1. Introduction
In 1940, Ulam [
1] inquired about the stability of groups of homomorphisms: “What is an additive mapping in close range to an additive mapping of a group and a metric group?” In the next year, Hyers [
2] responded affirmatively to the above query for more groups, assuming that Banach spaces are the groups. Rassias [
3] extended Hyers’ theorem by accounting for the unbounded Cauchy difference. Gavruta [
4] has demonstrated the stability of Hyers-Ulam-Rassias with its enhanced control function. This stability finding is the stability of Hyers-Ulam-Rassias functional equations. Baker [
5] utilized the Banach fixed point theorem to provide a Hyers-Ulam stability result.
Cădariu and Radu used the fixed point approach to prove the stability of the Cauchy functional equation in 2002. They planned to use the fixed-point alternative theorem [
6] in
-normed spaces to achieve an accurate solution and error estimate. In 2003, this novel method was used in two consecutive publications [
7,
8], to get general stability in Hyers-Ulam in the functional equation of Jensen. The paper [
9] also made the ECIT 2002 lecture possible. Many subsequent works employed the fixed point alternative to get generalized findings in many functional equations in various domains of Hyers-Ulam stability. The reader is given the following books and research articles that describe the progress made in the problem of Ulam over the last 70 years (see, for example [
10,
11,
12,
13,
14,
15,
16]). The functional equations
and
are known as additive functional equation and quadratic functional equation, respectively. Each additive and quadratic solution of a functional equation, in particular, must be an additive mapping and a quadratic mapping. Singh et al. [
17] discussed the asymptotic stability of fractional order differential equations in the framework of Banach spaces.
In [
18], Czerwik showed the stability of the quadratic functional Equation (
2). Skof has been shown for the function
, where
is normed space and
is Banach space (see [
19]), a stability issue in the Hyers-Ulam approach for Equation (
2). Skof’s theorem is still true if an Abelian group replaces the domain
, according to Cholewa [
20].
Grabiec has generalized the above results in [
21]. The quadratic functional equation is useful for distinguishing inner product spaces(for example, see [
22,
23,
24]). The further generalization of Th.M. Rassias’ theorem was provided by Găvruţa [
4]. Several papers and monographs on different generalizations and applications of stability of the Hyers–Ulam–Rassias have also been published over the last three decades for several functional equations and mappings (see [
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35]).
In this work, we introduce a new kind of generalized quadratic-additive functional equation is
where
, and obtain its general solutions. The main objective of this work is to examine the stability of a similar type of Hyers–Ulam theorem for the quadratic-additive functional equation in
-Banach modules on a Banach algebra by utilizing fixed point theory.
Throughout, in this work, we consider
refers either
or
and a real number
with
. We can directly utilize the definition of
-normed space in [
36] to proceed our main results.
Theorem 1 ([
6])
. If a complete generalized metric space is and is a strictly contractive function with the Lipschitz constant ,Then for each given , either or there is a positive integer satisfies- (1)
;
- (2)
the sequence converges to a fixed point of F;
- (3)
is the only one fixed point of F in ;
- (4)
.
3. Main Results
Here, we investigated the stability (in the sense of Hyers-Ulam stability) of (
3) in
-Banach modules by utilizing a fixed point approach for three different cases. Moreover, we can divide this section into three subsections. In
Section 3.1, we get the stability outcomes for odd case; in
Section 3.2, we get the stability outcomes for even case; in
Section 3.3, we examined our main outcomes of the function Equation (
3) for the mixed case.
Before proceed, let us consider is a unital Banach algebra with , , W is a -normed left Banach -module and V is a -normed left -module.
We utilize the below abbreviations for a mapping
:
for all
and
.
3.1. Stability Results: When Is Odd
Theorem 5. Let a mapping such that Let be an odd mapping such thatand . If there is (L is a Lipschitz constant) satisfiesandfor all , then there exists a unique additive mapping satisfies Moreover, if is continuous in for every , then is -linear, i.e., for all and all .
Proof. Letting
, and
and the remaining
in (
15), we get
Consider the set
and define the generalized metric on
as below:
Easily, we can verify that
is a complete generalized metric space (see [
20]).
Next, we define a function
by
Let
and an arbitrary constant
with
. Utilizing the definition of
d, we obtain
for all
. By the given hypothesis and the last inequality, one has
for all
. Hence,
From inequality (
18), we get
From Theorem 1,
F has an unique fixed point
in
satisfies
and
. Also, using (
23), we get
Hence, inequality (
17) valid for all
.
Now, we want to prove that the function
is additive. Using the inequalities (
14), (
15) and (
24), we obtain
that is,
for all
. Therefore, by Theorem 2, the function
is odd.
Finally, we have to show that the function
is unique. Let us consider that there exists an odd mapping
satisfies (
17). Since
and
is additive, we get
and
for all
, i.e.,
is a fixed point of
F in
. Clearly,
.
Moreover, if
is continuous in
for every
, then using the proof of [
3],
is
-linear.
Switching
and
in (
15), we get
for all
and all
. Thus, using definition of
and the inequalities (
14) and (
26), we get
for all
and all
. So,
for all
and all
. Since
is additive, we get
for all
and all
.
Since is -linear, let . Then for all and .
Hence, is -linear. □
Corollary 1. If an odd function such thatand , then there exists a unique additive mapping satisfieswhere , . Moreover, if is continuous in for all , then is -linear. Proof. By putting
and
in Theorem 5, we obtain our needed result. □
Corollary 2. Let such that and , and let be an odd mapping such thatand , then there exists a unique additive mapping satisfiesfor all . Moreover, if is continuous in for all , then is -linear. Proof. By letting
and
in Theorem 5, we obtain our needed result. □
Theorem 6. Let a mapping such thatfor all . Let be an odd mapping satisfies (15). If there is such thatandfor all , then there exists a unique additive mapping satisfies Moreover, if is continuous in for all , then is -linear.
Proof. Letting
and
and the remaining
in (
15), we get
for all
. Interchanging
v with
in (
32), we have
for all
. Assume the set
and define the generalized metric on
as below:
Easily, we can verify that
is a complete generalized metric space (see [
20]).
Next, we can define a function
by
Let and an arbitrary constant with .
Using the definition of
d, we obtain
for all
. By the given hypothesis and the last inequality, one has
for all
. Hence,
From inequality (
33), we get
From Theorem 1,
F has an unique fixed point
in
such that
and
. Also,
Hence, the inequality (
31) valid for all
.
Again, we want to show that the function
is additive. Using the inequalities (
29), (
15) and (
38), we obtain
for all
. Therefore, by Theorem 2, the function
is odd.
Finally, we have to show that the function
is unique. Let us consider that there exists an odd mapping
satisfies (
31). Since
and
is additive, we have
and
for all
, i.e.,
is a fixed point of
F in
. Clearly,
.
Moreover, if
is continuous in
for all
, then using the proof of [
3],
is
-linear.
Replacing
and the remaining
in (
15), we get
for all
and all
. Thus, using definition of
, the inequalities (
29) and (
40), we get
for all
and all
. So,
for all
and all
. Since
is additive, we get
for all
and all
.
Since
is
-linear, let
.
Hence, is -linear. □
Corollary 3. If is an odd mapping such thatand , then there exists a unique additive mapping satisfiesfor all , where and . Moreover, if is continuous in for all , then is -linear. Proof. By letting
and
in Theorem 6, we obtain our needed outcome. □
Corollary 4. If is an odd mapping such thatfor all and . Then there exists unique additive mapping satisfiesfor all , where and with . Moreover, if is continuous in for all , then is -linear. Proof. By taking
and
in Theorem 6, we obtain our needed outcome. □
3.2. Stability Results: When Is Even
Theorem 7. Let a mapping such thatfor all . Let be an even mapping with such that (15). If there is such thatandfor all , then there exists a unique quadratic mapping satisfiesfor all . Moreover, if is continuous in for all , then is -quadratic, i.e., for all and all . Proof. Letting
and
and the remaining
in (
15), we get
Consider the set
and define the generalized metric on
as below:
Clearly,
is a complete generalized metric space (see [
20]).
We can define a function
by
Let and an arbitrary constant with .
Using the definition of
d, we obtain
for all
. By the given hypothesis and the last inequality, one has
for all
. Hence,
By using the inequality (
46) that
Thus, by Theorem 1,
F has a unique fixed point
in
satisfies
and
for all
. Also,
Thus, inequality (
45) holds for all
.
Now, we show that
is quadratic. By (
43), (
15) and (
51), we have
that is,
for all
. Therefore, by Theorem 3, the function
is even. Next, we want to prove that the function
is unique. Consider there exists an another quadratic mapping
satisfies the inequality (
45). Then,
and
is quadratic, which gives
and
for all
, i.e., is a fixed point of F in . Hence, .
Moreover, if
is continuous in
for every
, then using the proof of [
3],
is
-quadratic.
Replacing
and the remaining
in (
15), we get
for every
and all
. Using definition of
, (
43) and (
53), we have
for all
and all
. So,
for all
and all
. Since
is quadratic, we get
for all
and all . Since is -quadratic, let , then for all and all . Hence, is -quadratic. □
Corollary 5. Let be an even function with such thatfor every and , then there is only one quadratic function fulfilswhere , . Moreover, if is continuous in for all , then is -quadratic. Proof. By letting
and
in Theorem 7, we obtain our needed result. □
Corollary 6. Let such that and , and let an even mapping and such thatfor all and , then there exists a unique quadratic mapping satisfiesfor all . Moreover, if is continuous in for all fixed , then is -quadratic. Proof. By letting
and
in Theorem 7, we obtain our needed result. □
Theorem 8. Let be a function such thatfor all . Let be an even function with such that (15). If there is satisfiesandfor all , then there exists a unique quadratic mapping satisfiesMoreover, if is continuous in for all , then is -quadratic. Proof. Letting
and
and the remaining
in (
15), we get
for all
. Switching
v by
in (
59), we have
for all
. Consider the set
and define the generalized metric on
as below:
Clearly,
is a complete generalized metric space (see [
20]). Now, we define a function
by
for all
and all
. Let
and an arbitrary constant
with
.
Using the definition of
d, we get
for all
. By the given hypothesis and the last inequality, one has
for all
. Hence,
By utilizing inequality (
60) that
Thus, by Theorem 1,
F has a only one fixed point
in
satisfies
and
. Also,
Thus, the inequality (
58) holds for all
.
Now, we show that
is quadratic. By (
56), (
15) and (
65), we have
Therefore, by Theorem 3, the function
is even. Next, we want to prove that the function
is unique. Consider there is a quadratic function
which fulfils the inequality (
58). Then,
and
is quadratic, which gives
and
for every
, i.e.,
is a fixed point of
F in
. Hence,
.
Moreover, if
is continuous in
for all
, then using the proof of [
3],
is
-quadratic. Interchanging
with
in (
15), we get
for all
and all
. Using definition of
, (
56) and (
67), we have
for all
and all
. So,
for all
and all
. Since
is quadratic, we get
for all
and all
. Since
is
-quadratic, let
,
and all
. Hence,
is
-quadratic. □
Corollary 7. Let be an even function with such thatand , then there exists a unique quadratic mapping satisfieswhere and . Moreover, if is continuous in for all , then is -quadratic. Proof. By letting
and
in Theorem 8, we achieve our needed result. □
Corollary 8. Let be an even function with such thatfor all and , then there exists a unique quadratic mapping satisfieswhere such that and . Moreover, if is continuous in for all , then is -quadratic. Proof. By putting
and
in Theorem 8, we obtain our needed outcome. □
3.3. Stability Results for the Mixed Case
Theorem 9. Let a mapping such thatfor all . If a mapping and such that (15). If there exists a constant satisfiesfor all , then there exists a unique additive mapping and a unique quadratic mapping satisfiesfor all . Moreover, if is continuous in for all , then is -linearand is -quadratic. Proof. If we divide the function
into two parts such as even and odd by letting
for
, then
. Let
then by (
70), (
71) and (
72), we have
Hence, by Theorem 5 and 7, there exists a unique additive mapping
and a unique quadratic mapping
satisfies
and
for all
. Therefore,
for all
. □
Corollary 9. Let be a function with such thatand every , then there exists a unique additive mapping and a unique quadratic mapping satisfiesfor all , where and . Moreover, if is continuous in for all , then is -linear and is -quadratic. Corollary 10. Let be a function with such thatfor all and , then there exists a unique additive mapping and a unique quadratic mapping satisfiesfor all , where and . Moreover, if is continuous in for all , then is -quadratic and is -linear. Theorem 10. Let a mapping such thatfor all . If a mapping with such that (15). If there is a constant such thatfor all , then there exists a unique additive mapping and a unique quadratic mapping satisfiesfor all . Moreover, if is continuous in for all , then is -quadratic and is -linear. Corollary 11. If is a function with such thatfor every and , then there exists a unique additive mapping and a unique quadratic mapping satisfiesfor every , where and . Moreover, if is continuous in for all , then is -quadratic and is -linear. Corollary 12. If is a function with such thatand , then there exists a unique additive mapping and a unique quadratic mapping satisfiesfor all , where and . Moreover, if is continuous in for all , then is -quadratic and is -linear. Remark 1. If an even mapping satisfies the functional Equation (3), then the below assertions holds: - (1)
and .
- (2)
if the function φ is continuous.
Example 1. Let an even mapping defined by: wherethen the mapping satisfiesfor all , but doesn’t exist a quadratic mapping satisfieswhere λ and δ is a constant. Remark 2. If an odd mapping satisfies the functional Equation (3), then the below assertions holds: - (1)
and .
- (2)
if the function φ is continuous.
Example 2. Let an odd mapping defined by: wherethen the mapping satisfiesfor all , but doesn’t exist a additive mapping satisfieswhere λ and δ is a constant. 4. Conclusions
As of our knowledge, our findings in this study are novel in the field of stability theory. This is our antecedent endeavor to deal with a new type of mixed QA-functional equation. It is shown that the Equation (
3) is equivalent to each other to conclude that their solution is both additive and quadratic mapping. The stability results of different forms of additive and quadratic functional equations are obtained by many mathematicians in various spaces. But, in this work, we have introduced mixed QA-functional Equation (
3) and obtained its general solution in
Section 2. The main aim of this work is to examine the Hyers-Ulam stability of (
3), which has been achieved in
Section 3.3 with the help of
Section 3.1, where the function
is odd; and
Section 3.2, where the function
is even, in
-Banach modules by using fixed point approach. By the Corollaries, we have discussed Hyers-Ulam stability for the factors of
sum of norms and
sum of the product of norms.