Communications on Applied Nonlinear Analysis 

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A Hyperstability Approach to Generalized Functional Equations on 

Involutive Semigroups 

Ismaail Essalih*, Nordine Bounader, and Ahmed Akkaoui 

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kénitra, Morocco. 

E-mail address: essalih.ismaail@gmail.com, n.bounader@live.fr, ahmed.maths78@gmail.com 

* Corresponding author 

 

Article History: 

Received:04/09/2025 

Revised:03/10/2025 

Accepted:27/11/2025 

Abstract:  

In this paper, we examine the hyperstability of a general functional equation involving 

involutions on semigroups, where 𝑓 ∶  𝑆2  →  𝑋 and (𝑆,·) is an arbitrary semigroup 

equipped with involutive mappings 𝜎 and 𝜏 . The functional equation studied in this 

paper emerges naturally from the theory of additive and quadratic equations, 

particularly as a hybrid form combining structural properties of both. Such equations 

typically appear when analyzing mappings that preserve symmetrical or involutive 

relationships within algebraic structures, especially in semigroups endowed with 

additional symmetries. By employing a technique inspired by Maksa and Páles, we 

derive sufficient asymptotic conditions ensuring hyperstability. Additionally, we 

extend the result to an inhomogeneous variant incorporating a perturbation term 𝐹 ∶

 𝑆4  →  𝑋. These findings contribute to the broader understanding of stability 

phenomena in functional equations on algebraic structures. 

Keywords: semigroups, involutions, stability, hyperstability, functional equations. 

Mathematics Subject Classification: 39B62, 39B82, 41A60, 46B06 

 

1. Introduction 

One important aspect of functional equations is their stability. In mathematics, the stability of an 

equation refers to the sensitivity of its solutions to small perturbations in the equation itself. In the 

context of functional equations, stability refers to the extent to which the solutions of an equation 

remain close to each other under small changes to the equation. Stability is an essential concept in 

functional analysis, as it can help ensure the existence and uniqueness of solutions to functional 

equations. For example, if a functional equation is stable, it may be possible to use fixed-point 

theorems to show that the equation has a unique solution. The study of the stability of functional 

equations is an active area of research in mathematics, with many open problems and challenges. 

Researchers are interested in understanding the stability properties of different classes of functional 

equations, as well as developing new techniques and methods for analyzing the stability of these 

equations. 

The idea of studying the stability of functional equations emerged in 1940 when the mathematician 

S. M. Ulam put together a list of unsolved problems that included the following problem: 

 

 

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Communications on Applied Nonlinear Analysis 

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Ulam’s problem: [17]   

Let (𝐺1,∗1) be a group and let (𝐺2,∗2) be a metric group with a metric 𝑑(.,.). 𝐺𝑖𝑣𝑒𝑛휀 > 0, does there 

exists a 𝛿 > 0 such that if a mapping ℎ: 𝐺1 → 𝐺2 satisfies the inequality 

𝑑(ℎ(𝑥 ∗1 𝑦), ℎ(𝑥) ∗2 ℎ(𝑦)) < 휀 

for all 𝑥, 𝑦 ∈ 𝐺1, then there exists a homomorphism 𝐻: 𝐺1 → 𝐺2 with 

𝑑(ℎ(𝑥), 𝐻(𝑥)) < 𝛿 

for all 𝑥 ∈ 𝐺1 ? 

This question aroused the attention of many mathematicians. Almost a year later, D. H. Hayes 

published a paper in which he presented an answer to Ulam's problem in the case where 𝐺1 and 𝐺2 

are Banach spaces to investigate the stability of Cauchy functional equation. 

Theorem 1.1. [14] ] Let 𝐸1 and 𝐸2 be two Banach spaces and 𝑓: 𝐸1 → 𝐸2 be a function such that 

‖𝑓(𝑥 + 𝑦) − 𝑓(𝑥) − 𝑓(𝑦)‖ ≤ 𝛿 

for some 𝛿 > 0 and for all 𝑥, 𝑦 ∈ 𝐸1. Then the limit 

𝐴(𝑥): = lim
𝑛→∞

 2−𝑛𝑓(2𝑛𝑥) 

exists for each 𝑥 ∈ 𝐸1, and 𝐴: 𝐸1 → 𝐸2 is the unique additive function such that 

‖𝑓(𝑥) − 𝐴(𝑥)‖ ≤ 𝛿 

for all 𝑥 ∈ 𝐸1. Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥 ∈ 𝐸1, then the function 𝐴 is 

linear. 

Overall, Hyers' contribution to the theory of functional equations had significantly impacted 

mathematics and opened up new areas of research in this field, and it was only the beginning of a 

long list of necessary studies and results in the stability of functional equations. Starting with D. G. 

Bourgin [8], [9] and T. Aoki [4] who addressed the stability problem with unbounded Cauchy 

differences, in the scenario where the relevant inequality is not bounded, Th. M. Rassias [16] 

generalized Hyers' theorem by proving the existence of singular linear mappings close to approximate 

additive mappings. These results can be incorporated into the following theorem. 

Theorem 1.2. Let 𝑋 and 𝑌 be normed space and Banach space, respectively. Let 𝑐 and 𝑝 be two real 

numbers such that 𝑐 ≥ 0 and 𝑝 ≠ 1. Consider the operator 𝑓: 𝑋 → 𝑌 that fulfills the inequality 

‖𝑓(𝑥 + 𝑦) − 𝑓(𝑥) − 𝑓(𝑦)‖ ≤ 𝑐(‖𝑥‖𝑝 + ‖𝑦‖𝑝) 

for all 𝑥, 𝑦 ∈ 𝑋 ∖ {0}. Then there exists an additive mapping 𝑇: 𝑋 → 𝑌 with 

‖𝑓(𝑥) − 𝑇(𝑥)‖ ≤
𝑐

|1 − 2𝑝−1|
‖𝑥‖𝑝 

for all 𝑥 ∈ 𝑋 ∖ {0}. 

In the spirit of Rassias' approach, G. L. Forti [12] and P. Găvruţă [13] generalized all of the 

aforementioned stability results by swapping out the Cauchy differences for a control function 𝜑. At 

the same time, a special kind of stability known as hyperstability emerged. The hyperstability of 



Communications on Applied Nonlinear Analysis 

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functional equations refers to the property that small perturbations of a given functional equation lead 

to solutions that are close to the original one. In other words, if a function satisfies a functional 

equation, then nearby functions also satisfy the same equation. 

The following definition helps us distinguish between three key ideas in the theory of the stability of 

functional equations. 

Definition 1.3. Let 𝑋 be a nonempty set, ( 𝑌, 𝑑 ) be a metric space, and 𝜑: 𝑋 × 𝑋 ⟶ [0, ∞) and 

𝜓: 𝑋 ⟶ [0, ∞) be two functions. Let 𝑔𝑖: 𝑋 × 𝑋 ⟶ 𝑋 for 𝑖 ∈ {1, … ,4} and 𝐹1, 𝐹2: 𝑌 × 𝑌 ⟶ 𝑌. 

Then: 

(1) The following equation 

𝐹1(𝑓(𝑔1(𝑥, 𝑦)), 𝑓(𝑔2(𝑥, 𝑦))) = 𝐹2(𝑓(𝑔3(𝑥, 𝑦)), 𝑓(𝑔4(𝑥, 𝑦))) (1.1) 

is called a functional equation that specifies the unknown function 𝑓: 𝑋 ⟶ 𝑌. 

(2) If for every function 𝑓: 𝑋 ⟶ 𝑌 satisfying the inequality 

𝑑 (𝐹1(𝑓(𝑔1(𝑥, 𝑦)), 𝑓(𝑔2(𝑥, 𝑦))), 𝐺2(𝑓(𝑔3(𝑥, 𝑦)), 𝑓(𝑔4(𝑥, 𝑦)))) ≤ 𝜑(𝑥, 𝑦) (1.2) 

for all 𝑥, 𝑦 ∈ 𝑋, there exists a function 𝑇: 𝑋 → 𝑌 that satisfies the equation (1.1) such that 

𝑑(𝑓(𝑥), 𝑇(𝑥)) ≤ 𝜓(𝑥) (1.3) 

for all 𝑥 ∈ 𝑋, then we say that the functional equation (1.1) is generalized Hyers-Ulam-Rassias stable 

on ( 𝑋, 𝑌 ) with control functions 𝜑 and 𝜓. When 𝜑(𝑥, 𝑦) in (1.2) and 𝜓(𝑥) in (1.3) are replaced by 

the reals 𝛿 > 0 and 휀 > 0 respectively, then we say that corresponding phenomenon of the functional 

equation (1.1) is Hyers-Ulam stable on ( 𝑋, 𝑌 ) with controls 휀 and 𝛿. 

(3) If for every function 𝑓: 𝑋 ⟶ 𝑌 satisfying the inequality (1.2), and either 𝑓 is bounded or it is a 

solution to Equation (1.1), then we say that the functional equation (1.1) is superstable on ( 𝑋, 𝑌 ) 

with control function 𝜑. 

(4) If for every function 𝑓: 𝑋 ⟶ 𝑌 satisfying the inequality (1.2) and 𝑓 is a solution of Equation (1.1), 

then we say that the functional equation (1.1) is hyperstable on ( 𝑋, 𝑌 ) with control 𝜑. 

The use of hyperstability in functional equations is mainly theoretical, as it allows us to extend the 

validity of known solutions to nearby cases that may be of interest. For example, if we have a solution 

to a functional equation that describes a certain phenomenon, we can use hyperstability to conclude 

that nearby phenomena are likely to exhibit similar behavior. 

Hyperstability can also be used in the study of the stability of numerical methods for solving 

functional equations. If we know that the equation is hyperstable, then small errors in the numerical 

solution will not significantly affect the accuracy of the result. 

Overall, hyperstability is a useful property of functional equations that allows us to extend the range 

of their applicability and better understand their behavior. 

Although the term "hyperstability" was first used in 2001 [15], the first hyperstability finding appears 

to have been published in 1949 [8]. Let 𝑋 be a real normed space and ( 𝑆,⋅ ) be an arbitrary semigroup. 

In 2001, one of the important results was shown in stability by Gy. Maksa and Zs. Páles [15]. They 

investigated the hyperstability of the following particular class of linear functional equations 



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𝑓(𝑥) + 𝑓(𝑦) =
1

𝑛
∑  

𝑛

𝑖=1

 𝑓(𝑥𝜑𝑖(𝑦)) (1.4) 

where 𝑓: 𝑆 → 𝑋 and where 𝜑1, ⋯ , 𝜑𝑛: 𝑆 → 𝑆 are pairwise distinct automorphisms of 𝑆. Let 𝜎: 𝑆 → 𝑆 

be an involution, that is, 𝜎(𝜎(𝑥)) = 𝑥 and 𝜎(𝑥 ⋅ 𝑦) = 𝜎(𝑥) ⋅ 𝜎(𝑦) for all 𝑥, 𝑦 ∈ 𝑆. In 2018, Jae-

Hyeong Bae and Won-Gil Park [7] introduced the following functional equation 

𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4), (1.5) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑋, where 𝑓: 𝑋 × 𝑋 → 𝑌 with 𝑌 is a Banach space and where 𝜎 and 𝜏 are 

involutions on 𝑋. 

When 𝜎(𝑥) = 𝜏(𝑥) = 𝑥−1, the following quadratic equation is obtained 

𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝑥2
−1, 𝑥3 ⋅ 𝑥4

−1) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4) (1.6) 

The stability of Eq. (1.6) was investigated in [5] and [6]. 

We observe that the functional equation (1.6), can be reformulated in a more compact and structured 

form by introducing the mapping 

𝜑(𝑥2, 𝑥4): = (𝜎(𝑥2), 𝜏(𝑥4)) 

where 𝜎 and 𝜏 are involutions on a semigroup 𝑆. Let 𝒮: = 𝑆 × 𝑆 denote the direct product semigroup 

equipped with the componentwise multiplication 

(𝑥1, 𝑥3)(𝑥2, 𝑥4): = (𝑥1𝑥2, 𝑥3𝑥4) 

For 𝑋 = (𝑥1, 𝑥3) and 𝑌 = (𝑥2, 𝑥4) in 𝒮, the original equation can then be expressed as 

𝑓(𝑋𝑌) + 𝑓(𝑋𝜑(𝑌)) = 2𝑓(𝑋) + 2𝑓(𝑌), ∀𝑋, 𝑌 ∈ 𝒮 (1.7) 

This equation exhibits a symmetric quadratic structure on the semigroup 𝒮 with an involutive 

automorphism 𝜑. Such reformulations are useful for highlighting the algebraic structure underlying 

the equation and for enabling more general approaches to stability analysis. The equation (1.7) is the 

quadratic type, its central solutions are determined on arbitrary semigroups and its all solutions on 

abelian semigroups in several works (see for example [1], [2], and [11]). 

The primary objective of this work is to investigate the hyperstability of a general functional equation 

of the form 

𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4), 

where 𝑓: 𝑆2 → 𝑋, and 𝜎, 𝜏 are involutive mappings on a semigroup 𝑆. Our approach is inspired by the 

asymptotic technique introduced by Maksa and Páles [15], which has proven effective in establishing 

hyperstability results for linear-type functional equations. The main contribution of this paper lies in 

extending this method to a more general setting involving products and involutions on semigroups. 

Specifically, we provide sufficient asymptotic conditions under which the equation admits 

hyperstability on arbitrary semigroups. 

Moreover, we go beyond the homogeneous case by analyzing the hyperstability of an associated 

inhomogeneous functional equation: 



Communications on Applied Nonlinear Analysis 

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𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4) + 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4), 

where 𝐹: 𝑆4 → 𝑋 is an arbitrary perturbation function. We prove that under suitable assumptions, any 

approximate solution of this perturbed equation must be an exact solution, thereby establishing its 

hyperstability as well. These results contribute to the ongoing development of the theory of stability 

in functional equations by broadening the class of equations and algebraic structures for which 

hyperstability can be rigorously validated. One of the main goals of this paper is to study the 

hyperstability of a general functional equation involving involutions on semigroups, using an 

asymptotic approach inspired by the method of Maksa and Páles[15]. Our analysis focuses on 

identifying sufficient conditions under which the approximate solutions of the functional equation 

converge to exact solutions, thereby establishing its hyperstability in the sense of Ulam. 

In addition to analyzing the hyperstability of the original functional equation in its standard form, this 

work also aims to recast the equation within a product semigroup framework. By identifying the 

underlying structure in terms of the product semigroup 𝒮 = 𝑆 × 𝑆 and an involutive automorphism 

𝜑: 𝒮 → 𝒮, we reformulate the original equation as 

𝑓(𝜉휁) + 𝑓(𝜉𝜑(휁)) = 2𝑓(𝜉) + 2𝑓(휁), 𝜉, 휁 ∈ 𝒮 

This alternative formulation, which reinterprets the functional equation within the Cartesian product 

semigroup 𝑆 × 𝑆, serves a dual purpose. First, it exposes the underlying algebraic symmetry of the 

equation by aligning it with a structured binary operation on paired elements, thus clarifying the 

involutive behavior induced by 𝜎 and 𝜏. Second, this reformulation enables the application of general 

semigroup techniques to a broader class of functional equations by transforming the problem into an 

equivalent one on the product structure. Such a representation is particularly advantageous when 

exploring stability properties, as it allows the equation to be treated in terms of single-variable 

operations on pairs, thereby facilitating the use of asymptotic and fixed-point methods. This 

perspective not only streamlines the analytical process but also opens pathways for further 

generalizations to multi-variable or higher-dimensional functional equations on composite algebraic 

systems. 

2. Main results 

In what follows, we denote by ℝ+the set of nonnegative real numbers and by ℕ the set of positive 

integers. Let 𝑋 be a real normed space, and let ( 𝑆,⋅ ) be a semigroup. Throughout this section, we 

consider two mappings 𝜎, 𝜏: 𝑆 → 𝑆 that are assumed to be involutive endomorphisms unless specified 

otherwise. 

The analysis presented in this section is motivated by the asymptotic technique introduced by Maksa 

and Páles [15], which has proven to be a powerful tool in establishing hyperstability results. Our 

approach begins with a key lemma that forms the basis for the subsequent stability results. 

Lemma 2.1. Let 𝑓: 𝑆2 ⟶ 𝑋 be an arbitrary function. Then the function 𝐷𝑓: 𝑆4 ⟶ 𝑋 that is defined 

by 

𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) = 𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4), (2.1) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆, satisfies the following functional equation 



Communications on Applied Nonlinear Analysis 

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2𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) +𝐷𝑓(𝑥1 ⋅ 𝑥2, 𝑎, 𝑥3 ⋅ 𝑥4, 𝑏) +𝐷𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑎, 𝑥3 ⋅ 𝜏(𝑥4), 𝑏)

=𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝑎, 𝑥3, 𝑥4 ⋅ 𝑏) +𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝜎(𝑎), 𝑥3, 𝑥4 ⋅ 𝜏(𝑏))

 +2𝐷𝑓(𝑥2, 𝑎, 𝑥4, 𝑏)                                                          (2.2)            

 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑎, 𝑏 ∈ 𝑆. 

Proof. When we evaluate the left and right sides of (2.2), we obtain 

2𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) + 𝐷𝑓(𝑥1 ⋅ 𝑥2, 𝑎, 𝑥3 ⋅ 𝑥4, 𝑏) + 𝐷𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑎, 𝑥3 ⋅ 𝜏(𝑥4), 𝑏)

 = 2𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 2𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 4𝑓(𝑥1, 𝑥3) − 4𝑓(𝑥2, 𝑥4)

 +𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝑎, 𝑥3 ⋅ 𝑥4 ⋅ 𝑏) + 𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝑥4 ⋅ 𝜏(𝑏))

 −2𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) − 2𝑓(𝑎, 𝑏) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝑎, 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝑏)

 +𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝜏(𝑏)) − 2𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓(𝑎, 𝑏)

 = −4𝑓(𝑥1, 𝑥3) − 4𝑓(𝑥2, 𝑥4) + 𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝑎, 𝑥3 ⋅ 𝑥4 ⋅ 𝑏) + 𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝑥4 ⋅ 𝜏(𝑏))

 −4𝑓(𝑎, 𝑏) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝑎, 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝑏) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝜏(𝑏)),

 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑎, 𝑏 ∈ 𝑆, and 

𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝑎, 𝑥3, 𝑥4 ⋅ 𝑏) + 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝜎(𝑎), 𝑥3, 𝑥4 ⋅ 𝜏(𝑏)) + 2𝐷𝑓(𝑥2, 𝑎, 𝑥4, 𝑏)

 = 𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝑎, 𝑥3 ⋅ 𝑥4 ⋅ 𝑏)

 +𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝜏(𝑏)) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2 ⋅ 𝑎, 𝑥4 ⋅ 𝑏)

 +𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝑥4 ⋅ 𝜏(𝑏)) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝑎, 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝑏) − 2𝑓(𝑥1, 𝑥3)

 −2𝑓(𝑥2 ⋅ 𝜎(𝑎), 𝑥4𝜏(𝑏)) + 2𝑓(𝑥2 ⋅ 𝑎, 𝑥4 ⋅ 𝑏) + 2𝑓(𝑥2 ⋅ 𝜎(𝑎), 𝑥4𝜏(𝑏)) − 4(𝑥2, 𝑥4) − 4𝑓(𝑎, 𝑏)

 = −4𝑓(𝑥1, 𝑥3) − 4𝑓(𝑥2, 𝑥4) + 𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝑎, 𝑥3 ⋅ 𝑥4 ⋅ 𝑏) + 𝑓(𝑥1 ⋅ 𝑥2 ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝑥4 ⋅ 𝜏(𝑏))

 −4𝑓(𝑎, 𝑏) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝑎, 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝑏) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2) ⋅ 𝜎(𝑎), 𝑥3 ⋅ 𝜏(𝑥4) ⋅ 𝜏(𝑏)),

 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑎, 𝑏 ∈ 𝑆. Therefore, (2.2) holds. 

In the next theorem, we study the hyperstability of the equation (1.5). Specifically, we prove that the 

functional equation (1.5) is hyperstable in the class of functions 𝑓: 𝑆 → 𝑋 for certain asymptotic 

properties of the control function 휀: 𝑆4 → ℝ+. 

Theorem 2.2. Let 휀: 𝑆4 → ℝ+be a function, and consider that there exists a sequence {𝑢𝑛}𝑛∈ℕ in 𝑆 

that satisfies the following two conditions 

lim
𝑛→∞

 inf휀(𝑥1, 𝑥2 ⋅ 𝑢𝑛 , 𝑥3, 𝑥4 ⋅ 𝑢𝑛) = 0 (2.3) 

and 

lim
𝑛→∞

 inf휀(𝑥1, 𝑥2 ⋅ 𝜎(𝑢𝑛), 𝑥3, 𝑥4 ⋅ 𝜏(𝑢𝑛)) = 0, (2.4) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆. Assume that 𝑓: 𝑆2 → 𝑋 satisfies the inequality 

‖𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4)‖ ≤ 휀(𝑥1, 𝑥2, 𝑥3, 𝑥4) (2.5) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆. Then the functional equation (1.5) is hyperstable on 𝑆. 

Proof. In view of the function 𝐷𝑓 defined by (2.1), the inequality (2.5) becomes 

𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) ≤ 휀(𝑥1, 𝑥2, 𝑥3, 𝑥4), 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 (2.6) 

Using Lemma 2.1, we get 



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2𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) + 𝐷𝑓(𝑥1 ⋅ 𝑥2, 𝑎, 𝑥3 ⋅ 𝑥4, 𝑏) + 𝐷𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑎, 𝑥3 ⋅ 𝜏(𝑥4), 𝑏)

 = 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝑎, 𝑥3, 𝑥4 ⋅ 𝑏) + 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝜎(𝑎), 𝑥3, 𝑥4 ⋅ 𝜏(𝑏)) + 2𝐷𝑓(𝑥2, 𝑎, 𝑥4, 𝑏)                        (2.7)
 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑎, 𝑏 ∈ 𝑆. Assume that there is a sequence {𝑢𝑛}𝑛∈ℕ of elements of 𝑆 that satisfying 

the conditions (2.3) and (2.4). Replacing 𝑥2 by 𝑥2 ⋅ 𝑢𝑛 and 𝑥4 by 𝑥4 ⋅ 𝑢𝑛 in (2.5), where 𝑛 ∈ ℕ, we 

get 

‖𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝑢𝑛, 𝑥3, 𝑥4 ⋅ 𝑢𝑛)‖ ≤ 휀(𝑥1, 𝑥2 ⋅ 𝑢𝑛, 𝑥3, 𝑥4 ⋅ 𝑢𝑛), 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 (2.8) 

In view of (2.3), we obtain 

lim
𝑛→∞

 inf 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝑢𝑛, 𝑥3, 𝑥4 ⋅ 𝑢𝑛) = 0,        𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 (2.9) 

Replacing 𝑥2 by 𝑥2 ⋅ 𝜎(𝑢𝑛) and 𝑥4 by 𝑥4 ⋅ 𝜏(𝑢𝑛) in (2.5), where 𝑛 ∈ ℕ, we find 

‖𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝜎(𝑢𝑛), 𝑥3, 𝑥4 ⋅ 𝜏(𝑢𝑛))‖ ≤ 휀(𝑥1, 𝑥2 ⋅ 𝜎(𝑢𝑛), 𝑥3, 𝑥4 ⋅ 𝜏(𝑢𝑛)),    𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 (2.10) 

By applying the condition (2.4), we get 

lim
𝑛→∞

 inf 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝜎(𝑢𝑛), 𝑥3, 𝑥4 ⋅ 𝜏(𝑢𝑛)) = 0, 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 (2.11) 

Let 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝛼, 𝛽 ∈ 𝑆 be fixe. By replacing 𝑎 by 𝛼 ⋅ 𝑢𝑛 and 𝑏 by 𝛽 ⋅ 𝑢𝑛 in (2.7), we conclude the 

following equality 

2𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) + 𝐷𝑓(𝑥1 ⋅ 𝑥2, 𝛼 ⋅ 𝑢𝑛, 𝑥3 ⋅ 𝑥4, 𝛽 ⋅ 𝑢𝑛)

+𝐷𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝛼 ⋅ 𝑢𝑛 , 𝑥3 ⋅ 𝜏(𝑥4), 𝛽 ⋅ 𝑢𝑛)  

 = 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝛼 ⋅ 𝑢𝑛, 𝑥3, 𝑥4 ⋅ 𝛽 ⋅ 𝑢𝑛) + 𝐷𝑓(𝑥1, 𝑥2 ⋅ 𝜎(𝛼 ⋅ 𝑢𝑛), 𝑥3, 𝑥4 ⋅ 𝜏(𝛽 ⋅ 𝑢𝑛))

 +2𝐷𝑓(𝑥2, 𝛼 ⋅ 𝑢𝑛, 𝑥4, 𝛽 ⋅ 𝑢𝑛)                                                                                                                 (2.12)

 

Letting 𝑛 → ∞ with using (2.9) and (2.11), we derive from (2.12) that 

𝐷𝑓(𝑥1, 𝑥2, 𝑥3, 𝑥4) = 0, 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 

That is, 𝑓 is a solution of equation (1.5) which means that (1.5) is hyperstable on 𝑆. 

We arrive at the following corollary as a direct result of the Theorem 2.2. 

Corollary 2.3. Let 휀: 𝑆4 ⟶ ℝ+be a function and we consider that there exist 𝛼 ∈ 𝑆, 0 ≤ 𝑠 < 1 and 

0 ≤ 𝑡 < 1 such that 

{
휀(𝑥1, 𝑥2 ⋅ 𝛼, 𝑥3, 𝑥4 ⋅ 𝛼) ≤ 𝑠휀(𝑥1, 𝑥2, 𝑥3, 𝑥4)

휀(𝑥1, 𝑥2 ⋅ 𝜎(𝛼), 𝑥3, 𝑥4 ⋅ 𝜏(𝛼)) ≤ 𝑡휀(𝑥1, 𝑥2, 𝑥3, 𝑥4)
(2.13) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆 and all 𝑛 ∈ ℕ. If 𝑓: 𝑆 ⟶ 𝑋 satisfies the inequality (2.5), then it is a solution 

to the functional equation (1.5). 

Proof. It is not hard to demonstrate, using induction on 𝑛 ∈ ℕ, that 

{
휀(𝑥1, 𝑥2 ⋅ 𝛼𝑛, 𝑥3, 𝑥4 ⋅ 𝛼𝑛) ≤ 𝑠𝑛휀(𝑥1, 𝑥2, 𝑥3, 𝑥4)

휀(𝑥1, 𝑥2 ⋅ 𝜎(𝛼𝑛), 𝑥3, 𝑥4 ⋅ 𝜏(𝛼𝑛)) ≤ 𝑡𝑛휀(𝑥1, 𝑥2, 𝑥3, 𝑥4)
(2.14) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆. Therefore, the conditions (2.3) and (2.4) hold with 𝑢𝑛 = 𝛼𝑛 for all 𝑛 ∈ ℕ. 

We conclude that the functional equation (1.5) is hyperstable on 𝑆 based on Theorem 2.2. 



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In the next theorem, we discuss the hyperstability of the following inhomogeneous functional 

equation: 

𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4) + 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4) (2.15) 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆, where 𝐹: 𝑆4 → 𝑋. 

Theorem 2.4. Let 𝑋, 𝑆, 𝜎, and 𝜏 remain unchanged from Theorem 2.2. Let 𝑓: 𝑆2 → 𝑋 and 𝐹: 𝑆4 → 𝑋 

be two functions satisfy the inequality 

‖𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) −2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4) − 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)‖

 ≤ 휀(𝑥1, 𝑥2, 𝑥3, 𝑥4) (2.16)
 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆. If the functional equation (2.15) admits a solution 𝑓0: 𝑆2 → 𝑋, then (2.15) is 

hyperstable on 𝑆. 

Proof. Let 𝑔: 𝑆2 → 𝑋 be a function defined by 𝑔(𝑥1, 𝑥2): = 𝑓(𝑥1, 𝑥2) − 𝑓0(𝑥1, 𝑥2) for all 𝑥1, 𝑥2 ∈ 𝑆. 

Then 

‖𝑔(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑔(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑔(𝑥1, 𝑥3) − 2𝑔(𝑥2, 𝑥4)‖

 = ‖𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − (𝑓0(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4)

+𝑓0(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4))) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4) − 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)

 −(2𝑓0(𝑥1, 𝑥3) + 2𝑓0(𝑥2, 𝑥4) + 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4))‖

 ≤ ‖𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4)

 −𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)‖

 +‖𝑓0(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓0(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓0(𝑥1, 𝑥3) − 2𝑓0(𝑥2, 𝑥4)

 −𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)‖

 = ‖𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4)

 −𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)‖

 ≤ 휀(𝑥1, 𝑥2, 𝑥3, 𝑥4),     𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆.

 

According to Theorem 2.2, the function 𝑔 = 𝑓 − 𝑓0 satisfies the homogeneous equation (1.5). 

Therefore, by linearity, we have: 

𝑓(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓(𝑥1, 𝑥3) − 2𝑓(𝑥2, 𝑥4) − 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)

 = 𝑔(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑔(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑔(𝑥1, 𝑥3) − 2𝑔(𝑥2, 𝑥4)

 +𝑓0(𝑥1 ⋅ 𝑥2, 𝑥3 ⋅ 𝑥4) + 𝑓0(𝑥1 ⋅ 𝜎(𝑥2), 𝑥3 ⋅ 𝜏(𝑥4)) − 2𝑓0(𝑥1, 𝑥3) − 2𝑓0(𝑥2, 𝑥4) − 𝐹(𝑥1, 𝑥2, 𝑥3, 𝑥4)

 = 0

 

for all 𝑥1, 𝑥2, 𝑥3, 𝑥4 ∈ 𝑆. Since both 𝑔 and 𝑓0 satisfy their respective equations, each component 

vanishes, and we conclude that 𝑓 is also a solution of (2.15). 

Before closing this section, we reformulate the main functional equation within the framework of the 

Cartesian product semigroup 𝑆 × 𝑆. This structural reinterpretation allows us to express the equation 

in a more symmetric and compact form, which not only reflects the underlying algebraic properties 

induced by the involutions 𝜎 and 𝜏, but also enables broader generalizations. By identifying elements 

of 𝑆2 with ordered pairs and defining the multiplication and involutive mappings accordingly, we 



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derive a natural equivalent of the original equation on the product semigroup. The following corollary 

confirms that the hyperstability property persists in this extended setting. 

Corollary 2.5. Let 𝒮: = 𝑆 × 𝑆 be the direct product of a semigroup 𝑆 with itself, equipped with the 

product operation 

(𝑥1, 𝑥3)(𝑥2, 𝑥4): = (𝑥1𝑥2, 𝑥3𝑥4) 

and let 𝜑: 𝒮 → 𝒮 be defined by 𝜑(𝑥2, 𝑥4): = (𝜎(𝑥2), 𝜏(𝑥4)), where 𝜎 and 𝜏 are involutions on 𝑆. 

Suppose that 𝑓: 𝒮 → 𝑋 satisfies the inequality 

‖𝑓(𝜉휁) + 𝑓(𝜉𝜑(휁)) − 2𝑓(𝜉) − 2𝑓(휁)‖ ≤ 휀(𝜉, 휁) 

for all 𝜉, 휁 ∈ 𝒮, where 휀: 𝒮2 → ℝ+satisfies the asymptotic conditions analogous to those in Theorem 

2.2. Then the functional equation 

𝑓(𝜉휁) + 𝑓(𝜉𝜑(휁)) = 2𝑓(𝜉) + 2𝑓(휁), ∀𝜉, 휁 ∈ 𝒮 (2.17) 

is hyperstable on 𝒮. 

Proof. Let 𝜉 = (𝑥1, 𝑥3) and 휁 = (𝑥2, 𝑥4) be arbitrary elements in 𝒮. Then: 

𝜉휁 = (𝑥1𝑥2, 𝑥3𝑥4)

𝜉𝜑(휁) = (𝑥1𝜎(𝑥2), 𝑥3𝜏(𝑥4))
 

Hence, the equation 

𝑓(𝜉휁) + 𝑓(𝜉𝜑(휁)) = 2𝑓(𝜉) + 2𝑓(휁) 

is equivalent to the original functional equation: 

𝑓(𝑥1𝑥2, 𝑥3𝑥4) + 𝑓(𝑥1𝜎(𝑥2), 𝑥3𝜏(𝑥4)) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4) 

which was shown in Theorem 2.2 to be hyperstable under the given asymptotic condition on 휀. 

Moreover, we can define a sequence {𝑈𝑛}𝑛∈ℕ ⊂ 𝒮 by 𝑈𝑛 = (𝛼𝑛, 𝛼𝑛) for some fixed 𝛼 ∈ 𝑆 satisfying 

the decay condition: 

휀(𝜉 ⋅ 𝑈𝑛 , 휁 ⋅ 𝑈𝑛) → 0  and  휀(𝜉 ⋅ 𝜑(𝑈𝑛), 휁 ⋅ 𝜑(𝑈𝑛)) → 0 

as 𝑛 → ∞, for all 𝜉, 휁 ∈ 𝒮. 

Therefore, by direct application of the same reasoning in Theorem 2.2, we conclude that 

𝑓(𝜉휁) + 𝑓(𝜉𝜑(휁)) = 2𝑓(𝜉) + 2𝑓(휁) 

holds for all 𝜉, 휁 ∈ 𝒮, i.e., the equation (2.17) is hyperstable on 𝒮. 

Conclusion 

In this paper, we have investigated the hyperstability of a general functional equation involving two 

involutive mappings defined on an arbitrary semigroup. Using an asymptotic approach based on the 

method of Maksa and Páles [15], we established sufficient conditions under which approximate 

solutions to the equation 

𝑓(𝑥1𝑥2, 𝑥3𝑥4) + 𝑓(𝑥1𝜎(𝑥2), 𝑥3𝜏(𝑥4)) = 2𝑓(𝑥1, 𝑥3) + 2𝑓(𝑥2, 𝑥4) 



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must in fact be exact solutions. These results extend prior stability theorems and contribute to the 

ongoing development of the theory of functional equations in algebraic settings involving involutions. 

 Furthermore, we provided a reformulation of the equation over the product semigroup 

 𝑆 =  𝑆 ×  𝑆, where we demonstrated via Corollary 2.5 that the hyperstability property is preserved 

under this transformation. This reformulation not only highlights the underlying algebraic symmetry 

of the problem but also broadens the structural scope of the stability analysis. 

Future work may explore analogous phenomena in more complex algebraic systems, including 

inverse and topological semigroups or frameworks involving multiple interacting involutions. 

Additional investigations may also target non-quadratic functional equations and their 

inhomogeneous counterparts under similar asymptotic stability conditions. 

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