1. Introduction
The stability of functional equations is a fundamental concept in mathematics with far-reaching applications. It was first proposed by Ulam in 1940 [
1] who posed the essential question: under what conditions does an approximate solution to a functional equation imply the existence of a nearby exact solution? This idea was further developed by Hyers [
2] in 1941 within the framework of Banach spaces, marking a major milestone in the study of functional equation stability. In 1978, Rassias [
3] extended this theory by introducing a generalized version of Hyers–Ulam stability, incorporating a control function of the form
, where
and
, particularly for additive mappings. Later, in 1994, Gǎvruta [
4] refined this approach by replacing Rassias’ control function with a more general function
, further broadening the scope of stability results. More researchers have been studying Hyers–Ulam stability in various structures, for example, non-Abelian semigroups [
5,
6], Banach algebras [
7], commutative normed algebras [
8], and
-algebras [
9,
10,
11]. In addition, many researchers have investigated Ulam–Hyers–Rassias stability via fixed point theory (see [
12,
13,
14,
15]).
Hyers–Ulam stability refers to a property of equations, where small changes in the equation result in only minor changes in its solutions. In 1993, Obloza [
16] extended Hyers–Ulam stability to linear differential equations. In 2003, Miura et al. [
17] studied this stability for first-order and higher-order linear constant coefficient differential equations. Following that, researchers examined Hyers–Ulam stability for linear operators on Hilbert spaces [
18] and differential operators on weighted Hardy spaces [
19].
The concept of hyperstability was first introduced in [
20], with an initial focus on specific ring homomorphisms. A functional equation is said to be hyperstable if every approximate solution is, in fact, an exact solution. This notion marks a notable development in the theory of functional equation stability, highlighting cases where stability is so strong that any deviation from the equation necessarily implies the function already satisfies it exactly. Extensive research has since been conducted in this area; for further details, see [
21,
22,
23].
In the 19th century, Cayley made pioneering contributions by introducing 3-ary operations through cubic matrices. This innovation paved the way for the development of
n-ary algebraic structures across various fields. Notably, the exploration of
n-ary algebras, with a specific focus on ternary algebraic structures, gained significant attention. In 2008, Amyari and Moslehian [
24] introduced the concept of ternary algebras, defining a ternary algebra
as a complex space equipped with a ternary product
from
into
. This product is required to be
-linear in the outer variables, conjugate
-linear in the middle variable, and associative. Additionally, it must satisfy certain norm properties, such as
and
. When a ternary algebra
is also a Banach space, it is referred to as a ternary Banach algebra. These extended operations satisfy key properties like associativity and linearity, enabling more complex algebraic structures. Both ternary and
n-ary algebras have applications in coding theory, computer science, physics, quantum mechanics, and Nambu mechanics. For more details, see [
25,
26,
27].
In the following, we solve an example of the ternary structure property.
Example 1 (Matrix Algebra with Ternary Product)
. Let be the algebra of all complex matrices, equipped with the norm (e.g., the Frobenius norm). Define a ternary product on byfor all . This defines a ternary Banach algebra, since the product is linear in each variable, and is complete. Now, define a linear mapping byfor some fixed scalar . Then, for all , we compute Consequently,showing that T is a ternary derivation. Next, consider the three-dimensional additive-type functional equation Suppose a mapping satisfies the inequalityfor some small and all . Definewhere is a perturbation satisfying . Then,and thus, Therefore, f is an approximate solution of the 3D additive-type functional equation.
In 2015, Park [
28] introduced additive
-functional inequalities and provided a comprehensive analysis of the Hyers–Ulam stability of these inequalities in both Banach spaces and non-Archimedean Banach spaces. Subsequently, leveraging the concept of additive-type functional equations, we establish a 3D additive-type functional equation between normed spaces. In this context, let
and
denote normed spaces. The mapping
from
to
is called a 3D additive-type functional equation, if
satisfies
In 2012, Eshaghi Gordji et al. [
29], defined a 3-Lie multiplier on a 3-Lie Banach algebra. In this sequel, by paying attention to the concept of ternary homomorphisms and ternary multipliers, we investigate a ternary hom-multiplier between ternary Banach algebras. For this work, let
be a ternary Banach algebra, and let the mappings
and
from
to
be called a ternary homomorphism and a ternary multiplier, respectively. Assume
and
satisfy
, and
for all
Let mappings
and
from
to
be a ternary homomorphism and a ternary multiplier, respectively. Then,
for all
.
As noted above, the study of higher-order algebraic structures such as ternary (or more generally, n-ary) algebras has gained attention due to their natural appearance in various branches of physics and abstract algebra, particularly in modeling systems with non-binary interactions. While binary (or double) hom-multipliers have been widely studied and serve as foundational tools in functional equations and homomorphism theory, they are often insufficient for capturing the full complexity of interactions inherent in ternary systems.
This motivates the aim to extend classical notions like the hom-multiplier and Jordan hom-multiplier into the ternary setting. A ternary hom-multiplier is designed to capture the behavior of mappings that preserve ternary operations under specific structural constraints. Unlike their binary counterparts, these mappings interact with a three-variable product, which is characteristic of ternary Banach algebras.
To illustrate the distinction, a double hom-multiplier deals with structures where associativity or compatibility is considered between two elements, whereas a ternary hom-multiplier extends this compatibility to ternaries of elements, making them more suitable for analyzing systems governed by ternary operations. Similarly, the ternary Jordan hom-multiplier extends the symmetric structure of the Jordan product to a ternary context, allowing for broader applications in areas where symmetry in ternary operations is central.
These extensions are not merely formal generalizations—they provide new insights and tools for studying stability phenomena in ternary functional equations, as well as modeling physical systems where ternary interactions are fundamental (e.g., in quark models in quantum chromodynamics or in ternary logic systems).
In the following section, we define a ternary hom-ternary multiplier (briefly, ternary hom-multiplier) and a ternary Jordan hom–Jordan multiplier (briefly, (ternary Jordan hom-multiplier) between ternary Banach algebras.
Definition 1. Let a mapping γ from to be a ternary homomorphism. A -linear mapping θ from to is called a ternary hom-multiplier if θ satisfiesfor all . Definition 2. Let a mapping γ from to be a ternary homomorphism. A -linear mapping θ from to is called a ternary Jordan hom-multiplier if θ satisfiesfor all . Theorem 1 ([
30])
. Consider a complete generalized metric space equipped with a strictly contractive mapping characterized by a Lipschitz constant . For any element , we examine the behavior of the sequence , where i ranges over non-negative integers. We observe that this sequence either diverges to infinity for all non-negative integers i, or there exists a positive integer such that- (1)
for all ;
- (2)
The sequence converges to a fixed point of F;
- (3)
is the unique fixed point of F within the set ;
- (4)
for all .
In
Section 2, we delve into the intricate details of the 3D additive-type functional equation and establish its properties as an additive mapping. Subsequently, we embark on an investigation into its
-linearity for each
. Finally, employing a fixed-point method, we rigorously demonstrate the Hyers–Ulam stability and hyperstability of the 3D additive-type functional equation within the realm of ternary Banach algebras.
In
Section 3, by using Theorem 1, we prove that both the ternary hom-multiplier and ternary Jordan hom-multiplier can be stable and hyperstable associated with the 3D additive-type functional equation under the control functions of Gǎvruta and Rassias. For this reason, we will mix the idea of control functions with the bounded/unbounded Cauchy difference.
2. Results of 3D Additive-Type Functional Equation
In this section, let be a member of the set , let and be two normed spaces, and let be a ternary Banach algebra. In the following lemma, we solve the 3D additive-type functional equation and investigate when it is an additive mapping. Please note that X, Y, and are Banach spaces endowed with norms (denoted by ); the context will clarify which space is meant.
Lemma 1. If a mapping ζ from to satisfies (
1)
, then ζ is an additive mapping. Proof. Suppose the mapping
satisfies Equation (
1). If
in (
1), then
. Letting
in (
1), we get
In the subsequent step, by using (
2) and letting
and
in (
1), we have
Finally, letting
once again and using Equations (
2) and (
3) in Equation (
1), we get
□
In the subsequent lemma, we introduce the element
into the 3D additive-type functional equation. By doing so, we proceed to confirm that the mapping
, is a
-linear mapping. This implies that
satisfies the lemma above and is homogeneous over the complex numbers (i.e., for any complex number
and any element
,
Lemma 2. If a mapping ζ from to satisfies the equationfor all and , then ζ is a -linear mapping. Proof. By utilizing Lemma 1, it follows that
is an additive mapping. Setting
in (
4), we obtain the following
By applying ([
31], Lemma 1), or ([
32], Corollary 2.2), we obtain that the mapping
J is linear with respect to complex numbers. □
Remark 1. Actually, it follows from ([31], Lemma 1), or ([32], Corollary 2.2) that it is enough to assume in Lemma 2 that α belongs to some "big" subsets of . In the upcoming theorem, we establish that the 3D additive-type functional equation can achieve stability by employing Gǎvruta’s control function.
Theorem 2. Let a function satisfy the inequalityfor some . If satisfiesthere exists a unique additive mapping such thatfor all . Proof. Using (
6) and taking
, we have
. Additionally from (
6) we get
Setting
and
in (
7), we obtain the relation
for all
.
Let
be the set of all mappings
with
. Define
d on
as
where, as usual,
. It is clear that
constitutes a generalized metric space (see [
33]). Next, we investigate the linear mappings
satisfying
for all
.
If
and
g are elements of
such that
, then
. Since
for each
and
, then
for all
. It follows from (
6) that
for all
. So,
. According to Theorem 1, there is a mapping
that satisfies the following.
- (1)
F is a fixed point of
, i.e.,
The mapping
F serves as the unique fixed point of
in the set
This implies that
F is the unique mapping that satisfies (
9) such that there exists a
satisfying
- (2)
as
. This implies the equality
- (3)
, which implies
Finally, it follows from (
7) that
for all
. Thus,
for all
According to Lemma 2, the mapping
F is an additive mapping. □
Considering Theorem 2, we delve into the investigation of the stability of the 3D additive-type functional equation on a ternary Banach algebra in the subsequent corollary, exploring stability using the control function proposed by Rassias.
Corollary 1. Let and κ be non-negative real numbers. If any mapping satisfiesthen there is a unique linear mapping such that Proof. Let
and
be non-negative real numbers. Consider Theorem 2 with
and
. Checking (
6), we have
By the assumption in (
10), we have that (
7) holds; so, the conditions of Theorem 2 are met. Therefore, the conclusion of Theorem 2 holds, which is the stated conclusion of this corollary. □
In the forthcoming theorem, we explore the hyperstability of the 3D additive-type functional equation on a ternary Banach algebra, focusing on Gǎvruta’s control function for a comprehensive analysis.
Theorem 3. Let a function satisfy Additionally, if the mapping satisfiesfor all , then ζ is an additive mapping. Proof. If
, and
in (
12), then
, and by using induction on
, we get
Hence,
for all
and
. Therefore, as
in (
13) and using (
11), we get
for all
. Therefore,
is an additive mapping. □
Referring to Theorem 3, in the subsequent corollary we investigate hyperstability concerning the 3D additive-type functional equation, focusing on the utilization of Rassias’ control function for a comprehensive analysis.
Corollary 2. Let and κ be non-negative real numbers. If a mapping satisfiesfor all , then ζ is a linear mapping. 3. Results of Ternary Hom-Multipliers
In the forthcoming theorem, we rigorously establish the stability of a ternary hom-multiplier on a ternary Banach algebra. Our proof methodology involves leveraging the fixed point method while carefully considering the insights provided by Theorem 2.
Theorem 4. Let a function η satisfyfor all . Assume that mappings ζ and h satisfy (
7)
and the inequalitiesand Then, there exist a unique ternary homomorphism and a unique ternary hom-multiplier such thatand Proof. Utilizing a methodology akin to the one elucidated in the proof of Theorem 2, we can define mappings
and
in the manner of
It follows from (
15), (
16), and (
19) that
So, the mapping is a ternary homomorphism.
In the following proof, by using relations (
15) and (
17)–(
19), we get
and
Consequently, the mapping is a ternary hom-multiplier. □
In the next corollary, we investigate whether the ternary hom-multiplier can be stable with the control function of Rassias.
Corollary 3. Consider non-negative real numbers κ and r such that . If mappings satisfy in (
10)
andandthen there exist a unique ternary homomorphism and unique ternary hom-multiplier such thatand Proof. For this, it is enough to put and instead of and into the proof of Theorem 4, respectively. □
The subsequent theorem and corollary delve into an in-depth exploration of the hyperstability exhibited by the ternary hom-multiplier on a ternary Banach algebra. By using two control functions Gǎvruta and Rassias, we investigate the hyperstability properties of the ternary hom-multiplier using the fixed point method.
Theorem 5. Assume that function η satisfies (
11)
. If mappings ζ and h from to satisfy (
12)
and the inequalitiesandthen the mappings γ and θ from to are a ternary homomorphism and a ternary hom-multiplier, respectively. Proof. Similar to the proof of Theorem 3 and employing the method of mathematical induction on
, we obtain
for all
. The proof continues as in the proof of Theorem 4. □
Corollary 4. Consider non-negative real numbers κ and r such that . If mappings satisfy (
14)
and the inequalitiesandthen the mappings h and ζ are a ternary homomorphism and a ternary hom-multiplier, respectively. In the concluding phase of our study, through the fixed point method, we delve into the exploration of the Hyers–Ulam stability and hyperstability of the ternary Jordan hom-multiplier with two control functions introduced by Gǎvruta and Rassias.
Theorem 6. Let a function η satisfy (
15)
. If mappings ζ and h satisfy (
7)
and the inequalitiesandthen there exist a unique ternary homomorphism and unique ternary Jordan hom-multiplier such thatand Proof. This is similar to the proof that is employed for Theorem 4. □
Corollary 5. Consider non-negative real numbers κ and r such that . If mappings satisfy in (
10)
andandthen there exist a unique ternary homomorphism and unique ternary Jordan hom-multiplier such thatand Remark 2. Let a function η satisfy (
11)
. If the mappings ζ and h satisfy (
12)
and the inequalitiesandthen the mappings γ and θ are a ternary homomorphism and ternary Jordan hom-multiplier, respectively. Remark 3. Let and κ be non-negative real numbers. If mappings satisfy (
14)
andandfor all , then the mappings h and ζ are a ternary homomorphism and ternary Jordan hom-multiplier, respectively.