1. Introduction
In the past few decades, the Hyers–Ulam stability analysis of functional equations has attracted many researchers, and a number of research articles (of good quality) can be found in the topic; for instance, see [
1,
2,
3,
4,
5,
6,
7,
8,
9] and the references cited therein. Being an emerging field for researchers, various methods (that is the direct method, the fixed point method, and so on) have been developed and applied to a variety of functional equations [
10,
11,
12,
13].
Generally, proving the stability results of functional equations by using the direct method requires one of the following two conditions:
or
. Depending on certain assumptions, we need different distinctions to choose one approximate approach to solve some particular problems. The author in [
14] studied the stability of a functional equation involving a single variable while imposing various conditions on the underlying constants. The results were then used to prove the stability of a pair of functional equations in several variables by using a direct technique. This work not only generalized and shortened the known available methods, but also improved the approximating constants to a great extent. Before studying the idea of [
14], Forti in [
15] presented a direct method for proving the Hyers–Ulam stability of functional equations. The latter literature is considered as a special case of the former article. In fact, these two research articles provided a solid background for the subject matter and opened different doors of research related to the field. These research papers can be used for obtaining stability results related to orthogonal additivity [
10], the functional equation of Drygas [
16], and the functional equation with the quadratic property [
17] without repeating the same procedure.
It is important to mention that the proposed problem arises in various applications, and the same functional equation may be extended to a class of functions involving several variables. However, the available approaches cannot be applied directly to these functional equations, and thus, modifications of these methods are necessary. Keeping in view the importance of the problem, in
Section 2, the authors try to solve another class of functional equations involving a single variable. We extend this approach, and various functional inequalities are constructed, which definitely helped us perform the stability analysis of a class of functional equations involving several variables.
The main theme of the present work is to analyze the stability of the solution of the functional equation form:
where
. The solution of the problem was checked for preserving the addition operation, and we show that the solution satisfies a few identities from which the optimum conditions can be obtained. Based on our main results, the authors solved a number of functional equations, and to the best of our knowledge, these problems are novel and not previously attempted by researchers. The authors have great concern about solving more general forms of the inequalities in the near future. For the interest of the readers, the authors provide a problem in the last paragraph of
Section 3.
3. Applications
An F-space is called
-homogeneous if
for all
and all
(see the definition in [
18]). In this section of Theorem 3,
are considered as positive real numbers with
and
Furthermore,
X is supposed as a
-homogeneous F-space, while
Y is assumed as a
-homogeneous F-space. In fact, there was also a similar solution of the functional inequality in the Banach space in [
14]. The following functional inequality was originally derived from the inner product space [
19,
20].
Theorem 3. Let be a function such that, for some and :for all . Then, there exists a unique mapping such that:for all . Moreover, is the unique solution of the following equation satisfying the above equality:for all . Proof. From
in
, we have:
and then:
From
in
, we have:
Combining
and
, we obtain:
Let
. We apply the results obtained in
Section 2 with
and
. We can easily compute that either
or
Thus:
Therefore, the series is convergent for all
Related to the above results, there exists a uniquely determined mapping satisfying (2) and (3).
In order to show that
g satisfies the last equation of Theorem 3, it is easy to see that:
which tends to zero as
by the convergence series
, and this completes the proof. □
In Theorem 3, the problem can also be derived in the Banach space with
and
. For the Euler–Lagrange equation, we provide another approach to prove its stability in comparison to that from [
11] in completing the stability of the title functional equation.
Theorem 4. Assume that is a group and is a Banach space. Suppose that the mapping such that for all and some : Then, there exists a uniquely determined function such that:and: In particular, if X is commutative, then g is also the solution of the following equality:for all Proof. We show that
in (16) and obtain:
From
in (16), we have:
Consequently, (17) and (18) yield that:
Without loss of generality, we may assume that
Let
. We apply the results obtained in
Section 2 with
, and
. We can easily compute that either
or
Thus:
Therefore, the series is convergent for all
It remains to interpret that if
X is commutative, then for all
and
we have by (2):
which tends to zero as
by the convergence series
and the relation between
and
. According to the commutativeness of
X, we prove that the last equation of Theorem 4 holds for all
. This completes the proof. □
In fact, the stability problem for the equation can also be solved in the β—homogeneous space and the more direct method derived in [
11]. The Drygas equation has been solved by many papers in [
16], and also, an interesting property of the equation is that it is symmetric to
y. Applying the property, we solve the following stability problem.
Theorem 5. Assume that is a group and is a Banach space. Suppose that the mapping such that for all and some : Then, there exists a unique mapping such that:and: In particular, if X is commutative, then g also satisfies: Proof. Without loss of generality, we may assume that
. Substitute in the sequel
in (19) in order to obtain:
From
in
, we have:
Consequently, (21) and (22) yield that:
From
in (19), we have:
Combined with the inequality (21), we have:
and this inequality together with (23) give:
The rest of the proof is the same as in the previous Theorem 3 if and . □
Theorem 6. Assume that is a unique two-divisible group and is a Banach space. Suppose that the mapping satisfies the condition:where d is an integer with and ε is a nonnegative constant. Then, there exists a uniquely determined function satisfying (2) and such that: In particular, if X is commutative, then g satisfies: Proof. Observe first that without loss of generality, we may assume that
. Currently, substitute
and
in the place of
in (27), in order to have:
From the above two inequalities, this leads to:
and this inequality together with the one with
x changed for
can obtain:
By virtue of the unique two-divisibility of
X, we have:
Substitute
in (27) in order to know:
The above equation multiplied by
d together with the above equation yield that:
We exchange
for
x in
:
From
in (27), we have:
and by virtue of (29), (30), and (31), we can obtain:
Let
. We apply the results obtained in
Section 2 with
and
. We can easily compute that either
or
Thus:
Therefore, the series is convergent for all
It remains to show that if
X is commutative, then
g satisfies the last equation of Theorem 6. By (27) applied for
and
, for all
and
, we can obtain:
Letting n tend to infinity, by the commutativity of X and the convergence sequences and , we prove the last equation of Theorem 6 holds true. □
In Equation
, the stability for the problem can be derived in [
14,
21]. In particular,
in Theorem 6 can be presented in the following as a corollary.
Corollary 2. Assume that is a unique two-divisible group and is a Banach space. Suppose that the mapping satisfies the condition:where . Then, there exists a uniquely determined function such that: In particular, if X is commutative, then g satisfies: Equation was treated, as well as the solutions of it were given in [14]. Theorem 7. Assume that is a group uniquely divisible by two and by three, and let be a Banach space. Given an , assume that satisfies for all the condition: Then, there exists a uniquely determined function satisfying (2) and such that: In particular, if X is commutative, then g satisfies:for all . Proof. Substitute
,
,
in the place of
, respectively, in (33) in order to obtain:
We do not repeat the calculation procedures, which are similar to the proof in Theorem 3. It remains to show that if
X is commutative, then
g satisfies the last equation. By (33) applied for
and
, for all
and
, we can obtain:
Letting n tend to infinity, by the commutativity of X and the convergence sequences and , we prove the last equation holds true. □
In the future, the functional inequality:
for
and some
with
will be further explored. In fact, the more general forms:
may be discussed to solve the stability problem for functional equations in several variables. Finally, a useful example of the quartic functional equation is stated:
for all
Substitute
,
,
in the above equation to obtain:
an obvious fact that eliminates
combining
and
and also later eliminates
. Consequently, we can solve the functional inequality:
for some integer
and
by using Theorem 1. In parallel with this method, we eliminate
together with
and
, and replacing
by
x, we obtain functional inequality:
for some integer
and
and solving this by the method in [
14]. We listed it as follows:
1. In Theorem 3, we achieve that the approximating constant is
for
. However, the approximating constant was
in [
1,
17] and
in [
14];
2. In Theorem 4, the approximating constant is more than the approximating constant obtained in [
11];
3. In Theorem 5, we achieve that the approximating constant is
. However, the approximating constant was
in [
1,
16,
22], and also, there was
in [
14];
4. In Theorem 6, we achieve that the approximating constant is
, and there was
in [
14]. Furthermore, the more concrete approximating constant were presented as
in [
23],
in [
14], as well as
in Corollary 2 in our work;
5. The approximating constant is
in [
14] and
in Theorem 7 in the literature and
in [
13,
24].
To summarize, comparing these five application results for the approximating constant, it is hard to say which method achieves the best approximating constant in the theory of functional equations. Now, this may also be considered as an open problem for this research field.