1. Literature Review
In [
1], Volkmann proved that every function
defined on an abelian group G can be described in the form
, which is the solution of the maximum FE
where
is an additive function. Consequently, in [
2], Toborg showed that Equation (
1) also characterises the absolute value of additive functions when
G be any group. Their’s main theorem is stated, as follows:
Theorem 1 (Toborg [
2]).
Let G be any group (G is an abelian group (Volkmann [1])), then a function fulfills (1) if and only if there exists an additive function , such that for any . We recommend the readers to see [
3,
4], and the references cited therein in order to obtain comprehensive results related to functional Equation (
1), which characterises the additive function’s absolute value.
According to Simon and Volkmann [
5], solutions of the MFE
with the additional assumption that G is an additive abelian group was exhibited in the theorem, which is stated as:
Theorem 2 ([
5], Theorem 2).
Assume that G is an additive abelian group, whose each element is divisible by 6 (divisible by 2 and 3), and then a function fulfillsif and only if there exists an additive function such that or for any . Jarczyck and Volkmann [
6] demonstrated the stability results of MFE (
1) on additive abelian groups. Consequently, Badora et al. [
7] also generalized their results in order to prove the stability for a certain class of groupoids. In [
3], Gilanyi et al. determined the stability results of maximum functional equation
on a square-symmetric groupoid. Consequently, they also examined the stability of maximum equation
on additive abelian groups. Various appropriate and useful results regarding stability can be found in papers by Przebieracz [
8,
9].
This paper is arranged, as follows: in
Section 2, we prove the functional Equation (
1) for any arbitrary group
G without any characterisation of an additive function’s absolute value. Besides, we also demonstrate some consistent and useful results concerning the normal subgroup of
G.
In
Section 3, we analyse the functional Equation (
2) in order to obtain its solution. For this purpose, we drop the additional assumption that
G is an additive abelian group and is divisible by 6. We present the generalization of Theorem 2 by proposing a discretely normed abelian group
G. Moreover, we investigate the functional Equation (
2) for any arbitrary group
G, which satisfies the Kannappan condition [
10].
Section 4 deals with the stability results of the MFE (
1) for a function
, where
G be any group and find some useful connection of stability with commutators and embedding of a group.
3. Solutions of the Functional Equation (2)
Theorem 5. Let G be any arbitrary group, then a function satisfiesif and only if simultaneously orholds and satisfies Proof. Suppose that
satisfies Equation (
9). Assume 1 as neutral element of group
G, then by putting
in (
9), we can obtain that
, then
or
. Let
, then by Equation (
9) we can compute
for any
. Assume that
, then from (
9), It is easy to see that
,
and
for any
. The proof of Equation (
10) consists of the following simple computation:
Conversely, suppose that the Equations (
10) and (
11) are satisfied and
. Subsequently, it can be determined that
Here, we have two cases, in the first case, using Equation (
10) and (
11), we derive the required result, as follows:
From second case, when , then we can also get that . □
Corollary 5. Let G be a group and function satisfies the maximum Equation (9) if and only ifand also . Corollary 6. Let G be a group and function is a solution of Equation (10) satisfies (11) if and only if there exists an additive function , such that for any . Definition 2 ([
11]).
Let G be an abelian group. Subsequently, a function is said to be a discrete norm on G if there exists some such that whenever x is not identity element of G. Afterwards, is called the discretely normed abelian group [11]. Simon and Volkmann [
5] proved Theorem 2 with additional assumption that
G is an additive abelian group and is divisible by 6, but we present the generalization of Theorem 2 by introducing the notion of discretely normed abelian group
G, as follows:
Theorem 6. Assume that be a discretely normed abelian group, then a function fulfills (9) if and only if there exists an additive function , such that for any . Proof. Because
G is a discretely normed abelian group, therefore there exists a discrete norm function
such that
whenever
. Assume that
, then applying the main theorem of Toborg [
2], a function
satisfies
if and only if there exist an additive function
such that
. Subsequently, we have
if and only if
, which implies that
thus, we conclude that
if and only if there exists an additive function
, such that
for any
. □
Corollary 7. Let be a discretely normed abelian group, then a function χ is a solution of Equation (10) satisfying (11) if and only if there exists an additive function , such that for any non-identity element . Proof. From Theorem 6, we concluded that maximum functional Equation (
9) holds; therefore, using Theorem 5, we can also obtain required proof. □
We have well-known theorem presented by Stepr
ns Juris in [
11] about a group
G, which is a discretely normed abelian group. Therefore, we have following corollaries.
Corollary 8. For free abelian group G, a function χ is a solution of Equation (9) if and only if there exists an additive function , such that for any . Corollary 9. Suppose that G is a free abelian group, then χ is a solution of functional Equation (10) satisfying (11) if and only if there exists an additive function , such that for any . Theorem 7. Let G be any group and let a function is a solution of Equation (10) and (11), which is not identically zero, then there exists a normal subgroup of G. Proof. Because the function
satisfying the Equations (
10) and (
11), then by Theorem 5, we can obtain that
For neutral element 1, we can write
, then
. Let
, then
. From Equation (
13), follows immediately that
is even; therefore,
, hence
. Let
, therefore, by property of
,
and
. We can deduce from maximum Equation (
13) that
From inequalities (
14) and (
15), we can calculate
, which implies that
. Therefore,
is a subroup of
G. Additionally, Equation (
13) yields that
is central, therefore
, for every
and
. Hence,
is a normal subgroup of
G. □
Corollary 10. For any group G, let a function χ on group G satisfying (13), which is not identically zero, then also satisfies Proof. Let
be an arbitrary and assume that
, then
. We can deduce from (
13) that
which provides that
. Applying condition (
11), we compute
. Additionally, from (
13), we have
, so we can evaluate
for any
and
.
Because
is a subgroup of
G, then
, so we can see that
for any
and
, therefore
. Writing
x instead of
y and
v instead of
x in (
13), we can conclude
for any
, then writing
instead of
v, we have
. □
Corollary 11. Let G be a group and function satisfies the Kannappan condition. If the maximum functional Equation (13) holds, then for any . Proof. Assume that
satisfies the Equation (
9) and Kannappan condition. Subsequently, from Theorem 5, we can conclude that
or
combining both cases, we can see that
, therefore
. In either case, it follows that
for every
. □