Jordan-von Neumann functional equation
See also [
7]. Fechner [
8] and Gilányi [
9] proved the Hyers-Ulam stability of the functional inequality (
1).
Park [
10,
11] defined additive
-functional inequalities and proved the Hyers-Ulam stability of the additive
-functional inequalities in Banach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors (see [
12,
13,
14,
15]).
The notion of a quasi-multiplier is a generalization of the notion of a multiplier on a Banach algebra, which was introduced by Akemann and Pedersen [
16] for
-algebras. McKennon [
17] extended the definition to a general complex Banach algebra with bounded approximate identity as follows.
Definition 1. [17] Let A be a complex Banach algebra. A -bilinear mapping is called a quasi-multiplier on A if P satisfiesfor all . Definition 2. Let A be a complex Banach ∗-algebra. A bi-additive mapping is called a quasi-∗-multiplier on A if P is -linear in the first variable and satisfiesfor all . It is easy to show that if P is a quasi-∗-multiplier, then P is conjugate -linear in the second variable and for all .
We recall a fundamental result in fixed point theory.
Theorem 1. [18,19] Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , eitherfor all nonnegative integers n or there exists a positive integer such that - (1)
;
- (2)
the sequence converges to a fixed point of J;
- (3)
is the unique fixed point of J in the set ;
- (4)
for all .
In 1996, Isac and Rassias [
20] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [
21,
22,
23,
24,
25]).
This paper is organized as follows: In
Section 2 and
Section 3, we prove the Hyers-Ulam stability of the following bi-additive
s-functional inequalities
in complex Banach spaces by using the fixed point method. Here
s is a fixed nonzero complex number with
. In
Section 4, we prove the Hyers-Ulam stability and the superstability of quasi-∗-multipliers on Banach ∗-algebras and unital
-algebras associated to the bi-additive
s-functional inequalities (
2) and (
3).
Throughout this paper, let X be a complex normed space and Y a complex Banach space. Let A be a complex Banach ∗-algebra. Assume that s is a fixed nonzero complex number with .
2. Bi-Additive -Functional Inequality (2)
Park [
26] solved the bi-additive
s-functional inequality (
2) in complex normed spaces.
Lemma 1. ([26] Lemma 2.1) If a mapping satisfies andfor all , then is bi-additive. Using the fixed point method, we prove the Hyers-Ulam stability of the bi-additive
s-functional inequality (
4) in complex Banach spaces.
Theorem 2. Let be a function such that there exists an withfor all . Let be a mapping satisfying andfor all . Then there exists a unique bi-additive mapping such thatfor all . Proof. Letting
and
in (
6), we get
for all
. So
for all
.
Consider the set
and introduce the generalized metric on
S:
where, as usual,
. It is easy to show that
is complete (see [
27]).
Now we consider the linear mapping
such that
for all
.
Let
be given such that
. Then
for all
. Hence
for all
. So
implies that
. This means that
for all
.
It follows from (
8) that
for all
. So
.
By Theorem 1, there exists a mapping satisfying the following:
(1)
B is a fixed point of
J, i.e.,
for all
. The mapping
B is a unique fixed point of
J in the set
This implies that
B is a unique mapping satisfying (
9) such that there exists a
satisfying
for all
;
(2)
as
. This implies the equality
for all
;
(3)
, which implies
for all
. So we obtain (
7).
It follows from (
5) and (
6) that
for all
, since
tends to zero as
. So
for all
. By Lemma 1, the mapping
is bi-additive. ☐
Corollary 1. Let and θ be nonnegative real numbers and let be a mapping satisfying andfor all . Then there exists a unique bi-additive mapping such thatfor all . Proof. The proof follows from Theorem 2 by taking for all . Choosing , we obtain the desired result. ☐
Theorem 3. Let be a function such that there exists an withfor all . Let be a mapping satisfying (
6)
and for all . Then there exists a unique bi-additive mapping such thatfor all . Proof. Let be the generalized metric space defined in the proof of Theorem 2.
Now we consider the linear mapping
such that
for all
.
It follows from (
8) that
for all
.
The rest of the proof is similar to the proof of Theorem 2. ☐
Corollary 2. Let and θ be nonnegative real numbers and let be a mapping satisfying (
10)
and for all . Then there exists a unique bi-additive mapping such thatfor all . Proof. The proof follows from Theorem 3 by taking for all . Choosing , we obtain the desired result. ☐
3. Bi-Additive -Functional Inequality (3)
Park [
26] solved the bi-additive
s-functional inequality (
3) in complex normed spaces.
Lemma 2. ([26] Lemma 3.1) If a mapping satisfies andfor all , then is bi-additive. Using the fixed point method, we prove the Hyers-Ulam stability of the bi-additive
s-functional inequality (
12) in complex Banach spaces.
Theorem 4. Let be a function satisfying (
5)
. Let be a mapping satisfying andfor all . Then there exists a unique bi-additive mapping such thatfor all . Proof. Letting
in (
13), we get
for all
.
Consider the set
and introduce the generalized metric on
S:
where, as usual,
. It is easy to show that
is complete (see [
27]).
Now we consider the linear mapping
such that
for all
.
Let
be given such that
. Then
for all
. Hence
for all
. So
implies that
. This means that
for all
.
It follows from (
15) that
for all
. So
.
By Theorem 1, there exists a mapping satisfying the following:
(1)
B is a fixed point of
J, i.e.,
for all
. The mapping
B is a unique fixed point of
J in the set
This implies that
B is a unique mapping satisfying (
16) such that there exists a
satisfying
for all
;
(2)
as
. This implies the equality
for all
;
(3)
, which implies
for all
. So we obtain (
14).
The rest of the proof is similar to the proof of Theorem 2. ☐
Corollary 3. Let and θ be nonnegative real numbers and let be a mapping satisfying andfor all . Then there exists a unique bi-additive mapping such thatfor all . Proof. The proof follows from Theorem 4 by taking for all . Choosing , we obtain the desired result. ☐
Theorem 5. Let be a function satisfying (11). Let be a mapping satisfying (13) and for all . Then there exists a unique bi-additive mapping such thatfor all . Proof. Let be the generalized metric space defined in the proof of Theorem 4.
Now we consider the linear mapping
such that
for all
.
It follows from (
15) that
for all
.
The rest of the proof is similar to the proofs of Theorems 2 and 4. ☐
Corollary 4. Let and θ be nonnegative real numbers and let be a mapping satisfying (17) and for all . Then there exists a unique bi-additive mapping such thatfor all . Proof. The proof follows from Theorem 5 by taking for all . Choosing , we obtain the desired result. ☐
4. Quasi-∗-Multipliers in -Algebras
In this section, we investigate quasi-∗-multipliers on complex Banach ∗-algebras and unital
-algebras associated to the bi-additive
s-functional inequalities (
4) and (
12).
Theorem 6. Let be a function such that there exists an withfor all . Let be a mapping satisfying andfor all and all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . Furthermore, If, in addition, the mapping satisfies andfor all , then the mapping is a quasi-∗-multiplier. Proof. Let
in (
20). By Theorem 2, there is a unique bi-additive mapping
satisfying (
21) defined by
for all
.
If for all , then we can easily show that for all .
Letting
and
in (
20), we get
for all
and all
. So
for all
and all
. Hence
and so
for all
and all
. By ([
28] Theorem 2.1), the bi-additive mapping
is
-linear in the first variable.
It follows from (
22) that
for all
. Thus
for all
.
It follows from (
23) that
for all
. Thus
for all
. Hence the mapping
is a quasi-∗-multiplier. ☐
Corollary 5. Let and θ be nonnegative real numbers, and let be a mapping satisfying andfor all and all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies andfor all , then the mapping is a quasi-∗-multiplier. Proof. The proof follows from Theorem 6 by taking for all . Choosing , we obtain the desired result. ☐
Theorem 7. Let be a function such that there exists an withfor all . Let be a mapping satisfying (20) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies (22), (23) and for all , then the mapping is a quasi-∗-multiplier. Proof. The proof is similar to the proof of Theorem 6. ☐
Corollary 6. Let and θ be nonnegative real numbers, and let be a mapping satisfying (24) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies (26), (27) and for all , then the mapping is a quasi-∗-multiplier. Proof. The proof follows from Theorem 7 by taking for all . Choosing , we obtain the desired result. ☐
Similarly, we can obtain the following results.
Theorem 8. Let be a function satisfying (19) and let be a mapping satisfying andfor all and all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies (22), (23) and for all , then the mapping is a quasi-∗-multiplier. Corollary 7. Let and θ be nonnegative real numbers, and let be a mapping satisfying andfor all and all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies (26), (27) and for all , then the mapping is a quasi-∗-multiplier. Proof. The proof follows from Theorem 8 by taking for all . Choosing , we obtain the desired result. ☐
Theorem 9. Let be a function satisfying (28). Let be a mapping satisfying (31) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies (22), (23) and for all , then the mapping is a quasi-∗-multiplier. Corollary 8. Let and θ be nonnegative real numbers, and let be a mapping satisfying (33) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable, such thatfor all . If, in addition, the mapping satisfies (26), (27) and for all , then the mapping is a quasi-∗-multiplier. Proof. The proof follows from Theorem 9 by taking for all . Choosing , we obtain the desired result. ☐
From now on, assume that A is a unital -algebra with unit e and unitary group .
Theorem 10. Let be a function satisfying (19) and let be a mapping satisfying (20) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (21). If, in addition, the mapping satisfies (23), andfor all and all , then the mapping is a quasi-∗-multiplier satisfying for all . Proof. By the same reasoning as in the proof of Theorem 6, there is a unique bi-additive mapping
satisfying (
21), which is
-linear in the first variable, defined by
for all
.
If for all , then we can easily show that for all .
By the same reasoning as in the proof of Theorem 6, for all and all .
Since
B is
-linear in the first variable and each
is a finite linear combination of unitary elements (see [
29]), i.e.,
,
for all
. So by the same reasoning as in the proof of Theorem 6,
is a quasi-∗-multiplier and satisfies
for all
. Thus
is a quasi-∗-multiplier and satisfies
for all
. ☐
Corollary 9. Let and θ be nonnegative real numbers, and let be a mapping satisfying (24) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (25). If, in addition, the mapping satisfies (27), andfor all and all , then the mapping is a quasi-∗-multiplier satisfying for all . Proof. The proof follows from Theorem 10 by taking for all . Choosing , we obtain the desired result. ☐
Theorem 11. Let be a function satisfying (28). Let be a mapping satisfying (20) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (35). If, in addition, the mapping satisfies (37), (23) and for all , then the mapping is a quasi-∗-multiplier satisfying for all . Proof. The proof is similar to the proofs of Theorems 7 and 10. ☐
Corollary 10. Let and θ be nonnegative real numbers, and let be a mapping satisfying (24) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (36). If, in addition, the mapping satisfies (38), (27) and for all , then the mapping is a quasi-∗-multiplier satisfying for all . Proof. The proof follows from Theorem 11 by taking for all . Choosing , we obtain the desired result. ☐
Similarly, we can obtain the following results.
Theorem 12. Let be a function satisfying (19) and let be a mapping satisfying (31) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (32). If, in addition, the mapping satisfies (37), (23) and for all , then the mapping is a quasi-∗-multiplier satisfying for all . Corollary 11. Let and θ be nonnegative real numbers, and let be a mapping satisfying (33) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (34). If, in addition, the mapping satisfies (38), (27) and for all , then the mapping is a quasi-∗-multiplier satisfying for all . Proof. The proof follows from Theorem 12 by taking for all . Choosing , we obtain the desired result. ☐
Theorem 13. Let be a function satisfying (28). Let be a mapping satisfying (31) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (35). If, in addition, the mapping satisfies (37), (23) and for all , then the mapping is a quasi-∗-multiplier satisfying for all . Corollary 12. Let and θ be nonnegative real numbers, and let be a mapping satisfying (33) and for all . Then there exists a unique bi-additive mapping , which is -linear in the first variable and satisfies (36). If, in addition, the mapping satisfies (38), (27) and for all , then the mapping is a quasi-∗-multiplier satisfying for all . Proof. The proof follows from Theorem 13 by taking for all . Choosing , we obtain the desired result. ☐