Communications on Applied Nonlinear Analysis 

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Stability of a General Quadratic-Cubic Functional Equation in Non-

Archimedean 2-Normed Spaces 
 

Elumalai P1, Sangeetha S2*  
1,2 Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, 

Kattankulathur, Chengalpattu, Tamil Nadu-603 203, India.  

ep5583@srmist.edu.in  

Corresponding author : sangeets@srmist.edu.in 

  

Article History: 

Received: 31-07-2024 

Revised: 21-09-2024 

Accepted: 02-10-2024 

Abstract  

This research aims to investigate the Hyers-Ulam stability of the mixed-type quadratic-

cubic functional equation in non-Archimedean 2-normed spaces using the fixed-point 

method. Additionally, some counter-examples are illustrated for instability. The 

exciting possibilities of this cutting-edge research and unlocking new frontiers in 

mathematical analysis are explored. 

Keywords: Fixed point method, Hyers-Ulam stability, Non-Archimedean 2-normed 

spaces, p-adic field, Quadratic-cubic functional equation. 

1. Introduction 

Functional equations play an essential and fascinating role in mathematics, employing simple algebraic 

procedures that lead to intriguing solutions. The theory of functional equations is also applied in 

developing other domains such as analysis, algebra, geometry, and more. New approaches and 

techniques are utilized in problem-solving across fields like IT, finance, geometry, wireless sensor 

networks, and beyond. Ulam stability is a crucial concept in studying functional equations and their 

solutions. This theory examines whether a function that approximately satisfies a certain functional 

equation is close to a function that exactly satisfies the equation. Numerous researchers across various 

fields have explored different types of functional equation stability, such as Hyers-Ulam (H-U) 

stability, Hyers-Ulam-Rassias (H-U-R) stability, and generalized Hyers-Ulam stability, particularly in 

the context of different functional equations and mixed types over recent decades. Additionally, many 

authors have investigated the stability of various functional equations, yielding fascinating results in 

the classical (Archimedean) case. In recent years, the stability problems of these functional equations 

in non-Archimedean (NA) spaces have also been examined. In the last few decades, researchers have 

become much more interested in the study of Hyers-Ulam stability of functional equations (FEs). 

Numerous high-quality research papers have been published on this topic, as evidenced by references 

[1-3] and the associated citations in references [4-6]. As the field continues to grow, various 

approaches, including direct methods, fixed-point techniques, and others, have been developed and 

applied to solve different types of FEs [7-9]. 

Typically, when employing the direct method to establish stability outcomes for FEs, one must satisfy 

either of the two conditions: 

‖𝔣(ω1) −
1

α
𝔣(αω1)‖ ≤ 𝔲(ω1)  or  ‖𝔣(ω1) − α𝔣 (

ω1

α
)‖ ≤ 𝔲 (

ω1

α
). 

The choice between these conditions depends on specific assumptions, necessitating distinctions to 

apply an appropriate approach to solving distinct problems [10-12]. It was Ulam [13] who first raised 

the stability problem in 1940, and it was Hyers [14] who responded affirmatively in 1941 for a Banach 

mailto:ep5583@srmist.edu.in
mailto:sangeets@srmist.edu.in


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space. According to Rassias, the bound for the norm of the Cauchy difference is weakened after the 

FE  

∥ 𝔤(ω1 + ω2) − 𝔤(ω1) − 𝔤(ω2) ∥ ≤ ε(∥ ω1 ∥p +∥ ω2 ∥p) 

which leads to the generalized H-U stability theorem for additive mapping. Concerning Hyers’ 

theorem of additive mappings, papers were published by Aoki [15] and Rassias [16]. Gavruta 

presented a generalization of the Rassias theorem in 1994 [17]. The FE  

                                                                      𝔤(ω1 + ω2) = 𝔤(ω1) + 𝔤(ω2)  1 

is referred to as an additive FE. Specifically, every solution of the additive FE is termed an additive 

mapping. In the work by Gahler [18], the theory of 2-norms and n-norms within a linear space was 

introduced. Subsequently, Gahler and White proposed the idea of 2-Banach spaces [19]. Kim and Park 

[20] investigated the generalized H-U stability of additive FEs in NA 2-normed space in 2014. In 2020, 

Wand, Park, and Shin [21] examined the H-U stability of additive ρ-functional equations in NA 2-

normed space. Recently, Ghali and Kabbaj [22] investigated the hyperstability of the Cauchy-Jensen 

FE in NA 2-Banach spaces and some of its applications. In 2020, Cho, Gordji, and Zolfaghari [23] 

investigated the solution and stability of a generalized mixed-type quadratic-cubic(𝑄2
́ − 𝒞3

́ ) functional 

equation in random normed spaces. In 2022, Mohiuddine, Tamilvannan, Mursaleen, and Alotaibi [24] 

examined the stability of a quartic functional equation in modular spaces using Hyers and fixed-point 

methods. 

For example, Hensel [25] discovered the p-adic numbers in 1897 as a number theory equivalent to 

power series in complex analysis. He created a field with a valuation standard that lacks the 

Archimedean property. Numbers with p-adic numbers are the best examples of NA spaces. Those who 

work with p-adic numbers understand that they do not adhere to Archimedean properties, which state 

that for any positive number n, there exists an integer x such that 𝑛𝑥 >  𝑦 for another positive integer 

y. In the past 30 years, physicists have shown increased interest in the theory of NA spaces, especially 

with problems in quantum physics, p-adic physics, and superstring theory. The NA version of several 

results in the usual normed spaces theory differs significantly in their proofs, requiring a new level of 

understanding. Notably, in every valuation field where |n| ≤ 1, every triangle is isosceles, and there 

may not be a unit vector in a non-Archimedean space. These details highlight the remarkable structure 

of NA spaces. 

In this present article, we will use the alternative fixed point method to investigate the H-U stability of 

the general quadratic-cubic FE over NA 2-normed spaces.  

2. Preliminaries 

Some basic definitions and theorems are presented in this section as a reminder. 

Definition 1: [21] Let 𝒲 be a vector space over a scalar field 𝕂 with an NA non-trivial valuation |. |. 
A function ∥. ∥ from  𝒲  to  ℝ is called an NA norm (valuation) if it satisfies the following conditions:   

    (1) ∥ ω1 ∥= 0 if and only if ω1 = 0;  

    (2)  ∥ rω1 ∥= |r| ∥ ω1 ∥ for all r ∈ 𝕂, ω1 ∈ 𝒲;  

    (3) ∥ ω1, ω2 ∥≤ max{∥ ω1 ∥, ∥ ω2 ∥} (satisfying the strong triangle inequality or ultra-metric 

property) 

for all ω1, ω2 ∈ 𝒲. Then (𝒲, ∥. ∥) is called an NA-normed space.   

Example 1:  Let ρ be a fixed prime number. For any non-zero rational number ω1, there is a unique 

integer nω1
∈ ℤ such that  



Communications on Applied Nonlinear Analysis 

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ω1 =
α

β
ρnω1 , 

where α and β are integers not divisible by ρ. Then, the function |. |ρ: ℚ
ρ

→ [0, +∞) defined by  

|ω1| = {
0, ω1 = 0,

ρ−nω1
 , ω1 ≠ 0

 

 is an NA valuation on ℚ
ρ
.  

Example 2: Let ω1 =
70

13
. In this case, let us find its 5-adic absolute value (ρ = 5) as given below 

ω1 =
70

13
= 51.

14

13
 

which means |ω1|5 =
1

5
  or  5−1. 

 It will be simple to |ω1|13 = 13, because 

ω1 = 13
−1. 75 

|ω1|13 =
1

13
−1

= 13. 

which means |ω1|13 = 13.   

Definition 2: [18] Let 𝒲 be a vector space over a scalar field 𝕂 with an NA non-trivial valuation |. | 
with dim 𝒲 > 1. A function ∥. , . ∥   from  𝒲  to  ℝ is called an NA 2-norm (valuation) if it satisfies 

the following conditions:   

    (1)  ∥ ω1, ω2 ∥= 0 if and only if ω1, ω2 are linearly dependent;  

    (2)  ∥ ω1, ω2 ∥=∥ ω2, ω1 ∥;  

    (3)  ∥ r  ω1, ω2 ∥= |r| ∥ ω1, ω2 ∥ for  all  r ∈ 𝕂, ω1, ω2 ∈ 𝒲;  

    (4)  ∥ ω1, ω2 + υ ∥≤ max{∥ ω1, ω2 ∥, ∥ ω1, υ ∥}  for  all  ω1, ω2, υ ∈ 𝒲  

Then (𝒲, ∥. , . ∥) is called an NA 2-normed space.  

The following lemma follows from Definition 2. 

Lemma 1: [18] Let (𝒲, ∥. , . ∥) be an NA 2-normed space. If ω1 ∈ 𝒲 and ∥ ω1, ω2 ∥= 0 for all ω2 ∈
𝒲, then ω1 = 0.  

Definition 3: [18] A sequence {ω1𝔫
} in an NA 2-normed space (𝒲, ∥. , . ∥) is called a Cauchy sequence 

if there are two linearly independent points ω1, ω2 ∈ 𝒲 such that  

 lim
𝔪,𝔫

∥ ω1𝔫
− ω1𝔪

, ω2 ∥= 0  and  lim
𝔪,𝔫

∥ ω1𝔫
− ω1𝔪

, ω2 ∥= 0.  

Definition 4: [18] A sequence {ω1𝔫
} in an NA 2-normed space (𝒲, ∥. , . ∥) is called a convergent 

sequence if there exists an ω1 ∈ 𝒲 such that  

 lim
𝔫

∥ ω1𝔫
− ω1, ω2 ∥= 0      for  all  ω1, ω2 ∈ 𝒲. 

In this case, recall that {ω1𝔫
} converges to ω1 or that ω1 is the limit of {ω1𝔫

}, write {ω1𝔫
} → ω1  as  𝔫 →

∞  or  lim
𝔫→∞

ω1𝔫
= ω1. 



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By Definition 2 (4), we have  

 ∥ ω1𝔫
− ω1𝔪

, ω2 ∥≤ max{∥ ω1ȷ+1
− ω1ȷ

, ω2 ∥: 𝔪 ≤ ȷ ≤ 𝔫 − 1},    (𝔫 > 𝔪),     

for all 𝜔1, ω2  ∈ 𝒲. Hence, a sequence {ω1𝔫
} in Cauchy in (𝒲, ∥. , . ∥) if and only if {ω1𝔫+1

− ω1𝔫
} 

converges to 0 in an NA 2-normed space (𝒲, ∥. , . ∥).  

Remark 1: [18] Let (𝒲, ∥. , . ∥) be an NA 2-normed space. One can show that conditions (2) and (4) 

in Definition 2 imply that  

 ∥ ω1 + ω2, υ ∥≤∥ ω1, υ ∥ +∥ ω2, υ ∥     and    | ∥ ω1 − υ ∥ −∥ ω2, υ ∥ | ≤∥ ω1 − ω2, υ ∥ 

for all ω1, ω2, υ ∈ 𝒲. It is effortless to get the following lemma by using Remark 1. 

Lemma 2: [18] For a convergent sequence {ω1𝔫
} in an NA 2-normed space (𝒲, ∥. , . ∥),  

 lim
𝔫→∞

∥ ω1𝔫
, ω2 ∥=∥ lim

𝔫→∞
ω1𝔫

, ω2 ∥     for  all  ω2 ∈ 𝒲. 

Definition 5: [18] If every Cauchy sequence in 𝒲 converges, then the NA 2-normed space 𝒲 is called 

an NA 2-Banach space or an ultrametric 2-Banach space. 

Definition 6: [26] Let 𝒲 be a set. A function ð: 𝒲 × 𝒲 → [0, ∞] is called a generalized metric(GM) 

on 𝒲 if ð satisfies the following conditions:   

    (1)  ð(ω1, ω2) = 0 if and only if ω1 = ω2;  

    (2)  ð(ω1, ω2) = ð(ω2, ω1)    for  all  ω1, ω2 ∈ 𝒲;  

    (3)  ð(ω1, υ) ≤ ð(ω1, ω2) + ð(ω2, υ)  for all ω1, ω2, υ ∈ 𝒲. 

Theorem 1: [26] Let (𝒲, ð) be complete and Λ: 𝒲 → 𝒲 be strictly contractive mapping with  0 <
£ < 1. Then for each given element ω1 ∈ 𝒲, either  

ð(Λ
n
ω1, Λn+1

ω1) = ∞  for  all  n ≥ 0 

or there exists a natural number n0 such that   

    (1)  ð(Λ
n
ω1, Λ

n+1
ω1) < ∞, for all n ≥ n0;  

    (2)  the sequence {Λ
n
ω1} converges to a fixed point ω2

∗ of Λ;  

    (3)  ω2
∗ is the unique fixed point of Λ in the set  

 Ψ = {ω2 ∈ 𝒲 ∶   ð(Λ
n0ω1, ω2) < ∞}; 

    (4)  ð(ω2, ω2
∗) ≤

1

1−£
ð(  ω2, Λω2), for all ω2 ∈ Ψ.” 

In this article, let 𝒲 be an NA 2-normed space with dim 𝒲 > 1 and 𝒵 be an NA 2-Banach space 

with dim 𝒵 > 1.  Suppose for a mapping 𝔤: 𝒲 → 𝒵, 

       D𝔤(ω1, ω2) ∶= 𝔤(ω1 + 𝔪ω2) + 𝔤(ω1 − 𝔪ω2 ) − 𝔪2[𝔤(ω1 + ω2) + 𝔤(ω1 − ω2)] −

                       
2(𝔪2−1)

𝔪2(𝔪−2)
𝔤(𝔪ω1) + 

𝔪3−𝔪2−𝔪+1

2(𝔪−2)
  𝔤(2ω1) − (𝔤(2ω2) − 𝔤(−2ω2)) + 8(𝔤(ω2) − 𝔤(−ω2))2 

for all ω1, ω2 ∈ 𝒲. 

 

 

 



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3. Results and Discussion 

Stability of the FE – Even case 

One can easily demonstrate this by applying the conditons for even and odd functions in this section. 

The condition for an even function is 𝔤(−a) = 𝔤(a), and for and odd function, it is 𝔤(−a) = −𝔤(a). 
An even mapping 𝔤: 𝒲 → 𝒵 with 𝔤(0) = 0 satisfies Eq 2, if and only if the even mapping 𝔤: 𝒲 → 𝒵 

is a quadratic (𝑄2
́ ) mapping, that is,  

 𝔤(ω1 + ω2) + 𝔤(ω1 + ω2) = 2𝔤(ω1) + 2𝔤(ω2) 

and an odd mapping 𝔤: 𝒲 → 𝒵 satisfies Eq 2 if and only if the odd mapping 𝔤: 𝒲 → 𝒵 is a cubic (𝐶3
́ ) 

mapping, that is,  

 𝔤(2ω1 + ω2) + 𝔤(2ω1 + ω2) = 2𝔤(ω1 + ω2) + 2𝔤(ω1 + ω2) + 12𝔤(ω1) 

It was proved in [23], that 𝔤(ω1) = f(2ω1) − 4f(ω1) and 𝔤(ω1) = f(2ω1) − 8f(ω1) are 𝑄2
́   and 𝐶3

́  

mappings respectively. 

In this section, to prove the generalized H-U stability of the FE D𝔤(ω1, ω2) = 0 in NA 2-normed space 

is discussed for even case. 

Theorem 2:  Let Φ: 𝒲 × 𝒲 → [0, ∞) be an even function such that there exists a constant 0 < £ < 1 

with  

                                      Φ(𝔪ω1, 𝔪ω2) ≤ |𝔪|2£ Φ(ω1, ω2) 3 

for all  ω1, ω2 ∈ 𝒲. Let 𝔤: 𝒲 → 𝒵 be an even mapping satisfying  

                                          ‖D𝔤(ω1, ω2), υ‖ ≤ Φ(ω1, ω2) 4 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵. Then there is a unique quadratic mapping 𝑄2
́ : 𝒲 → 𝒵 such that  

                             ‖𝔤(ω1) − 𝑄2
́ (ω1), υ‖ ≤

1

|2| |𝔪|2 (1−£)
Φ(0, ω1) 5 

for all ω1 ∈ 𝒲, υ ∈ 𝒵.  

Proof: Putting ω1 = 0 in Eq 2, implies 

                              ‖2𝔤(𝔪ω2) − 2𝔪2𝔤(ω2), υ‖ ≤ Φ(0, ω2) 6 

for all ω2 ∈ 𝒲, υ ∈ 𝒵. Interchanging ω2 by ω1 in Eq 6 and dividing on both sides of Eq 6 by 2, gives  

‖𝔤(𝔪ω1) − 𝔪2𝔤(ω1), υ‖ ≤
1

|2|
Φ(0, ω1) 

                              ‖
𝔤(𝔪ω1)

𝔪2
  − 𝔤(ω1), υ‖  ≤

1

|2|

1

|𝔪|2
Φ(0, ω1)  7 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Consider the set  

                                               Γ = {𝔣: 𝒲 → 𝒵} 8 

and define the generalized metric ð in Γ by  

 ð(𝔣, 𝔥) = inf{σ ∈ (0, ∞): ‖𝔣(ω1) − 𝔥(ω1), υ‖ ≤   σ  Φ(0, ω1), ∀  ω1 ∈ 𝒲}. 9 

It is simple to prove that (Γ, ð) is complete [26]. 

 Now, define the function Λ ∶ Γ → Γ such that  



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                                             Λ𝔣(ω1) =
1

𝔪2
  𝔣(𝔪ω1) 10 

for all ω1 ∈ 𝒲. Let 𝔣, 𝔥 ∈ Γ be given such that ð(𝔣, 𝔥) = ϵ. Then  

                                                          ‖𝔣(ω1) − 𝔥(ω1), υ‖ ≤ ϵ Φ(0, ω1) 11 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Hence  

‖Λ𝔣(𝔪ω1) − Λ𝔥(𝔪ω1), υ‖ = ‖
1

𝔪2
𝔣(𝔪ω1) −

1

𝔪2
𝔥(𝔪ω1), υ‖ 

         ≤
1

|𝔪|2
. ϵ. Φ(0, 𝔪ω1) 

                                    ≤
1

|𝔪|2
. ϵ. |𝔪|2. £. Φ(0, ω1) 

            ≤ ϵ. £. Φ(0, ω1) 

for all ω1 ∈ 𝒲, υ ∈ 𝒵, that is ð(Λ𝔣, Λh) ≤ £ε. Therefore  

ð(Λ𝔣, Λh) ≤ £  ð(𝔣, h) 

for all 𝔣, 𝔥 ∈ Γ. According to Eq 7  

 ð(𝔤, Λ𝔤) ≤
1

|2|
.

1

|𝔪|2
< +∞.  12 

 By Theorem 1, there is a mapping 𝑄2
́ : 𝒲 → 𝒵 satisfying the following conditions: 

    (1)  𝑄2
́  is a fixed point of Λ, that is,  

 𝑄2
́ (𝔪ω1) = 𝔪2  𝑄2

́ (ω1) 13 

       𝑄2
́  is a unique fixed point of the set denoted by Λ.  

S = {𝔥 ∈ Γ ∶ ð(𝔣, 𝔥) < ∞}. 

       This indicates that 𝑄2
́  is a unique mapping satisfying Eq 13 such that there is a σ ∈ (0, ∞) satisfying  

‖𝔤(ω1) − 𝑄2
́ (ω1), υ‖ ≤ σ  .  Φ(0, ω1),      ∀  ω1 ∈ 𝒲,   υ ∈ 𝒵. 

   (2)  ð(Λ
n𝔤, 𝑄2

́ ) → 0 as n → ∞. This gives that,  

 lim
n→∞

(Λ
n𝔤)(ω1) = lim

n→∞

𝔤(𝔪nω1)

𝔪2n
= 𝑄2

́ (ω1),      ∀  ω1 ∈ 𝒲. 

    (3)  ð(𝔤, 𝑄2
́ ) ≤

1

1−£
ð(𝔤, Λ

n𝔤), which gives the inequality  

ð(𝔤, 𝑄2
́ ) ≤

1

1 − £
ð(𝔤, Λ𝔤) ≤

1

|2|

1

|𝔪|2

1

(1 − £)
.   

This indicates that the inequality Eq 5 remains valid.  

According to Eq 3 and Eq 4  

∥ D𝑄2
́ (ω1, ω2), υ ∥= lim

n→∞
∥ 𝔪−2nD𝔤(𝔪nω1, 𝔪nω2), υ ∥ 

                       ≤ lim
n→∞

1

|𝔪|2n
Φ(𝔪nω1, 𝔪nω2) 



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                    ≤ lim
n→∞

1

|𝔪|2n
£

n|𝔪|2nΦ(ω1, ω2) 

      ≤ lim
n→∞

£
n
Φ(ω1, ω2) = 0 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵, and n ∈ ℕ. So,∥ D𝑄2
́ (ω1, ω2), υ ∥= 0.  

Thus the mapping 𝑄2
́ : 𝒲 → 𝒵 is 𝑄2

́  as desired.  

Corollary 1: Let θ ≥ 0 and τ = s + t be a positive real number with τ < 2. Let 𝔤: 𝒲 → 𝒵 be an even 

mapping with 𝔤(0) = 0 satisfying  

∥ D𝔤(ω1, ω2), υ ∥≤ θ(∥ ω1 ∥τ +∥ ω2 ∥τ +∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵, and 𝑄2
́ : 𝒲 → 𝒵 is a quadratic mapping such that  

∥ 𝔤(ω1) − 𝑄2
́ (ω1) ∥≤

|𝔪|τ

|𝔪|2(|𝔪|τ − |𝔪|2)

θ ∥ ω1 ∥τ

|2|
 

for all ω1 ∈ 𝒲.  

Proof: Assuming 

 Φ(ω1, ω2) ∶= θ  (‖ω1‖τ + ‖ω2‖τ+∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, and by choosing £ = |𝔪|2−τ, the expected result can be obtained by Theorem 2. 

Theorem 3:  Let Φ: 𝒲 × 𝒲 → [0, ∞) be an even function such that there is a constant 0 < £ < 1 with  

Φ (
ω1

𝔪
,
ω2

𝔪
) ≤

£

|𝔪|2
Φ(ω1, ω2) 

for all ω1, ω2 ∈ 𝒲. Let 𝔤: 𝒲 → 𝒵 be an even mapping satisfying Eq 2. Then there is a unique quadratic 

mapping 𝑄2
́ : 𝒲 → 𝒵 such that  

                                                   ‖𝔤(ω1) − 𝑄2
́ (ω1), υ‖ ≤

1

|2|

1

|𝔪|2
£

(1−£)
 Φ(0, ω1)  14 

for all ω1 ∈ 𝒲, υ ∈ 𝒵.  

Proof: Putting ω1 = 0 in Eq 2 implies  

                                                          ‖2𝔤(𝔪ω2) − 2𝔪2𝔤(ω2), υ‖ ≤ Φ(0, ω2), 15 

for all ω2 ∈ 𝒲, υ ∈ 𝒵. Replacing ω2 by (
ω1

𝔪
) in Eq 15  and dividing on both sides of Eq 6 by 2, it gives  

                                                          ‖𝔤(ω1) − 𝔪2𝔤 (
ω1

𝔪
) , υ‖ ≤

1

|2|
Φ (0,

ω1

𝔪
), 16 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Let (Γ, ð) be the GMS as defined by Theorem 2.  

Now, define the function Λ: Γ → Γ such that  

                                                                    Λ𝔣(ω1) = 𝔪2𝔣 (
ω1

𝔪
) 17 

for all ω1 ∈ 𝒲. So ð(𝔤, Λ𝔤) ≤
£

|2||𝔪|2
.  According to Eq 16 

ð(𝔤, 𝑄2
́ ) ≤

1

|2|
.

£

|𝔪|2(1 − £)
 



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for all ω1 ∈ 𝒲. This proof follows the same pattern as Theorem 2. 

Corollary 2: Let θ ≥ 0 and τ = s + t be a positive real number with τ > 2. Let 𝔤: 𝒲 → 𝒵 be an even 

mapping with 𝔤(0) = 0 satisfying  

∥ D𝔤(ω1, ω2), υ ∥≤ θ(∥ ω1 ∥τ +∥ ω2 ∥τ +∥ ω1 ∥s. ∥ ω2 ∥t), 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Then there is a unique 𝑄2
́  mapping 𝑄2

́ : 𝒲 → 𝒵 such that  

∥ 𝔤(ω1) − 𝑄2
́ (ω1), υ ∥≤

|𝔪|τ

|𝔪|2(|𝔪|2 − |𝔪|τ)

θ ∥ ω1 ∥τ

|2|
 

for all ω1 ∈ 𝒲, υ ∈ 𝒵.  

Proof: Assuming  

Φ(ω1, ω2) ∶= θ  (‖ω1‖τ + ‖ω2‖τ+∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, and by choosing £ = |𝔪|τ−2,  the expected result can be obtained by Theorem 3. 

Example 3: Let ρ > 2 be a prime number and 𝒲 = 𝒵 = ℚ
𝔭
. Define 𝔤: 𝒲 → 𝒵 by 𝔤(ω_1) = ω1

2 + 1 

for all ω1 ∈ 𝒲. Since |2n|ρ = 1. 

|D𝔤(ω1, ω2)| = |
88

9
| ≤ θ(‖ω1‖τ + ‖ω2‖τ+∥ ω1 ∥s. ∥ ω2 ∥t)   (∀  ω1, ω2 ∈ 𝒲)    

and  

‖
h(2

n
ω1)

2
2n

−
h(2

n−1
ω1)

2
2(n−1)

‖ = |9| ≠ 0. 

Hence {2
−2n

h(2
n
ω1)} is not a Cauchy sequence. Where h(ω1)=𝔤(2ω1) − 4𝔤(ω1).  

Stability of the FE Eq 2: Odd case 

Theorem 4:  Let Φ: 𝒲 × 𝒲 → [0, ∞) be an odd function such that there is a constant 0 < £ < 1 with  

                                                          Φ(𝔪ω1, 𝔪ω2) ≤ |𝔪|3£Φ(ω1, ω2) 18 

for all ω1, ω2 ∈ 𝒲. Let 𝔤: 𝒲 → 𝒵 be an odd mapping satisfying  

                                                             ‖D𝔤(ω1, ω2), υ‖ ≤ Φ(ω1, ω2) 19 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵. Then there is a unique C3
́  mapping C3

́ : 𝒲 → 𝒵 such that  

                                                    ‖𝔤(ω1) − C3
́ (ω1), υ‖ ≤

1

|𝔪|3(1−£)
  Φ(ω1), 20 

where 

Φ(ω1) = max {
|𝔪2(𝔪 − 1)|

|2|. |4|
Φ(0, ω1),

|𝔪2(𝔪 − 2)|

|2(1 − 𝔪2)|
Φ(ω1, 0)} 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. 

Proof: Replacing ω1 = 0 in Eq 2, obtains 

                                                            ‖2𝔤(2ω2) − 16𝔤(ω2), υ‖ ≤ Φ(0, ω2) 21 

for all ω2 ∈ 𝒲, υ ∈ 𝒵. Interchanging ω2 by ω1 in Eq 21 and dividing on both sides of Eq 21 by 2, 

which gives  



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                                                            ‖𝔤(2ω1) − 8𝔤(ω1), υ‖ ≤
1

|2|
  Φ(0, ω1) 22 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Letting ω2 = 0 in Eq 2, it gives  

                               ‖2(1 − 𝔪2)𝔤(ω1) +
2(1−𝔪2)

𝔪2(𝔪−2)
𝔤(𝔪ω1) +

𝔪3−𝔪2−𝔪+1

2(𝔪−2)
𝔤(2ω1), υ‖  ≤ Φ(ω1, 0)  23 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Therefore  

                                ‖𝔪2(𝔪 − 2)𝔤(ω1) + 𝔤(𝔪ω1) −
𝔪2(𝔪−1)

4
𝔤(2ω1), υ‖ ≤

|𝔪2(𝔪−2)|

|2(1−𝔪2)|
Φ(ω1, 0) 24 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. According to  Eq 22 and Eq 24  

 ∥ 𝔤(𝔪ω1) − 𝔪3𝔤(ω1), υ ∥≤ max {
|𝔪2(𝔪−1)|

|2|.|4|
Φ(0, ω1),

|𝔪2(𝔪−2)|

|2(1−𝔪2)|
Φ(ω1, 0)}   25 

where  

Φ(ω1) = max {
|𝔪2(𝔪 − 1)|

|2|. |4|
Φ(0, ω1),

|𝔪2(𝔪 − 2)|

|2(1 − 𝔪2)|
Φ(ω1, 0)}. 

Then, 

                                                             ∥ 𝔤(𝔪ω1) − 𝔪3𝔤(ω1), υ ∥≤ Φ(ω1) 26 

 for all ω1 ∈ 𝒲, υ ∈ 𝒵. According to Eq 26, obtains  

                                                        ‖
𝔤(𝔪ω1)

𝔪3
− 𝔤(ω1), υ‖ ≤

1

|𝔪|3
Φ(ω1) 27 

 for all ω1 ∈ 𝒲, υ ∈ 𝒵. Consider the set  

                                                                  Γ = {𝔣: 𝒲 → 𝒵} 28 

 and define the generalized metric ð in Γ by  

 ð(𝔣, 𝔥) = inf{σ ∈ (0, ∞): ‖𝔣(ω1) − 𝔥(ω1), υ‖ ≤ σ  Φ(ω1), ∀  ω1 ∈ 𝒲, υ ∈ 𝒵}. 29 

 It is simple to prove that (Γ, ð) is complete [26].  

Now, define the function Λ: Γ → Γ such that  

                                                                 Λ𝔣(ω1) =
1

𝔪3
 𝔣(𝔪ω1) 30 

 for all ω1 ∈ 𝒲. Let 𝔣, 𝔥 ∈ Γ be given such that ð(𝔣, 𝔥) = ϵ. Then  

                                                           ‖𝔣(ω1) − 𝔥(ω1), υ‖ ≤ ϵ  Φ(ω1) 31 

 for all ω1 ∈ 𝒲, υ ∈ 𝒵. Hence  

‖Λ𝔣(𝔪ω1) − Λ𝔥(𝔪ω1), υ‖ = ‖
1

𝔪3
𝔣(𝔪ω1) −

1

𝔪3
𝔥(𝔪ω1), υ‖ 

                      ≤
1

|𝔪|3
  ϵ  Φ(𝔪ω1) 

                                 ≤
1

|𝔪|3
  ϵ  |𝔪|3£  Φ(ω1) 

             ≤ ϵ  £  Φ(ω1) 



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for all ω1 ∈ 𝒲, υ ∈ 𝒵, that is ð(Λ𝔣, Λ𝔥) ≤ £ε. Therefore  

ð(Λ𝔣, Λ𝔥) ≤ £  ð(𝔣, 𝔥) 

for all 𝔣, 𝔥 ∈ Γ. According to Eq 27  

                                                                     ð(𝔤, Λ𝔤) ≤
1

|𝔪|3
< +∞. 32 

 By Theorem 1, there is a function C3
́ : 𝒲 → 𝒵 satisfying the following conditions: 

    (1)  C3
́  is a fixed point of Λ, that is,  

                                                                    C3
́ (𝔪ω1) = 𝔪3  C3

́ (ω1) 33 

         for all ω1 ∈ 𝒲. C3
́  is a unique fixed point of the set denoted by Λ   

 S = {𝔥 ∈ Γ ∶ ð(𝔣, 𝔥) < ∞}. 

      This indicates that C3
́  is a unique mapping satisfying Eq 33 such that there is a σ ∈ (0, ∞) satisfying  

 ‖𝔤(ω1) − C3
́ (ω1), υ‖ ≤ σ Φ(ω1)                                  ∀  ω1 ∈ 𝒲, υ ∈ 𝒵. 

    (2)  ð(Λ
n𝔤, C3

́ ) → 0 as n → ∞. This indicates the equality,  

 lim
n→∞

(Λ
n𝔤)(ω1) = lim

n→∞

𝔤(𝔪nω1)

𝔪3n
= C3

́ (ω1),                   ∀  ω1 ∈ 𝒲. 

    (3)  ð(𝔤, C3
́ ) ≤

1

1−£
ð(𝔤, Λn𝔤), which implies  

ð(𝔤, C3
́ ) ≤

1

1 − £
ð(𝔤, Λ𝔤) ≤

1

|𝔪|3

1

(1 − £)
.          

         This indicates that the inequality Eq 20 remains valid.  

 According to Eq 18 and Eq 19  

∥ DC3
́ (ω1, ω2), υ ∥= lim

n→∞
∥ 𝔪−3nD𝔤(𝔪nω1, 𝔪nω2), υ ∥ 

                 ≤ lim
n→∞

1

|𝔪|3n
Φ(𝔪nω1, 𝔪nω2) 

                    ≤ lim
n→∞

1

|𝔪|3n
£

n|𝔪|3nΦ(ω1, ω2) 

     ≤ lim
n→∞

£
n
Φ(ω1, ω2) = 0. 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵 and n ∈ ℕ. So ∥ DC3
́ (ω1, ω2), υ ∥= 0.  

Thus the mapping C3
́ : 𝒲 → 𝒵 is C3

́  as desired. 

Corollary 3: Let θ ≥ 0 and τ = s + t be a positive real number with τ < 3. Let 𝔤: 𝒲 → 𝒵 be an odd 

mapping with 𝔤(0) = 0 satisfying  

 ∥ D𝔤(ω1, ω2), υ ∥≤ θ(∥ ω1 ∥τ +∥ ω2 ∥τ +∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵. Then there is a unique C3
́  mapping C3

́ : 𝒲 → 𝒵 such that  

 ∥ 𝔤(ω1) − C3
́ (ω1), υ ∥≤

|𝔪|τ

|𝔪|3(|𝔪|τ−|𝔪|3)
max {

|𝔪2(𝔪−1)|

|2|.|4|
θ ∥ ω1 ∥τ,

|𝔪2(𝔪−2)|

|2(1−𝔪2)|
θ ∥ ω1 ∥τ} 

for all ω1 ∈ 𝒲, υ ∈ 𝒵.  



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Proof: Assuming  

 Φ(ω1, ω2) ∶= θ  (‖ω1‖τ + ‖ω2‖τ+∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, and by choosing £ = |𝔪|3−τ, the expected result can be obtained by Theorem 4. 

Theorem 5:  Let Φ: 𝒲 × 𝒲 → [0, ∞) be an odd function such that there is a constant 0 < £ < 1 with  

                                                                   Φ (
ω1

𝔪
,

ω2

𝔪
) ≤

£

|𝔪|3
Φ(ω1, ω2)  34 

for all ω1, ω2 ∈ 𝒲. Let 𝔤: 𝒲 → 𝒵 be an odd mapping satisfying Eq 19. Then there is a unique C3
́  

mapping C3
́ : 𝒲 → 𝒵 such that  

                                                         ‖𝔤(ω1) − C3
́ (ω1), υ‖ ≤

£

|𝔪|3(1−£)
    Φ(ω1) 35 

where 

Φ(ω1) = max {
|𝔪2(𝔪 − 1)|

|2|. |4|
Φ(0, ω1),

|𝔪2(𝔪 − 2)|

|2(1 − 𝔪2)|
Φ(ω1, 0)} 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. 

Proof: According to Eq 26,  

                                                          ‖𝔤(ω1) − 𝔪3𝔤 (
ω1

𝔪
) , υ‖ ≤ Φ (

ω1

𝔪
) 36 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. 

Let (Γ, ð) be the GMS as defined by Theorem 2. 

 Now, define the function Λ: Γ → Γ such that  

                                                                   Λ𝔣(ω1) = 𝔪3𝔣 (
ω1

𝔪
) 37 

for all ω1 ∈ 𝒲. Let 𝔣, 𝔥 ∈ Γ, be given such that ð(𝔣, 𝔥) = ϵ. Then  

                                                         ∥ 𝔣(ω1) − 𝔥(ω1), υ ∥≤ ϵ Φ (
ω1

𝔪
) 38 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Hence 

∥ Λ𝔣(ω1) − Λ𝔥(ω1), υ ∥= ‖𝔪3𝔣 (
ω1

𝔪
) − 𝔪3𝔥 (

ω1

𝔪
) , υ‖ 

              ≤ |𝔪|3ϵ Φ (
ω1

𝔪
) 

                     ≤ |𝔪|3ϵ 
£

|𝔪|3
. Φ(ω1) 

 for all ω1 ∈ 𝒲, υ ∈ 𝒵, that is ð(Λ𝔣, Λ𝔥) ≤ £ϵ. Therefore 

        ð(Λ𝔣, Λ𝔥) ≤ £ð(𝔣, 𝔥) 39 

 for all 𝔣, 𝔥 ∈ Γ. According to Eq 36,  

                                                                ð(𝔤, Λ𝔤) ≤
£

|𝔪|3
< ∞. 40 

 So  



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                                     ð(𝔤, C3
́ ) ≤

£

|𝔪|3(1−£)
. 

This gives us the inequality Eq 35. This proof follows the same pattern as Theorem 4. 

Corollary 4: Let θ ≥ 0 and τ = s + t be a positive real number with τ > 3. Let 𝔤: 𝒲 → 𝒵 be an odd 

mapping with 𝔤(0) = 0 satisfying  

 ∥ D𝔤(ω1, ω2), υ ∥≤ θ(∥ ω1 ∥τ +∥ ω2 ∥τ +∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵. Then there is a unique C3
́  mapping C3

́ : 𝒲 → 𝒵 such that  

 ∥ 𝔤(ω1) − C3
́ (ω1), υ ∥≤

|𝔪|τ

|𝔪|3(|𝔪|3−|𝔪|τ)
max {

|𝔪2(𝔪−1)|

|2|.|4|
θ ∥ ω1 ∥τ,

|𝔪2(𝔪−2)|

|2(1−𝔪2)|
θ ∥ ω1 ∥τ}  

for all ω1 ∈ 𝒲, υ ∈ 𝒵.  

Proof: Assuming  

 Φ(ω1, ω2): = θ  (‖ω1‖τ + ‖ω2‖τ+∥ ω1 ∥s. ∥ ω2 ∥t) 

for all ω1, ω2 ∈ 𝒲, and by choosing £ = |𝔪|τ−3, the expected result can be obtained by Theorem 5. 

Example 4: Let ρ > 2 be a prime number and 𝒲 = 𝒵 = ℚ
𝔭
. Define 𝔤: 𝒲 → 𝒵 by 𝔤(ω1) = ω1

3 + 1 

for all ω1 ∈ 𝒲. Since |2n|ρ = 1. 

 |D𝔤(ω1, ω2)| = |
88

9
| ≤ θ(‖ω1‖τ + ‖ω2‖τ+∥ ω1 ∥s. ∥ ω2 ∥t)             (∀  ω1, ω2 ∈ 𝒲), 

and  

 ‖
h(2nω1)

23n −
h(2n−1ω1)

23(n−1) ‖ = |49| ≠ 0. 

Hence {2
−3n

h(2
n
ω1)} is not a Cauchy sequence. Where h(ω1)=𝔤(2ω1) − 8𝔤(ω1).  

 

Stability of the FE Eq 2: Mixed case 

Our goal in this section will be to establish the generalized H-U stability of the 𝑄2
́ − C3

́  FE Eq 2, in 

NA 2-normed spaces. For a given mapping 𝔤: 𝒲 → 𝒵, let  

 𝔤o(ω1) =
𝔤(ω1)−𝔤(−ω1)

2
    and   𝔤e(ω1) =

𝔤(ω1)+𝔤(−ω1)

2
. 

Then 𝔤o is odd and 𝔤e is even. 

Theorem 6:  Let Φ: 𝒲 × 𝒲 → [0, ∞) be a function such that there is a constant 0 < £ < 1 with   

                                                         Φ(𝔪ω1, 𝔪ω2) ≤ |𝔪|3£  Φ(ω1, ω2)                                       41 

for all ω1, ω2 ∈ 𝒲. Suppose 𝔤: 𝒲 → 𝒵 is a mapping satisfying the  inequality  

                                     ‖D𝔤(ω1, ω2), υ‖ ≤ Φ(ω1, ω2)    42 

for all ω1, ω2 ∈ 𝒲, υ ∈ 𝒵. Then there are a unique 𝑄2
́  mapping 𝑄2

́ : 𝒲 → 𝒲 and a unique C3
́  mapping 

C3
́ : 𝒲 → 𝒵 such that  

∥ 𝔤(ω1) − 𝑄2
́ (ω1) − C3

́ (ω1), υ ∥ 

      ≤ max {
1

|2|
max {

1

|2|.|𝔪|2.(1−£)
Φ(0, ω1) Φ(0, −ω1)} ,

1

|2|
max {

1

|𝔪|3(1−£)
  Φ(ω1),

1

|𝔪|3(1−£)
  Φ(−ω1)}} 43 



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where  Φ(ω1) = max {
|𝔪2(𝔪−1)|

|2|.|4|
Φ(0, ω1),

|𝔪2(𝔪−2)|

|2(1−𝔪2)|
Φ(ω1, 0)} 

for all ω1 ∈ 𝒲, υ ∈ 𝒵.  

Proof: Assume that 𝔤(ω1) = 𝔤e(ω1) + 𝔤o(ω1). Let  

 Φ(ω1, ω2) =
1

|2|
max{Φ(ω1, ω2), ϕ(−ω1, −ω2)} 

then by Eq 41, and Eq 42, which gives  

 Φ(𝔪ω1, 𝔪ω2) ≤ |𝔪|3  £  Φ(ω1, ω2) ≤ |𝔪|2  £  Φ(ω1, ω2) 

 ∥ D𝔤o(ω1, ω2  ), υ ∥≤ Φ(ω1, ω2) ,    ∥ D𝔤e(ω1, ω2), υ ∥≤ Φ(ω1, ω2), 

Hence by Theorem 2 and Theorem 4, there are a unique 𝑄2
́  mapping 𝑄2

́ : 𝒲 → 𝒵 and a unique C3
́  

mapping C3
́ : 𝒲 → 𝒵 such that  

∥ 𝔤o(ω1) − 𝑄2
́ (ω1), υ ∥≤

1

|2|. |𝔪|2. (1 − £)
Φ(0, ω1) 

 and  

∥ 𝔤e(ω1) − C3
́ (ω1), υ ∥≤

1

|𝔪|3(1 − £)
  Φ(ω1) 

for all ω1 ∈ 𝒲, υ ∈ 𝒵. Therefore  

∥ 𝔤(ω1) − C3
́ (ω1) − 𝑄2

́ (ω1), υ ∥= ‖𝔤o(ω1) + 𝔤e(ω1) − C3
́ (ω1) − 𝑄2

́ (ω1), υ‖ 

              ≤ max{∥ 𝔤o(ω_1) − C3
́ (ω_1), υ ∥, ∥ 𝔤e(ω_1) − 𝑄2

́ (ω_1), υ ∥} 

               ≤

max {
1

|2|
max {

1

|2|.|𝔪|2.(1−£)
Φ(0, ω1),

1

|2|.|𝔪|2.(1−£)
Φ(0, −ω1)} ,

1

|2|
max {

1

|𝔪|3(1−£)
  Φ(ω1),

1

|𝔪|3(1−£)
  Φ(−ω1)}}   

for all ω1 ∈ 𝒲, υ ∈ 𝒵. This completes the proof. 

4. Conclusion 

Lorem In this article, we investigated the H-U stability of the quadratic-cubic functional equation (Eq 

2) in NA 2-normed spaces using fixed-point methods and provided a suitable counterexample. In light 

of this work, it has become possible to gain a more comprehensive understanding of the stability 

problems of functional equations within the context of NA 2-normed spaces. 

Conflict of interest  

The authors declare that they have no competing interests. 

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https://doi.org/10.3390/math10111938
http://dx.doi.org/10.1007/s40863-021-00220-9
https://dx.doi.org/10.21123/bsj.2023.8230
https://doi.org/10.4236/oalib.1109821
http://dx.doi.org/10.21123/bsj.2021.18.1(Suppl.).0812
https://doi.org/10.1515/dema-2020-0009
https://doi.org/10.1515/dema-2020-0009
https://www.hrpub.org/download/20221030/MS7-13428783.pdf
https://doi.org/10.22055/jamm.2020.28513.1685
https://doi.org/10.1073/pnas.27.4.222
https://projecteuclid.org/journalArticle/Download?urlId=10.2969%2Fjmsj%2F00210064
http://dx.doi.org/10.1090/S0002-9939-1978-0507327-1
https://doi.org/10.1006/jmaa.1994.1211
http://dx.doi.org/10.1002/mana.19640280102
https://doi.org/10.1002/mana.19690420104
https://doi.org/10.11568/kjm.2014.22.2.339
http://www.aimspress.com/article/doi/10.3934/math.2021116
https://pjm.ppu.edu/sites/default/files/papers/PJM_May_%283%292022_586_to_597.pdf
https://journalofinequalitiesandapplications.springeropen.com/counter/pdf/10.1155/2010/403101.pdf
https://doi.org/10.1155/2010/403101
https://doi.org/10.3390/math10111938
http://resolver.sub.uni-goettingen.de/purl?PPN37721857X_0006
http://www.emis.de/journals/JIPAM/images/075_02_JIPAM/075_02.pdf
https://projecteuclid.org/journalArticle/Download?urlId=bams%2F1183529535