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Affine Type Aq Functional Equation In Various Banach Spaces

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01 July 2024

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04 July 2024

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Abstract
In this paper, we analyze the generalized Ulam-Hyers stability of affine type AQ Functional Equation in various Banach Spaces using Direct and Fixed Methods
Keywords: 
Subject: Computer Science and Mathematics  -   Analysis

1. Introduction

A inspirational and popular talk presented by S.M Ulam [37] in 1940, refreshed the reading of stability problems for various functional equations. He gave a wide range of talk before a Mathematical Colloquium at the University of Wisconsin in which he presented a list of unsolved problems.
The first assertive answer to Ulam’s question concerning the problem of stability of functional equations was given by D.H. Hyers [20] for the case of additive mappings in Banach spaces. In growth of time, the theorem delivered by Hyers was generalized by T. Aoki [3], Th.M Rassias [30], J.M. Rassias [28], P. Gavruta [19] for additive mappings and K. Ravi [32] for quadratic mappings.
The famous additive and quadratic functional equations are
F ( w 1 + w 2 ) = F ( w 1 ) + F ( w 2 ) ,
and
F ( w 1 + w 2 ) + F ( w 1 − w 2 ) = 2 F ( w 1 ) + 2 F ( w 2 ) .
The general solution and generalized Ulam - Hyers stability of several types of functional equations in various normed spaces were discussed by many authors one can see [2,16,18,21,22,31] and references there in.
Also, the general solution and Hyers-Ulam-Rassias stability of the several affine functional equations are discussed by L. Lucht, C. Methfessel [23], L. Cadariu, L. Gavruta, P. Gavruta [15], Md. Nasiruzzaman [26], M. Mursaleen, KJ. Ansari[25].
Infact, the general solution and generalized Hyers-Ulam stability of the several AQ functional equations are established in [4,5,6,7,8,9,10,11,12,14,29].
In this paper, the we analyze the generalized Ulam-Hyers stability of affine type AQ Functional Equation of the form
F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) = 6 F ∑ ψ = 1 3 w ψ + 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ − ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ )
in various Banach Spaces using Direct and Fixed Methods .
Lemma 1.1. 
[26] Let A and B be real vector spaces. Suppose F : A → B be an odd mapping satisfies (3) then F is additive.
Lemma 1.2. 
[17] Let A and B be real vector spaces. Suppose F : A → B be an even mapping satisfies (3) then F is quadratic.
Now, we present the result due to Margolis, Diaz [24] and Radu [27] for fixed point theory.
Theorem 1.3. 
[24,27] Suppose that for a complete generalized metric space ( Ω , δ ) and a strictly contractive mapping T : Ω ⟶ Ω with Lipschitz constant L. Then, for each given x ∈ Ω , either
d ( T n x , T n + 1 x ) = ∞ ∀ n ≥ 0 ,
or there exists a natural number n 0 such that
  • (FPC1) d ( T n x , T n + 1 x ) < ∞ for all n ≥ n 0 ;
  • (FPC2) The sequence ( T n x ) is convergent to a fixed point y * of T
  • (FPC3) y * is the unique fixed point of T in the set Δ = { y ∈ Ω : d ( T n 0 x , y ) < ∞ } ;
  • (FPC4) d ( y * , y ) ≤ 1 1 − L d ( y , T y ) for all y ∈ Δ .

2. Stability In Banach Space of (3)

In this section, we explore the generalized Ulam - Hyers stability of the functional equation (3) in Banach space. To prove stability results, let us take W 1 be a normed space and W 2 be a Banach space. Suppose that F : W 1 → W 2 and Ψ : W 1 3 → [ 0 , ∞ ) satisfying the following functional inequalities
∥ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ − 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ ) ∥ ≤ Ψ w 1 , w 2 , w 3 ,
and
∥ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ − 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ ) ∥ ≤ δ , δ ∑ ψ = 1 3 w ψ φ , δ ∑ ψ = 1 3 w ψ φ   ψ , δ ∏ ψ = 1 3 w ψ φ , δ ∏ ψ = 1 3 w ψ φ   ψ , δ ∑ ψ = 1 3 w ψ 3 φ + ∏ ψ = 1 3 w ψ φ ,
for all w 1 , w 2 , w 3 ∈ W 1 and δ be a positive constant.

2.1. Oddness of F : Additive Case Stability Results : Direct Method

Theorem 2.1. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the condition
lim ℓ → ∞ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 5 ℓ μ = 0 ; μ = ± 1
for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − A ( w 1 ) ≤ 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ Ψ A 5 η μ w 1
= 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 3 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1
and the mapping A ( w 1 ) is obtained by
A ( w 1 ) = lim ℓ → ∞ 1 5 ℓ μ F 5 ℓ μ w 1
for all w 1 ∈ W 1 .
Proof. 
Using oddness of F in (1), we get
∥ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) ∥ ≤ Ψ w 1 , w 2 , w 3 , ∀ w 1 , w 2 , w 3 ∈ W 1 .
Interchanging w 1 , w 2 , w 3 by w 1 , w 1 , w 1 in (7), we obtain
∥ 3 F ( 5 w 1 ) − 6 F 3 w 1 + 3 F ( w 1 ) ∥ ≤ Ψ w 1 , w 1 , w 1 , ∀ w 1 ∈ W 1 .
Again interchanging w 1 , w 2 , w 3 by w 1 , w 1 , − w 1 in (7), we have
∥ 2 F ( 3 w 1 ) − 6 F w 1 ∥ ≤ Ψ w 1 , w 1 , − w 1 ⇒ ∥ 6 F ( 3 w 1 ) − 18 F w 1 ∥ ≤ 3 Ψ w 1 , w 1 , − w 1 , ∀ w 1 ∈ W 1 .
Combining (8) and (9), we arrive
∥ 3 F ( 5 w 1 ) − 15 F ( w 1 ) ∥ ≤ ∥ 3 F ( 5 w 1 ) − 6 F 3 w 1 + 3 F ( w 1 ) ∥ + ∥ 6 F ( 3 w 1 ) − 18 F w 1 ∥ ≤ Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 , ∀ w 1 ∈ W 1 .
One can see from (10) that
∥ F ( 5 w 1 ) − 5 F ( w 1 ) ∥ ≤ 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = Ψ A w 1 , ∀ w 1 ∈ W 1 .
It follows from (11) that
∥ 1 5 F ( 5 w 1 ) − F ( w 1 ) ∥ ≤ 1 5 Ψ A w 1 , ∀ w 1 ∈ W 1 .
Generalizing for a positive integer ℓ, we get
∥ 1 5 ℓ F ( 5 ℓ w 1 ) − F ( w 1 ) ∥ ≤ 1 5 ∑ η = 0 ℓ 1 5 η Ψ A 5 η w 1 , ∀ w 1 ∈ W 1 .
Now, changing w 1 by 5 ℓ 1 w 1 in (13), we obtain
∥ 1 5 ℓ + ℓ 1 F ( 5 ℓ + ℓ 1 w 1 ) − 1 5 ℓ 1 F ( 5 ℓ 1 w 1 ) ∥ = 1 5 ℓ 1 ∥ 1 5 ℓ F ( 5 ℓ + ℓ 1 w 1 ) − F ( 5 ℓ 1 w 1 ) ∥ ≤ 1 5 ∑ η = 0 ℓ 1 5 η + ℓ 1 Ψ A 5 η + ℓ 1 w 1 → 0 a s ℓ 1 → ∞ , ∀ w 1 ∈ W 1 .
Therefore, the sequence
1 5 ℓ F ( 5 ℓ w 1 ) ,
is a Cauchy sequence and it converges to A ( w 1 ) in W 2 . So, we define
A ( w 1 ) = lim ℓ → ∞ 1 5 ℓ F 5 ℓ w 1 , ∀ w 1 ∈ W 1 .
Taking limit ℓ → ∞ in (13), we have
∥ A ( w 1 ) − F ( w 1 ) ∥ ≤ 1 5 ∑ η = 0 ∞ 1 5 η Ψ A 5 η w 1 , ∀ w 1 ∈ W 1 .
Thus, (4) and (5) holds for μ = 1 . Interchanging
w 1 , w 2 , w 3 = 5 ℓ w 1 , 5 ℓ w 2 , 5 ℓ w 3 ,
we arrive
1 5 ℓ ∥ F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) − 6 F ∑ ψ = 1 3 5 ℓ w ψ − 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ + ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) ∥ ≤ 1 5 ℓ Ψ 5 ℓ w 1 , 5 ℓ w 2 , 5 ℓ w 3 , ∀ w 1 , w 2 , w 3 ∈ W 1 .
Taking limit ℓ → ∞ in (17), using (15) and (3), we get
A ( 3 w 1 + w 2 + w 3 ) + A ( w 1 + 3 w 2 + w 3 ) + A ( w 1 + w 2 + 3 w 3 ) = 6 A ∑ ψ = 1 3 w ψ + 1 2 A ∑ ψ = 1 3 w ψ + A − ∑ ψ = 1 3 w ψ − ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ )
for all w 1 , w 2 , w 3 ∈ W 1 . So, A ( w 1 ) satisfies (3). In order to confirm that A ( w 1 ) is unique, suppose B ( w 1 ) be another mapping (3), (15) and (16), we obtain
∥ A ( w 1 ) − B ( w 1 ) ∥ = ∥ 1 5 ℓ A 5 ℓ w 1 − 1 5 ℓ B 5 ℓ w 1 ∥ ≤ 1 5 ℓ ∥ A 5 ℓ w 1 − F 5 ℓ w 1 ∥ + 1 5 ℓ ∥ F 5 ℓ w 1 − B 5 ℓ w 1 ∥ ≤ 2 5 ∑ η = 0 ∞ 1 5 η + ℓ Ψ A 5 η + ℓ w 1 → 0 a s ℓ 1 → ∞ ,
for all w 1 ∈ W 1 . Therefore A ( w 1 ) is unique. So, the Theorem holds for μ = 1 .
Changing w 1 = w 1 5 in (11), we have
∥ F ( w 1 ) − 5 F w 1 5 ∥ ≤ Ψ A w 1 5 , ∀ w 1 ∈ W 1 .
Generalizing for a positive integer ℓ, we get
∥ F ( w 1 ) − 5 ℓ F w 1 5 η ∥ ≤ 1 5 ∑ η = 1 ℓ 5 η Ψ A w 1 5 η , ∀ w 1 ∈ W 1 .
The rest of the proof is similar to that of above case. So, the Theorem holds for μ = − 1 . Hence the proof is complete □
Corollary 2.2. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − A ( w 1 ) ≤ δ | 3 | , 4 δ | w 1 | φ | 5 − 5 φ | ; φ ≠ 1 , 4 δ 3 ∑ ψ = 1 3 | w ψ | φ   ψ | 5 − 5 φ   ψ | ; φ 1 , φ 2 , φ 3 ≠ 1 , 4 δ | w 1 | 3 φ 3 | 5 − 5 3 φ | ; 3 φ ≠ 1 , 4 δ | w ψ | ∑ ψ = 1 3 φ   ψ 3 | 5 − 5 ∑ ψ = 1 3 φ   ψ | ; ∑ ψ = 1 3 φ   ψ ≠ 1 , 16 δ | w 1 | 3 φ 3 | 5 − 5 3 φ | ; 3 φ ≠ 1 ,
for all w 1 ∈ W 1 .

2.2. Evenness of F : Quadratic Case Stability Results : Direct Method

Theorem 2.3. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the condition
lim ℓ → ∞ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 25 ℓ μ = 0 ; μ = ± 1
for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − Q ( w 1 ) ≤ 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ Ψ Q 5 η μ w 1
= 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 7 2 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1
and the mapping Q ( w 1 ) is obtained by
Q ( w 1 ) = lim ℓ → ∞ 1 25 ℓ μ F 5 ℓ μ w 1
for all w 1 ∈ W 1 .
Proof. 
Using evenness of F in (1), we get
∥ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 7 F ∑ ψ = 1 3 w ψ − 4 ∑ ψ = 1 3 F ( w ψ ) ∥ ≤ Ψ w 1 , w 2 , w 3 , ∀ w 1 , w 2 , w 3 ∈ W 1 .
Interchanging w 1 , w 2 , w 3 by w 1 , w 1 , w 1 in (25), we obtain
∥ 3 F ( 5 w 1 ) − 7 F 3 w 1 − 12 F ( w 1 ) ∥ ≤ Ψ w 1 , w 1 , w 1 , ∀ w 1 ∈ W 1 .
Again interchanging w 1 , w 2 , w 3 by w 1 , w 1 , − w 1 in (25), we have
∥ 2 F ( 3 w 1 ) − 18 F w 1 ∥ ≤ Ψ w 1 , w 1 , − w 1 ⇒ ∥ 7 F ( 3 w 1 ) − 63 F w 1 ∥ ≤ 7 2 Ψ w 1 , w 1 , − w 1 , ∀ w 1 ∈ W 1 .
Combining 26 and (27), we arrive
∥ 3 F ( 5 w 1 ) − 75 F ( w 1 ) ∥ ≤ ∥ 3 F ( 5 w 1 ) − 7 F 3 w 1 − 12 F ( w 1 ) ∥ + ∥ 7 F ( 3 w 1 ) − 63 F w 1 ∥ ≤ Ψ w 1 , w 1 , w 1 + 7 2 Ψ w 1 , w 1 , − w 1 , ∀ w 1 ∈ W 1 .
One can see from (28) that
∥ F ( 5 w 1 ) − 25 F ( w 1 ) ∥ ≤ 1 3 Ψ w 1 , w 1 , w 1 + 7 2 Ψ w 1 , w 1 , − w 1 = Ψ Q w 1 , ∀ w 1 ∈ W 1 .
It follows from (29) that
∥ 1 25 F ( 5 w 1 ) − F ( w 1 ) ∥ ≤ 1 25 Ψ Q w 1 , ∀ w 1 ∈ W 1 .
The rest of the proof is similar to that of Theorem 2.1. Hence the proof is complete. □
Corollary 2.4. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − Q ( w 1 ) ≤ 3 δ 2 | 24 | , 27 δ | w 1 | φ 6 | 25 − 5 φ | ; φ ≠ 2 , 9 δ 6 ∑ ψ = 1 3 | w ψ | φ   ψ | 25 − 5 φ   ψ | ; φ 1 , φ 2 , φ 3 ≠ 2 , 9 δ | w 1 | 3 φ 6 | 25 − 5 3 φ | ; 3 φ ≠ 2 , 9 δ | w ψ | ∑ ψ = 1 3 φ   ψ 6 | 25 − 5 ∑ ψ = 1 3 φ   ψ | ; ∑ ψ = 1 3 φ   ψ ≠ 2 , 36 δ | w 1 | 3 φ 6 | 25 − 5 3 φ | ; 3 φ ≠ 2 ,
for all w 1 ∈ W 1 .

2.3. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Direct Method

Theorem 2.5. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the conditions (3) and (21) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) ≤ 1 2 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ Ψ A 5 η μ w 1 + Ψ A − 5 η μ w 1 + 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ Ψ Q 5 η μ w 1 + Ψ Q − 5 η μ w 1 ≤ 1 2 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 3 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1 + 1 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , − 5 η μ w 1 + 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , 5 η μ w 1 + 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 7 2 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1 + 1 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , − 5 η μ w 1 + 7 2 Ψ − 5 η μ w 1 , − 5 η μ w 1 , 5 η μ w 1
and the mapping A ( w 1 ) and Q ( w 1 ) are given in (6) and (24) for all w 1 ∈ W 1 .
Proof. 
Consider a function F o d d ( w 1 ) by
F o d d ( w 1 ) = 1 2 F ( w 1 ) − F ( − w 1 ) , ∀ w 1 ∈ W 1 ,
which gives
F o d d ( 0 ) = 0 ; F o d d ( − w 1 ) = − F o d d ( w 1 ) , ∀ w 1 ∈ W 1 .
By Theorem 2.1, it follows from (33), (1), (5) and (6), we arrive
F o d d ( w 1 ) − A ( w 1 )
≤ 1 2 · 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ Ψ A 5 η μ w 1 + Ψ A − 5 η μ w 1 = 1 2 · 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 3 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1
+ 1 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , − 5 η μ w 1 + 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , 5 η μ w 1
for all w 1 , w 2 , w 3 ∈ W 1 . Consider a function F e v e n ( w 1 ) by
F e v e n ( w 1 ) = 1 2 F ( w 1 ) + F ( − w 1 ) , ∀ w 1 ∈ W 1 ,
which gives
F e v e n ( 0 ) = 0 ; F e v e n ( − w 1 ) = F e v e n ( w 1 ) , ∀ w 1 ∈ W 1 .
By Theorem 2.3, it follows from (37), (1), (22) and (23), we see
F e v e n ( w 1 ) − Q ( w 1 )
≤ 1 2 · 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ Ψ Q 5 η μ w 1 + Ψ Q − 5 η μ w 1 = 1 2 · 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 7 2 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1
+ 1 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , − 5 η μ w 1 + 7 2 Ψ − 5 η μ w 1 , − 5 η μ w 1 , 5 η μ w 1
for all w 1 , w 2 , w 3 ∈ W 1 . Assume a function F ( w 1 ) by
F ( w 1 ) = F o d d ( w 1 ) + F e v e n ( w 1 ) , ∀ w 1 ∈ W 1 .
Now, it follows from (35), (36), (39), (40) and (41), we have
F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) ≤ F o d d ( w 1 ) − A ( w 1 ) + F e v e n ( w 1 ) − Q ( w 1 ) ≤ 1 2 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ Ψ A 5 η μ w 1 + Ψ A − 5 η μ w 1 + 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ Ψ Q 5 η μ w 1 + Ψ Q − 5 η μ w 1 ≤ 1 2 1 5 ∑ η = 1 − μ 2 ∞ 1 5 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 3 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1 + 1 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , − 5 η μ w 1 + 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , 5 η μ w 1 + 1 25 ∑ η = 1 − μ 2 ∞ 1 25 η μ 1 3 Ψ 5 η μ w 1 , 5 η μ w 1 , 5 η μ w 1 + 7 2 Ψ 5 η μ w 1 , 5 η μ w 1 , − 5 η μ w 1 + 1 3 Ψ − 5 η μ w 1 , − 5 η μ w 1 , − 5 η μ w 1 + 7 2 Ψ − 5 η μ w 1 , − 5 η μ w 1 , 5 η μ w 1
for all w 1 , w 2 , w 3 ∈ W 1 . □
Corollary 2.6. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) ≤ δ | 3 | + 3 δ 2 | 24 | , 4 δ | w 1 | φ | 5 − 5 φ | + 27 δ | w 1 | φ 6 | 25 − 5 φ | ; φ ≠ 1 , 2 , 4 δ 3 ∑ ψ = 1 3 | w ψ | φ   ψ | 5 − 5 φ   ψ | + 9 δ 6 ∑ ψ = 1 3 | w ψ | φ   ψ | 25 − 5 φ   ψ | ; φ 1 , φ 2 , φ 3 ≠ 1 , 2 , 4 δ | w 1 | 3 φ 3 | 5 − 5 3 φ | + 9 δ | w 1 | 3 φ 6 | 25 − 5 3 φ | ; 3 φ ≠ 1 , 2 , 4 δ | w ψ | ∑ ψ = 1 3 φ   ψ 3 | 5 − 5 ∑ ψ = 1 3 φ   ψ | + 9 δ | w ψ | ∑ ψ = 1 3 φ   ψ 6 | 25 − 5 ∑ ψ = 1 3 φ   ψ | ; ∑ ψ = 1 3 φ   ψ ≠ 1 , 2 , 16 δ | w 1 | 3 φ 3 | 5 − 5 3 φ | + 36 δ | w 1 | 3 φ 6 | 25 − 5 3 φ | ; 3 φ ≠ 1 , 2 ,
for all w 1 ∈ W 1 .

2.4. Oddness of F : Additive Case Stability Results : Fixed Method

Theorem 2.7. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the condition
lim ℓ → ∞ Ψ τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 τ ν ℓ = 0 ; τ ν = 5 ; ν = 0 1 5 ; ν = 1 , ∀ w 1 , w 2 , w 3 ∈ W 1 .
If there exists L = L ( ν ) be a function have the property
Ψ A ( w 1 ) = Ψ A w 1 5 a n d 1 τ ν Ψ A τ ν w 1 = L Ψ A ( w 1 ) , ∀ w 1 ∈ W 1 .
Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1
= L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1
and the mapping A ( w 1 ) is obtained by
A ( w 1 ) = lim ℓ → ∞ 1 τ ν ℓ F τ ν ℓ w 1
for all w 1 ∈ W 1 .
Proof. 
Assume a set
G = { F / F : W 1 → W 2 , F ( 0 ) = 0 }
and introduce the generalized metric on the above set G as
d ( F , F 1 ) = inf { K ∈ ( 0 , ∞ ) : F ( w 1 ) − F 1 ( w 1 ) ≤ K Ψ ( w 1 , w 1 , w 1 ) , w 1 ∈ W 1 } .
It is easy to see that ( G , d ) is complete. Define a function H : G → G by
H F ( w 1 ) = 1 τ ν F ( τ ν w 1 ) , f o r a l l w 1 ∈ W 1 .
Now F , F 1 ∈ G and w 1 ∈ W 1 , we see
d ( F , F 1 ) ≤ K ⇒ ‖ F ( w 1 ) − F 1 ( w 1 ) ‖ ≤ K Ψ ( w 1 , w 1 , w 1 ) , ⇒ 1 τ ν F ( τ ν w 1 ) − 1 τ ν F 1 ( τ ν w 1 ) ≤ 1 τ ν K Ψ ( τ ν w 1 , τ ν w 1 , τ ν w 1 ) , ⇒ ‖ H F ( w 1 ) − H F 1 ( w 1 ) ‖ ≤ L K Ψ ( w 1 , w 1 , w 1 ) , ⇒ d ( H F , H F 1 ) ≤ L K ,
i.e., H is a strictly contractive mapping on G with Lipschitz constant L (see [24]).
For the case ν = 0 , it follows from (12) and with the help of (44), (50), (49), we get
∥ 1 5 F ( 5 w 1 ) − F ( w 1 ) ∥ ≤ 1 5 Ψ A w 1 , ⇒ d ( H F , F ) ≤ L = L 1 − ν , ∀ w 1 ∈ W 1 .
For the case ν = 1 , it follows from (18) and with the help of (44), (50), (49), we obtain
∥ F ( w 1 ) − 5 F w 1 5 ∥ ≤ Ψ A w 1 5 , ⇒ d ( F , H F ) ≤ 1 = L 1 − ν , ∀ w 1 ∈ W 1 .
Combining (51) and (52), we have
d ( F , H F ) ≤ 1 = L 1 − ν .
Therefore ( F P C 1 ) of Theorem 1.3 holds. The rest of the proof follows by Theorem 1.3. Hence the proof is complete. □
Corollary 2.8. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality (20) for all w 1 ∈ W 1 .
Proof. 
If we take
Ψ w 1 , w 2 , w 3 = δ , δ ∑ ψ = 1 3 w ψ φ , δ ∑ ψ = 1 3 w ψ φ   ψ , δ ∏ ψ = 1 3 w ψ φ , δ ∏ ψ = 1 3 w ψ φ   ψ , δ ∑ ψ = 1 3 w ψ 3 φ + ∏ ψ = 1 3 w ψ φ ,
in Theorem 2.7 and changing w 1 , w 2 , w 3 by τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 and dividing by τ ν ℓ in (54), one can see
1 τ ν ℓ Ψ τ ν ℓ w 1 , τ ν w 2 ℓ , τ ν ℓ w 3 = δ τ ν ℓ → 0 a s ℓ t o ∞ , δ τ ν ℓ ∑ ψ = 1 3 τ ν ℓ w ψ φ , → 0 a s ℓ t o ∞ , δ τ ν ℓ ∑ ψ = 1 3 τ ν ℓ w ψ φ   ψ , → 0 a s ℓ t o ∞ , δ τ ν ℓ ∏ ψ = 1 3 τ ν ℓ w ψ φ , → 0 a s ℓ t o ∞ , δ τ ν ℓ ∏ ψ = 1 3 τ ν ℓ w ψ φ   ψ , → 0 a s ℓ t o ∞ , δ τ ν ℓ ∑ ψ = 1 3 τ ν ℓ w ψ 3 φ + ∏ ψ = 1 3 τ ν ℓ w ψ φ , → 0 a s ℓ t o ∞ .
Therefore (43) holds for all w 1 , w 2 , w 3 ∈ W 1 . Now, from (44), we have
Ψ A ( w 1 ) = Ψ A w 1 5 = 1 3 Ψ w 1 5 , w 1 5 , w 1 5 + 3 Ψ w 1 5 , w 1 5 , − w 1 5 = 4 δ 3 , 12 | w 1 5 | φ 3 , 4 δ 3 ∑ ψ = 1 3 w 1 5 φ   ψ , 4 δ | w 1 5 | 3 φ 3 , 4 δ | w 1 5 | ∑ ψ = 1 3 φ   ψ 3 , 16 δ | w 1 5 | 3 φ 3 ,
1 τ ν Ψ A τ ν w 1 = 1 τ ν 1 3 Ψ τ ν w 1 , τ ν w 1 , τ ν w 1 + 3 Ψ τ ν w 1 , τ ν w 1 , − τ ν w 1 = 4 δ τ ν · 3 , 12 δ | τ ν w 1 | φ τ ν · 3 , 4 δ τ ν · 3 ∑ ψ = 1 3 τ ν w ψ φ   ψ , 4 δ | τ ν w 1 | 3 φ τ ν · 3 , 4 δ | τ ν w 1 | ∑ ψ = 1 3 φ   ψ τ ν · 3 , 16 δ | τ ν w 1 | 3 φ τ ν · 3 , = τ ν − 1 Ψ A ( w 1 ) , τ ν φ − 1 Ψ A ( w 1 ) , ∑ ψ = 1 3 τ ν φ   ψ − 1 Ψ A ( w 1 ) , τ ν 3 φ − 1 Ψ A ( w 1 ) , τ ν ∑ ψ = 1 3 φ   ψ − 1 Ψ A ( w 1 ) , τ ν 3 φ − 1 Ψ A ( w 1 ) , = L Ψ A ( w 1 ) L Ψ A ( w 1 ) L Ψ A ( w 1 ) L Ψ A ( w 1 ) L Ψ A ( w 1 ) L Ψ A ( w 1 )
for all w 1 ∈ W 1 .
For the case ν = 0 , we have L = τ 0 − 1 = 5 − 1 and from (46), we arrive
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1 = L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = ( 5 − 1 ) 1 − 0 1 − 5 − 1 4 δ 3 = δ 3 .
For the case ν = 1 , we have L = τ 1 − 1 = ( 1 5 ) − 1 = 5 and from (46), we obtain
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1 = L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = ( 5 ) 1 − 1 1 − 5 4 δ 3 = δ − 3 .
For the case ν = 0 , we have L = τ 0 φ − 1 = 5 φ − 1 and from (46), we arrive
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1 = L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = ( 5 φ − 1 ) 1 − 0 1 − 5 φ − 1 12 δ | w 1 5 | φ 3 = 4 δ 5 − 5 φ .
For the case ν = 1 , we have L = τ 1 φ − 1 = ( 1 5 ) φ − 1 = 5 1 − φ and from (46), we get
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1 = L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = ( 5 1 − φ ) 1 − 1 1 − 5 1 − φ 12 δ | w 1 5 | φ 3 = 4 δ 5 φ − 5 .
For the case ν = 0 , we have L = τ 0 3 φ − 1 = 5 3 φ − 1 and from (46), we arrive
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1 = L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = ( 5 3 φ − 1 ) 1 − 0 1 − 5 3 φ − 1 4 δ | w 1 5 | 3 φ 3 = 4 δ | w 1 | 3 φ 3 ( 5 − 5 3 φ ) .
For the case ν = 1 , we have L = τ 1 3 φ − 1 = ( 1 5 ) 3 φ − 1 = 5 1 − 3 φ and from (46), we obtain
F ( w 1 ) − A ( w 1 ) ≤ L 1 − ν 1 − L Ψ A w 1 = L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 = ( 5 1 − 3 φ ) 1 − 1 1 − 5 1 − 3 φ 4 δ | w 1 5 | 3 φ 3 = 4 δ | w 1 | 3 φ 3 ( 5 3 φ − 5 ) .
Similarly, we can prove for rest of the cases. □

2.5. Evenness of F : Quadratic Case Stability Results : Fixed Method

Theorem 2.9. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the condition
lim ℓ → ∞ Ψ τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 τ ν 2 ℓ = 0 ; τ ν = 5 ; ν = 0 1 5 ; ν = 1 , ∀ w 1 , w 2 , w 3 ∈ W 1 .
If there exists L = L ( ν ) be a function have the property
Ψ Q ( w 1 ) = Ψ Q w 1 5 a n d 1 τ ν 2 Ψ Q τ ν w 1 = L Ψ Q ( w 1 ) , ∀ w 1 ∈ W 1 .
for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − Q ( w 1 ) ≤ L 1 − ν 1 − L Ψ Q w 1
= L 1 − ν 1 − L 1 3 Ψ w 1 , w 1 , w 1 + 7 2 Ψ w 1 , w 1 , − w 1
and the mapping Q ( w 1 ) is obtained by
Q ( w 1 ) = lim ℓ → ∞ 1 τ ν 2 ℓ F τ ν ℓ w 1
for all w 1 ∈ W 1 .
Proof. 
By Theorem 2.7, define a function H : G → G by
H F ( w 1 ) = 1 τ ν 2 F ( τ ν w 1 ) , f o r a l l w 1 ∈ W 1 .
Now F , F 1 ∈ G and w 1 ∈ W 1 , we see
d ( F , F 1 ) ≤ K ⇒ ‖ F ( w 1 ) − F 1 ( w 1 ) ‖ ≤ K Ψ ( w 1 , w 1 , w 1 ) , ⇒ 1 τ ν 2 F ( τ ν w 1 ) − 1 τ ν 2 F 1 ( τ ν w 1 ) ≤ 1 τ ν 2 K Ψ ( τ ν w 1 , τ ν w 1 , τ ν w 1 ) , ⇒ ‖ H F ( w 1 ) − H F 1 ( w 1 ) ‖ ≤ L K Ψ ( w 1 , w 1 , w 1 ) , ⇒ d ( H F , H F 1 ) ≤ L K ,
i.e., H is a strictly contractive mapping on G with Lipschitz constant L (see [24]). The rest of the proof is similar to that of Theorem 2.7. Hence the proof is complete. □
Corollary 2.10. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality (31) for all w 1 ∈ W 1 .

2.6. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Fixed Method

Theorem 2.11. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the conditions (43) and (57) for all w 1 , w 2 , w 3 ∈ W 1 . If there exists L = L ( ν ) be function have the properties (44) and (58) Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
F ( w 1 ) − A ( w 1 ) − Q ( w 1 )
≤ 1 2 · L 1 − ν 1 − L Ψ A w 1 + Ψ A − w 1 + Ψ Q w 1 + Ψ Q − w 1 = 1 2 · L 1 − ν 1 − L 1 3 { Ψ w 1 , w 1 , w 1 + 3 Ψ w 1 , w 1 , − w 1 + Ψ − w 1 , − w 1 , − w 1 + 3 Ψ − w 1 , − w 1 , w 1
+ Ψ w 1 , w 1 , w 1 + 7 2 Ψ w 1 , w 1 , − w 1 + Ψ − w 1 , − w 1 , − w 1 + 7 2 Ψ − w 1 , − w 1 , w 1 }
and the mapping A ( w 1 ) and Q ( w 1 ) are given in (47) and (61) for all w 1 ∈ W 1 .
Proof. 
The proof is similar ideas to that of Theorem 2.5. □
Corollary 2.12. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality (42) for all w 1 ∈ W 1 .

3. Stability In Intuitionistic Fuzzy Banach Space of (3)

In this section, we explore the generalized Ulam - Hyers stability of the functional equations (3) in Intuitionistic Fuzzy Banach Space.
In order to prove stability results, assume W 1 , μ , ν and W 2 , μ ′ , ν ′ are Intuitionistic Fuzzy normed space and Intuitionistic Fuzzy Banach space respectively. Suppose that F : W 1 → W 2 and Ψ : W 1 3 → [ 0 , ∞ ) satisfying the following functional inequalities
μ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ − 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ ) , Λ ≥ μ ′ Ψ w 1 , w 2 , w 3 , Λ ν F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ − 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ ) , Λ ≤ ν ′ Ψ w 1 , w 2 , w 3 , Λ
and
μ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ − 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ ) , Λ ≥ μ ′ δ , Λ , μ ′ δ ∑ ψ = 1 3 w ψ φ , Λ , μ ′ δ ∑ ψ = 1 3 w ψ φ   ψ , Λ , μ ′ δ ∏ ψ = 1 3 w ψ φ , Λ , μ ′ δ ∏ ψ = 1 3 w ψ φ   ψ , Λ , μ ′ δ ∑ ψ = 1 3 w ψ 3 φ + ∏ ψ = 1 3 w ψ φ , Λ , ν F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ − 1 2 F ∑ ψ = 1 3 w ψ + F − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) − 5 2 F ( w ψ ) + F ( − w ψ ) , Λ ≤ ν ′ δ , Λ , ν ′ δ ∑ ψ = 1 3 w ψ φ , Λ , ν ′ δ ∑ ψ = 1 3 w ψ φ   ψ , Λ , ν ′ δ ∏ ψ = 1 3 w ψ φ , Λ , ν ′ δ ∏ ψ = 1 3 w ψ φ   ψ , Λ , ν ′ δ ∑ ψ = 1 3 w ψ 3 φ + ∏ ψ = 1 3 w ψ φ , Λ ,
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 with δ be a positive constant.

3.1. Definitions and Notations of Intuitionistic Fuzzy Banach Space

Now, we recall the basic definitions and notations in the setting of intuitionistic fuzzy normed space given in [33].
Definition 3.1. 
[33] A binary operation * : 0 , 1 × 0 , 1 ⟶ 0 , 1 is said to be continuous t-norm if * satisfies the following conditions:
(*1) 
* is commutative and associative;
(*2) 
* is continuous;
(*3) 
a * 1 = a for all a ∈ 0 , 1 ;
(*4) 
a * b ≤ c * d whenever a ≤ c and b ≤ d for all a , b , c , d ∈ 0 , 1 .
Definition 3.2. 
[33] A binary operation ⋄ : 0 , 1 × 0 , 1 ⟶ 0 , 1 is said to be continuous t-conorm if ⋄ satisfies the following conditions:
(⋄1) 
⋄ is commutative and associative;
(⋄2) 
⋄ is continuous;
(⋄3) 
a ⋄ 0 = a for all a ∈ 0 , 1 ;
(⋄4) 
a ⋄ b ≤ c ⋄ d whenever a ≤ c and b ≤ d for all a , b , c , d ∈ 0 , 1 .
Definition 3.3. 
[33] The five-tuple ( X , μ , ν , * , ⋄ ) is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space, * is a continuous t-norm, ⋄ is a continuous t − conorm, and μ , ν are fuzzy sets on X × ( 0 , ∞ ) satisfying the following conditions. For every x , y ∈ X and s , t > 0
(IFN1) 
μ ( x , t ) + ν ( x , t ) ≤ 1 ;
(IFN2) 
μ ( x , t ) > 0 ;
(IFN3) 
μ ( x , t ) = 1 , if and only if x = 0 ;
(IFN4) 
μ ( d x , t ) = μ x , t d for each d ≠ 0 ;
(IFN5) 
μ ( x , t ) * μ ( y , s ) ≤ μ ( x + y , t + s ) ;
(IFN6) 
μ ( x , · ) : ( 0 , ∞ ) → [ 0 , 1 ] is continuous;
(IFN7) 
lim t → ∞ μ ( x , t ) = 1 and lim t → 0 μ ( x , t ) = 0 ;
(IFN8) 
ν ( x , t ) < 1 ;
(IFN9) 
ν ( x , t ) = 0 , if and only if x = 0 ;
(IFN10) 
ν ( d x , t ) = ν x , t d for each d ≠ 0 ;
(IFN11) 
ν ( x , t ) ⋄ ν ( y , s ) ≥ ν ( x + y , t + s ) ;
(IFN12) 
ν ( x , · ) : ( 0 , ∞ ) → [ 0 , 1 ] is continuous;
(IFN13) 
lim t → ∞ ν ( x , t ) = 0 and lim t → 0 ν ( x , t ) = 1 .
In this case, ( μ , ν ) is called an intuitionistic fuzzy norm.
Example 3.4. 
[33] Let X , · be a normed space. Let a * b = a b and a ⋄ d = min a + b , 1 for all a , b ∈ [ 0 , 1 ] . For all x ∈ X and every t > 0 , consider
μ ( x , t ) = t t + x i f t > 0 ; 0 i f t ≤ 0 ; a n d ν ( x , t ) = x t + x i f t > 0 ; 0 i f t ≤ 0 .
Then X , μ , ν , * , ⋄ is an IFN-space.
Definition 3.5. 
[33] Let X , μ , ν , * , ⋄ be an IFNS. Then, a sequence x = { x k } is said to be intuitionistic fuzzy convergent to a point L ∈ X if
lim μ ( x k − L , t ) = 1 a n d lim ν ( x k − L , t ) = 0
for all ρ > 0 . In this case, we write
x k ⟶ I F L a s k → ∞
Definition 3.6. 
[33] Let X , μ , ν , * , ⋄ be an IFN-space. Then, x = { x k } is said to be intuitionistic fuzzy Cauchy sequence if
μ x k + p − x k , t = 1 a n d ν x k + p − x k , t = 0
for all ρ > 0 , and p = 1 , 2 ⋯ .
Definition 3.7. 
[33] Let X , μ , ν , * , ⋄ be an IFN-space. Then X , μ , ν , * , ⋄ is said to be complete if every intuitionistic fuzzy Cauchy sequence in X , μ , ν , * , ⋄ is intuitionistic fuzzy convergent X , μ , ν , * , ⋄ .

3.2. Oddness of F : Additive Case Stability Results : Direct Method

Theorem 3.8. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the conditions
μ ′ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 , Λ ≥ μ ′ I ℓ μ Ψ w 1 , w 2 , w 3 , Λ ν ′ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 , Λ ≤ ν ′ I ℓ μ Ψ w 1 , w 2 , w 3 , Λ
and
lim ℓ → ∞ μ ′ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 , 5 ℓ μ Λ = 1 lim ℓ → ∞ ν ′ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 , 5 ℓ μ Λ = 0
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 with μ = ± 1 and 0 < I 5 μ < 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ A ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ Ψ A w 1 , 3 Λ 4 | 5 − I | = μ ′ Ψ w 1 , w 1 , w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ w 1 , w 1 , − w 1 , 3 Λ 4 | 5 − I | ν A ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ Ψ A w 1 , 3 Λ 4 ( 5 − I ) = ν ′ Ψ w 1 , w 1 , w 1 , 3 Λ 4 ( 5 − I ) ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , 3 Λ 4 ( 5 − I )
and the mapping A ( w 1 ) is obtained by
lim ℓ → ∞ μ 1 5 ℓ μ F 5 ℓ μ w 1 − A ( w 1 ) , Λ = 1 lim ℓ → ∞ ν 1 5 ℓ μ F 5 ℓ μ w 1 − A ( w 1 ) , Λ = 0
for all w 1 ∈ W 1 and all Λ > 0 .
Proof. 
Using oddness of F in (1), we get
μ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) , Λ ≥ μ ′ Ψ w 1 , w 2 , w 3 , Λ ν F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 6 F ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 F ( w ψ ) , Λ ≤ ν ′ Ψ w 1 , w 2 , w 3 , Λ
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Interchanging w 1 , w 2 , w 3 by w 1 , w 1 , w 1 in (7), we obtain
μ 3 F ( 5 w 1 ) − 6 F 3 w 1 + 3 F ( w 1 ) , Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ ν 3 F ( 5 w 1 ) − 6 F 3 w 1 + 3 F ( w 1 ) , Λ ≤ ν ′ Ψ w 1 , w 1 , w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Again interchanging w 1 , w 2 , w 3 by w 1 , w 1 , − w 1 in (7) and using (IFN4), (IFN10), we have
μ 2 F ( 3 w 1 ) − 6 F w 1 , Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ ν 2 F ( 3 w 1 ) − 6 F w 1 , Λ ≤ ν ′ Ψ w 1 , w 1 , − w 1 , Λ ⇒ μ 6 F ( 3 w 1 ) − 18 F w 1 , 3 Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ ν 6 F ( 3 w 1 ) − 18 F w 1 , 3 Λ ≤ ν ′ Ψ w 1 , w 1 , − w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Combining (8) and (9) using (IFN5), (IFN11), we arrive
μ 3 F ( 5 w 1 ) − 15 F ( w 1 ) , 4 Λ ≥ μ 3 F ( 5 w 1 ) − 6 F 3 w 1 + 3 F ( w 1 ) , Λ * μ 6 F ( 3 w 1 ) − 18 F w 1 , 3 Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ * μ ′ Ψ w 1 , w 1 , − w 1 , Λ = μ ′ Ψ A w 1 , Λ ν 3 F ( 5 w 1 ) − 15 F ( w 1 ) , 4 Λ ≤ ν 3 F ( 5 w 1 ) − 6 F 3 w 1 + 3 F ( w 1 ) , Λ ⋄ ν 6 F ( 3 w 1 ) − 18 F w 1 , 3 Λ ≤ ν ′ Ψ w 1 , w 1 , w 1 , Λ ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , Λ = ν ′ Ψ A w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Using (IFN4), (IFN10), one can see from (10) that
μ 1 5 F ( 5 w 1 ) − F ( w 1 ) , 4 3 · 1 5 Λ ≥ μ ′ Ψ A w 1 , Λ ν 1 5 F ( 5 w 1 ) − F ( w 1 ) , 4 3 · 1 5 Λ ≤ ν ′ Ψ A w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Changing w 1 by 5 ℓ w 1 in (11), and using (IFN4), (IFN10), (3), we get
μ 1 5 ℓ + 1 F ( 5 ℓ + 1 w 1 ) − 1 5 ℓ F ( 5 ℓ w 1 ) , 4 3 · 5 · 1 5 ℓ Λ ≥ μ ′ Ψ A 5 ℓ w 1 , Λ ≥ μ ′ I ℓ Ψ A w 1 , Λ = μ ′ Ψ A w 1 , 1 I ℓ Λ ν 1 5 ℓ + 1 F ( 5 ℓ + 1 w 1 ) − 1 5 ℓ F ( 5 ℓ w 1 ) , 4 3 · 5 · 1 5 ℓ Λ ≤ ν ′ Ψ A 5 ℓ w 1 , Λ ≤ ν ′ I ℓ Ψ A w 1 , Λ = ν ′ Ψ A w 1 , 1 I ℓ Λ
for all w 1 ∈ W 1 and all Λ > 0 also ℓ > 0 . Changing Λ by I ℓ Λ in (112), we see
μ 1 5 ℓ + 1 F ( 5 ℓ + 1 w 1 ) − 1 5 ℓ F ( 5 ℓ w 1 ) , 4 3 · 5 · I 5 ℓ Λ ≥ μ ′ Ψ A w 1 , Λ ν 1 5 ℓ + 1 F ( 5 ℓ + 1 w 1 ) − 1 5 ℓ F ( 5 ℓ w 1 ) , 4 3 · 5 · I 5 ℓ Λ ≤ ν ′ Ψ A w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . It is easy to check that
1 5 ℓ F ( 5 ℓ w 1 ) − F ( w 1 ) = ∑ η = 0 ℓ − 1 1 5 η + 1 F ( 5 η + 1 w 1 ) − 1 5 η F ( 5 η w 1 )
for all w 1 ∈ W 1 . Using (IFN5), (IFN11), it follows from (13) and (14), we obtain
μ 1 5 ℓ F ( 5 ℓ w 1 ) − F ( w 1 ) , ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η Λ = μ ∑ η = 0 ℓ − 1 1 5 η + 1 F ( 5 η + 1 w 1 ) − 1 5 η F ( 5 η w 1 ) , ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η Λ ≥ ∏ η = 0 ℓ − 1 μ 1 5 η + 1 F ( 5 η + 1 w 1 ) − 1 5 η F ( 5 η w 1 ) , 4 3 · 5 · I 5 η Λ ≥ ∏ η = 0 ℓ − 1 μ ′ Ψ A w 1 , Λ = μ ′ Ψ A w 1 , Λ ν 1 5 ℓ F ( 5 ℓ w 1 ) − F ( w 1 ) , ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η Λ = ν ∑ η = 0 ℓ − 1 1 5 η + 1 F ( 5 η + 1 w 1 ) − 1 5 η F ( 5 η w 1 ) , ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η Λ ≤ ∐ η = 0 ℓ − 1 ν 1 5 η + 1 F ( 5 η + 1 w 1 ) − 1 5 η F ( 5 η w 1 ) , 4 3 · 5 · I 5 η Λ ≤ ∐ η = 0 ℓ − 1 ν ′ Ψ A w 1 , Λ = ν ′ Ψ A w 1 , Λ
where
∏ η = 0 ℓ − 1 μ = μ * μ * μ * . . . a n d ∐ η = 0 ℓ − 1 ν = ν ⋄ ν ⋄ ν ⋄ . . .
for all w 1 ∈ W 1 and all Λ > 0 . Again changing w 1 by 5 ℓ 1 w 1 in (15), and using (IFN4), (IFN10), (3) in that changing Λ by I ℓ 1 Λ , we have
μ 1 5 ℓ + ℓ 1 F ( 5 ℓ + ℓ 1 w 1 ) − 1 5 ℓ 1 F ( ℓ 1 w 1 ) , ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η + ℓ 1 Λ ≥ μ ′ Ψ A w 1 , Λ ν 1 5 ℓ + ℓ 1 F ( 5 ℓ + ℓ 1 w 1 ) − 1 5 ℓ 1 F ( ℓ 1 w 1 ) , ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η + ℓ 1 Λ ≤ ν ′ Ψ A w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 also ℓ , ℓ 1 > 0 . It follows from (16) that
μ 1 5 ℓ + ℓ 1 F ( 5 ℓ + ℓ 1 w 1 ) − 1 5 ℓ 1 F ( ℓ 1 w 1 ) , Λ ≥ μ ′ Ψ A w 1 , Λ ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η + ℓ 1 ν 1 5 ℓ + ℓ 1 F ( 5 ℓ + ℓ 1 w 1 ) − 1 5 ℓ 1 F ( ℓ 1 w 1 ) , Λ ≤ ν ′ Ψ A w 1 , Λ ∑ η = 0 ℓ − 1 4 3 · 5 · I 5 η + ℓ 1
for all w 1 ∈ W 1 and all Λ > 0 . By data, the Cauchy criterion for convergence in Intuitionistic Fuzzy normed space gives that the sequence 1 5 ℓ F ( 5 ℓ w 1 ) , is Cauchy in W 2 , μ ′ , ν ′ and it is a complete Intuitionistic Fuzzy normed space, this sequence converges to some point A ( w 1 ) in W 2 , μ ′ , ν ′ for all w 1 ∈ W 1 . So, by notation, we write
lim ℓ → ∞ μ 1 5 ℓ F 5 ℓ w 1 − A ( w 1 ) , Λ = 1 lim ℓ → ∞ ν 1 5 ℓ F 5 ℓ w 1 − A ( w 1 ) , Λ = 0
for all w 1 ∈ W 1 and all Λ > 0 . Letting ℓ 1 = 0 and ℓ → ∞ in (17) and using (18), we arrive
μ A ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ Ψ A w 1 , 3 Λ 4 ( 5 − I ) = μ ′ Ψ w 1 , w 1 , w 1 , 3 Λ 4 ( 5 − I ) * μ ′ Ψ w 1 , w 1 , − w 1 , 3 Λ 4 ( 5 − I ) ν A ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ Ψ A w 1 , 3 Λ 4 ( 5 − I ) = ν ′ Ψ w 1 , w 1 , w 1 , 3 Λ 4 ( 5 − I ) ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , 3 Λ 4 ( 5 − I )
for all w 1 ∈ W 1 and all Λ > 0 . Thus, (5) and (6) holds for μ = 1 . Interchanging
w 1 , w 2 , w 3 = 5 ℓ w 1 , 5 ℓ w 2 , 5 ℓ w 3 ,
in (1) and using (IFN4), (IFN10), we have
μ 1 5 ℓ { F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) − 6 F ∑ ψ = 1 3 5 ℓ w ψ − 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ + ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) } , Λ ≥ μ ′ Ψ 5 ℓ w 1 , 5 ℓ w 2 , 5 ℓ w 3 , 5 ℓ Λ ν 1 5 ℓ { F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) − 6 F ∑ ψ = 1 3 5 ℓ w ψ − 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ + ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) } , Λ ≤ ν ′ Ψ 5 ℓ w 1 , 5 ℓ w 2 , 5 ℓ w 3 , 5 ℓ Λ
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Now,
μ A ( 3 w 1 + w 2 + w 3 ) + A ( w 1 + 3 w 2 + w 3 ) + A ( w 1 + w 2 + 3 w 3 ) − 6 A ∑ ψ = 1 3 w ψ − 1 2 A ∑ ψ = 1 3 w ψ − A − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ ) , Λ ≥ μ A ( 3 w 1 + w 2 + w 3 ) − 1 5 ℓ F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) , Λ 7 * μ A ( w 1 + 3 w 2 + w 3 ) − 1 5 ℓ F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) , Λ 7 * μ A ( w 1 + w 2 + 3 w 3 ) − 1 5 ℓ F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) , Λ 7 * μ − 6 A ∑ ψ = 1 3 w ψ + 1 5 ℓ 6 F ∑ ψ = 1 3 5 ℓ w ψ , Λ 7 * μ 1 2 A ∑ ψ = 1 3 w ψ + A − ∑ ψ = 1 3 w ψ − 1 5 ℓ 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ , Λ 7 * μ ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ ) − 1 5 ℓ ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) , Λ 7 * μ 1 5 ℓ { F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) − 6 F ∑ ψ = 1 3 5 ℓ w ψ − 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ + ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) } , Λ 7 ν A ( 3 w 1 + w 2 + w 3 ) + A ( w 1 + 3 w 2 + w 3 ) + A ( w 1 + w 2 + 3 w 3 ) − 6 A ∑ ψ = 1 3 w ψ − 1 2 A ∑ ψ = 1 3 w ψ − A − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ ) , Λ ≤ ν A ( 3 w 1 + w 2 + w 3 ) − 1 5 ℓ F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) , Λ 7 ⋄ ν A ( w 1 + 3 w 2 + w 3 ) − 1 5 ℓ F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) , Λ 7 ⋄ ν A ( w 1 + w 2 + 3 w 3 ) − 1 5 ℓ F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) , Λ 7 ⋄ ν − 6 A ∑ ψ = 1 3 w ψ + 1 5 ℓ 6 F ∑ ψ = 1 3 5 ℓ w ψ , Λ 7 ⋄ ν 1 2 A ∑ ψ = 1 3 w ψ + A − ∑ ψ = 1 3 w ψ − 1 5 ℓ 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ , Λ 7 ⋄ ν ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ ) − 1 5 ℓ ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) , Λ 7 ⋄ ν 1 5 ℓ { F ( 5 ℓ ( 3 w 1 + w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + 3 w 2 + w 3 ) ) + F ( 5 ℓ ( w 1 + w 2 + 3 w 3 ) ) − 6 F ∑ ψ = 1 3 5 ℓ w ψ − 1 2 F ∑ ψ = 1 3 5 ℓ w ψ + F − ∑ ψ = 1 3 5 ℓ w ψ + ∑ ψ = 1 3 F ( 5 ℓ w ψ ) − 5 2 F ( 5 ℓ w ψ ) + F ( − 5 ℓ w ψ ) } , Λ 7
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Taking limit ℓ → ∞ in (21), using (18) and (20), we get
μ A ( 3 w 1 + w 2 + w 3 ) + A ( w 1 + 3 w 2 + w 3 ) + A ( w 1 + w 2 + 3 w 3 ) − 6 A ∑ ψ = 1 3 w ψ − 1 2 A ∑ ψ = 1 3 w ψ + A − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ ) , Λ = 1 ν A ( 3 w 1 + w 2 + w 3 ) + A ( w 1 + 3 w 2 + w 3 ) + A ( w 1 + w 2 + 3 w 3 ) − 6 A ∑ ψ = 1 3 w ψ − 1 2 A ∑ ψ = 1 3 w ψ + A − ∑ ψ = 1 3 w ψ + ∑ ψ = 1 3 A ( w ψ ) − 5 2 A ( w ψ ) + A ( − w ψ ) , Λ = 0
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Using (IFN3), (IFN9) in (22), we see, A ( w 1 ) satisfies (3). In order to confirm that A ( w 1 ) is unique, suppose B ( w 1 ) be another mapping (3), (18) and (19), we obtain
μ A ( w 1 ) − B ( w 1 ) , 2 Λ = μ A 5 ℓ w 1 − B 5 ℓ w 1 , 5 ℓ 2 Λ ≥ μ A 5 ℓ w 1 − F 5 ℓ w 1 , 5 ℓ Λ * μ F 5 ℓ w 1 − B 5 ℓ w 1 , 5 ℓ Λ ≥ μ ′ Ψ A 5 ℓ w 1 , 3 Λ 4 5 ℓ ( 5 − I ) * μ ′ Ψ A 5 ℓ w 1 , 3 Λ 4 5 ℓ ( 5 − I ) ≥ μ ′ Ψ A w 1 , 3 Λ 4 5 ℓ I ℓ ( 5 − I ) ν A ( w 1 ) − B ( w 1 ) , 2 Λ = ν A 5 ℓ w 1 − B 5 ℓ w 1 , 5 ℓ 2 Λ ≤ ν A 5 ℓ w 1 − F 5 ℓ w 1 , 5 ℓ Λ ⋄ ν F 5 ℓ w 1 − B 5 ℓ w 1 , 5 ℓ Λ ≤ ν ′ Ψ A 5 ℓ w 1 , 3 Λ 4 5 ℓ ( 5 − I ) ⋄ ν ′ Ψ A 5 ℓ w 1 , 3 Λ 4 5 ℓ ( 5 − I ) ≤ ν ′ Ψ A w 1 , 3 Λ 4 5 ℓ I ℓ ( 5 − I )
for all w 1 ∈ W 1 and all Λ > 0 . Taking limit ℓ → ∞ in (23), and using (IFN7), (IFN13), we arrive
μ A ( w 1 ) − B ( w 1 ) , 2 Λ = 1 ν A ( w 1 ) − B ( w 1 ) , 2 Λ = 0
for all w 1 ∈ W 1 and all Λ > 0 . By (IFN4) and (IFN10), we get A ( w 1 ) is unique. So, the Theorem holds for μ = 1 .
Changing w 1 = w 1 5 in (10) and using (IFN4), (IFN10), (3), in that changing Λ by Λ I , we have
μ F ( w 1 ) − 5 F w 1 5 , 4 3 · I Λ ≥ μ ′ Ψ A w 1 , Λ ν F ( w 1 ) − 5 F w 1 5 , 4 3 · I Λ ≤ ν ′ Ψ A w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Changing w 1 by w 1 5 ℓ in (25), and using (IFN4), (IFN10), (3) in that changing Λ by Λ I ℓ , we get
μ 5 ℓ F w 1 5 ℓ − 5 ℓ + 1 F w 1 5 ℓ + 1 , 4 3 · I 5 I ℓ Λ ≤ μ ′ Ψ A w 1 , Λ ν 5 ℓ F w 1 5 ℓ − 5 ℓ + 1 F w 1 5 ℓ + 1 , 4 3 · I 5 I ℓ Λ ≤ ν ′ Ψ A w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 also ℓ > 0 . It is easy to check that
F ( w 1 ) − 5 ℓ F w 1 5 ℓ = ∑ η = 0 ℓ − 1 5 η F w 1 5 η − 5 η + 1 F w 1 5 η + 1
for all w 1 ∈ W 1 . The rest of the proof is similar to that of above case. So, the Theorem holds for μ = − 1 . Hence the proof is complete □
Corollary 3.9. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ A ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ δ , | 3 | Λ , μ ′ δ | w 1 | φ , Λ 4 | 5 − 5 φ | , φ ≠ 1 , μ ′ δ ∑ ψ = 1 3 | w 1 | φ   ψ , 3 Λ 4 ∑ ψ = 1 3 | 5 − 5 φ   ψ | , φ 1 , φ 2 , φ 3 ≠ 1 , μ ′ δ | w 1 | 3 φ , 3 Λ 4 | 5 − 5 3 φ | , 3 φ ≠ 1 , μ ′ δ | w ψ | ∑ ψ = 1 3 φ   ψ , 3 Λ 4 | 5 − 5 ∑ ψ = 1 3 φ   ψ | , ∑ ψ = 1 3 φ   ψ ≠ 1 , μ ′ 2 δ | w 1 | 3 φ , 3 Λ 4 | 5 − 5 3 φ | , 3 φ ≠ 1 , ν A ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ δ , | 3 | Λ , ν ′ δ | w 1 | φ , Λ 4 | 5 − 5 φ | , φ ≠ 1 , ν ′ δ ∑ ψ = 1 3 | w 1 | φ   ψ , 3 Λ 4 ∑ ψ = 1 3 | 5 − 5 φ   ψ | , φ 1 , φ 2 , φ 3 ≠ 1 , ν ′ δ | w 1 | 3 φ , 3 Λ 4 | 5 − 5 3 φ | , 3 φ ≠ 1 , ν ′ δ | w ψ | ∑ ψ = 1 3 φ   ψ , 3 Λ 4 | 5 − 5 ∑ ψ = 1 3 φ   ψ | , ∑ ψ = 1 3 φ   ψ ≠ 1 , ν ′ 2 δ | w 1 | 3 φ , 3 Λ 4 | 5 − 5 3 φ | , 3 φ ≠ 1 ,
for all w 1 ∈ W 1 .

3.3. Evenness of F : Quadratic Case Stability Results : Direct Method

Theorem 3.10. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the conditions (3) and
lim ℓ → ∞ μ ′ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 , 25 ℓ μ Λ = 1 lim ℓ → ∞ ν ′ Ψ 5 ℓ μ w 1 , 5 ℓ μ w 2 , 5 ℓ μ w 3 , 25 ℓ μ Λ = 0
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 with μ = ± 1 and 0 < I 25 μ < 1 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ Q ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | = μ ′ Ψ w 1 , w 1 , w 1 , 7 Λ 3 | 25 − I | * μ ′ Ψ w 1 , w 1 , − w 1 , 7 Λ 3 | 25 − I | ν Q ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ Ψ Q w 1 , 7 Λ 3 ( 25 − I ) = ν ′ Ψ w 1 , w 1 , w 1 , 7 Λ 3 ( 25 − I ) ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , 7 Λ 3 ( 25 − I )
and the mapping Q ( w 1 ) is obtained by
lim ℓ → ∞ μ 1 25 ℓ μ F 5 ℓ μ w 1 − Q ( w 1 ) , Λ = 1 lim ℓ → ∞ ν 1 25 ℓ μ F 5 ℓ μ w 1 − Q ( w 1 ) , Λ = 0
for all w 1 ∈ W 1 and all Λ > 0 .
Proof. 
Using evenness of F in (1), we get
μ F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 7 F ∑ ψ = 1 3 w ψ − 4 ∑ ψ = 1 3 F ( w ψ ) , Λ ≥ μ ′ Ψ w 1 , w 2 , w 3 , Λ ν F ( 3 w 1 + w 2 + w 3 ) + F ( w 1 + 3 w 2 + w 3 ) + F ( w 1 + w 2 + 3 w 3 ) − 7 F ∑ ψ = 1 3 w ψ − 4 ∑ ψ = 1 3 F ( w ψ ) , Λ ≤ ν ′ Ψ w 1 , w 2 , w 3 , Λ
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Interchanging w 1 , w 2 , w 3 by w 1 , w 1 , w 1 in (32), we obtain
μ 3 F ( 5 w 1 ) − 7 F 3 w 1 − 12 F ( w 1 ) , Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ ν 3 F ( 5 w 1 ) − 7 F 3 w 1 − 12 F ( w 1 ) , Λ ≤ ν ′ Ψ w 1 , w 1 , w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Again interchanging w 1 , w 2 , w 3 by w 1 , w 1 , − w 1 in (32) and using (IFN4), (IFN10), we have
μ 2 F ( 3 w 1 ) − 18 F w 1 , Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ ν 2 F ( 3 w 1 ) − 18 F w 1 , Λ ≤ ν ′ Ψ w 1 , w 1 , − w 1 , Λ ⇒ μ 7 F ( 3 w 1 ) − 63 F w 1 , 2 7 Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ ν 7 F ( 3 w 1 ) − 63 F w 1 , 2 7 Λ ≤ ν ′ Ψ w 1 , w 1 , − w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Combining (33) and (34) using (IFN5), (IFN11), we arrive
μ 3 F ( 5 w 1 ) − 75 F ( w 1 ) , 9 7 Λ ≥ μ 3 F ( 5 w 1 ) − 7 F 3 w 1 − 12 F ( w 1 ) , Λ * μ 7 F ( 3 w 1 ) − 63 F w 1 , 2 7 Λ ≥ μ ′ Ψ w 1 , w 1 , w 1 , Λ * μ ′ Ψ w 1 , w 1 , − w 1 , Λ = μ ′ Ψ Q w 1 , Λ ν 3 F ( 5 w 1 ) − 15 F ( w 1 ) , 9 7 Λ ≤ ν 3 F ( 5 w 1 ) − 7 F 3 w 1 − 12 F ( w 1 ) , Λ ⋄ ν 7 F ( 3 w 1 ) − 63 F w 1 , 2 7 Λ ≤ ν ′ Ψ w 1 , w 1 , w 1 , Λ ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , Λ = ν ′ Ψ Q w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . Using (IFN4), (IFN10), one can see from (35) that
μ 1 25 F ( 5 w 1 ) − F ( w 1 ) , 9 7 · 3 · 1 25 Λ ≥ μ ′ Ψ Q w 1 , Λ ν 1 25 F ( 5 w 1 ) − F ( w 1 ) , 9 7 · 3 · 1 25 Λ ≤ ν ′ Ψ Q w 1 , Λ
for all w 1 ∈ W 1 and all Λ > 0 . The rest of the proof is similar to that of Theorem 3.8. Hence the proof is complete. □
Corollary 3.11. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ Q ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ δ , | 8 | 7 Λ , μ ′ δ | w 1 | φ , 7 Λ 9 | 25 − 5 φ | , φ ≠ 2 , μ ′ δ ∑ ψ = 1 3 | w 1 | φ   ψ , 7 Λ 3 ∑ ψ = 1 3 | 25 − 5 φ   ψ | , φ 1 , φ 2 , φ 3 ≠ 2 , μ ′ δ | w 1 | 3 φ , 7 Λ 3 | 25 − 5 3 φ | , 3 φ ≠ 2 , μ ′ δ | w ψ | ∑ ψ = 1 3 φ   ψ , 7 Λ 3 | 25 − 5 ∑ ψ = 1 3 φ   ψ | , ∑ ψ = 1 3 φ   ψ ≠ 2 , μ ′ 2 δ | w 1 | 3 φ , 7 Λ 3 | 25 − 5 3 φ | , 3 φ ≠ 2 , ν Q ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ δ , | 8 | 7 Λ , ν ′ δ | w 1 | φ , 7 Λ 9 | 25 − 5 φ | , φ ≠ 2 , ν ′ δ ∑ ψ = 1 3 | w 1 | φ   ψ , 7 Λ 3 ∑ ψ = 1 3 | 25 − 5 φ   ψ | , φ 1 , φ 2 , φ 3 ≠ 2 , ν ′ δ | w 1 | 3 φ , 7 Λ 3 | 25 − 5 3 φ | , 3 φ ≠ 2 , ν ′ δ | w ψ | ∑ ψ = 1 3 φ   ψ , 7 Λ 3 | 25 − 5 ∑ ψ = 1 3 φ   ψ | , ∑ ψ = 1 3 φ   ψ ≠ 2 , ν ′ 2 δ | w 1 | 3 φ , 7 Λ 3 | 25 − 5 3 φ | , 3 φ ≠ 2 ,
for all w 1 ∈ W 1 and all Λ > 0 .

3.4. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Direct Method

Theorem 3.12. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the conditions (3), (4), and (29) for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 with μ = ± 1 and 0 < I 5 μ < 1 , 0 < I 25 μ < 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≥ μ ′ Ψ A w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ A − w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | * μ ′ Ψ Q − w 1 , 7 Λ 3 | 25 − I | = μ ′ Ψ w 1 , w 1 , w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ w 1 , w 1 , − w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ − w 1 , − w 1 , − w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ − w 1 , − w 1 , w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ w 1 , w 1 , w 1 , 7 Λ 3 | 5 − I | * μ ′ Ψ w 1 , w 1 , − w 1 , 7 Λ 3 | 5 − I | * μ ′ Ψ − w 1 , − w 1 , − w 1 , 7 Λ 3 | 5 − I | * μ ′ Ψ − w 1 , − w 1 , w 1 , 7 Λ 3 | 5 − I | ν F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≤ ν ′ Ψ A w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ A − w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | ⋄ ν ′ Ψ Q − w 1 , 7 Λ 3 | 25 − I | = ν ′ Ψ w 1 , w 1 , w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ − w 1 , − w 1 , − w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ − w 1 , − w 1 , w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ w 1 , w 1 , w 1 , 7 Λ 3 | 5 − I | ⋄ ν ′ Ψ w 1 , w 1 , − w 1 , 7 Λ 3 | 5 − I | ⋄ ν ′ Ψ − w 1 , − w 1 , − w 1 , 7 Λ 3 | 5 − I | ⋄ ν ′ Ψ − w 1 , − w 1 , w 1 , 7 Λ 3 | 5 − I |
and the mapping A ( w 1 ) and Q ( w 1 ) are given in (6) and (31) for all w 1 ∈ W 1 .
Proof. 
By Theorem 3.8, it follows from (33), (1) and (5), we arrive
μ A ( w 1 ) − F o d d ( w 1 ) , 2 Λ ≥ μ ′ Ψ A w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ A − w 1 , 3 Λ 4 | 5 − I | ν A ( w 1 ) − F o d d ( w 1 ) , 2 Λ ≤ ν ′ Ψ A w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ A − w 1 , 3 Λ 4 | 5 − I |
for all w 1 ∈ W 1 and all Λ > 0 . By Theorem 3.10, it follows from (37), (1), and (30), we see
μ Q ( w 1 ) − F e v e n ( w 1 ) , 2 Λ ≥ μ ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | * μ ′ Ψ Q − w 1 , 7 Λ 3 | 25 − I | ν Q ( w 1 ) − F e v e n ( w 1 ) , 2 Λ ≤ ν ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | ⋄ ν ′ Ψ Q − w 1 , 7 Λ 3 | 25 − I |
for all w 1 ∈ W 1 and all Λ > 0 . Now, it follows from (39), (40) and (40), we have
μ F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≥ μ A ( w 1 ) − F o d d ( w 1 ) , 2 Λ * μ Q ( w 1 ) − F e v e n ( w 1 ) , 2 Λ ≥ μ ′ Ψ A w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ A − w 1 , 3 Λ 4 | 5 − I | * μ ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | * μ ′ Ψ Q − w 1 , 7 Λ 3 | 25 − I | ν F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≤ ν A ( w 1 ) − F o d d ( w 1 ) , 2 Λ ⋄ ν Q ( w 1 ) − F e v e n ( w 1 ) , 2 Λ ≤ ν ′ Ψ A w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ A − w 1 , 3 Λ 4 | 5 − I | ⋄ ν ′ Ψ Q w 1 , 7 Λ 3 | 25 − I | ⋄ ν ′ Ψ Q − w 1 , 7 Λ 3 | 25 − I |
for all w 1 ∈ W 1 and all Λ > 0 . □
Corollary 3.13. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≥ μ ′ 2 δ , | 3 | + 7 | 8 | Λ μ ′ 2 δ | w 1 | φ , Λ 1 4 | 5 − 5 φ | + 7 9 | 25 − 5 φ | , φ ≠ 1 , 2 , μ ′ 2 δ ∑ ψ = 1 3 | w ψ | φ   ψ , 3 4 ∑ ψ = 1 3 | 5 − 5 φ   ψ | + 7 3 ∑ ψ = 1 3 | 25 − 5 φ   ψ | , φ 1 , φ 2 , φ 3 ≠ 1 , 2 , μ ′ 2 δ | w 1 | 3 φ , Λ 3 4 | 5 − 5 3 φ | + 7 3 | 25 − 5 3 φ | , 3 φ ≠ 1 , 2 , μ ′ 4 δ | w ψ | ∑ ψ = 1 3 φ   ψ , 2 Λ 3 4 | 5 − 5 ∑ ψ = 1 3 φ   ψ | + 7 3 | 25 − 5 ∑ ψ = 1 3 φ   ψ | ∑ ψ = 1 3 φ   ψ ≠ 1 , 2 , μ ′ 4 δ | w 1 | 3 φ , Λ 3 4 | 5 − 5 3 φ | + 7 3 | 25 − 5 3 φ | 3 φ ≠ 1 , 2 , ν F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≤ ν ′ 2 δ , | 3 | + 7 | 8 | Λ ν ′ 2 δ | w 1 | φ , Λ 1 4 | 5 − 5 φ | + 7 9 | 25 − 5 φ | , φ ≠ 1 , 2 , ν ′ 2 δ ∑ ψ = 1 3 | w ψ | φ   ψ , 3 4 ∑ ψ = 1 3 | 5 − 5 φ   ψ | + 7 3 ∑ ψ = 1 3 | 25 − 5 φ   ψ | , φ 1 , φ 2 , φ 3 ≠ 1 , 2 , ν ′ 2 δ | w 1 | 3 φ , Λ 3 4 | 5 − 5 3 φ | + 7 3 | 25 − 5 3 φ | , 3 φ ≠ 1 , 2 , ν ′ 4 δ | w ψ | ∑ ψ = 1 3 φ   ψ , 2 Λ 3 4 | 5 − 5 ∑ ψ = 1 3 φ   ψ | + 7 3 | 25 − 5 ∑ ψ = 1 3 φ   ψ | ∑ ψ = 1 3 φ   ψ ≠ 1 , 2 , ν ′ 4 δ | w 1 | 3 φ , Λ 3 4 | 5 − 5 3 φ | + 7 3 | 25 − 5 3 φ | 3 φ ≠ 1 , 2 ,
for all w 1 ∈ W 1 .

3.5. Oddness of F : Additive Case Stability Results : Fixed Method

Theorem 3.14. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the condition
lim ℓ → ∞ μ ′ Ψ τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 , τ ν ℓ Λ = 1 lim ℓ → ∞ ν ′ Ψ τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 , τ ν ℓ Λ = 0 ; τ ν = 5 ; ν = 0 1 5 ; ν = 1
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 If there exists L = L ( ν ) be a function have the property
μ Ψ A ( w 1 ) , Λ = μ Ψ A w 1 5 , Λ ν Ψ A ( w 1 ) , Λ = ν Ψ A w 1 5 , Λ a n d μ 1 τ ν Ψ A τ ν w 1 , Λ = μ L Ψ A ( w 1 ) , Λ ν 1 τ ν Ψ A τ ν w 1 , Λ = ν L Ψ A ( w 1 ) , Λ ,
for all w 1 ∈ W 1 and all Λ > 0 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ A ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ L 1 − ν 1 − L Ψ A w 1 , 3 Λ 4 = μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 3 Λ 4 ν A ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ L 1 − ν 1 − L Ψ A w 1 , 3 Λ 4 = ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 3 Λ 4
and the mapping A ( w 1 ) is obtained by
lim ℓ → ∞ μ 1 τ ν ℓ F τ ν ℓ w 1 − A ( w 1 ) , Λ = 1 lim ℓ → ∞ ν 1 τ ν ℓ F τ ν ℓ w 1 − A ( w 1 ) , Λ = 0
for all w 1 ∈ W 1 and all Λ > 0 .
Proof. 
Assume a set G as in Theorem 2.7 of (48) and introduce the generalized metric on the above set G as
d ( F , F 1 ) = inf K ∈ ( 0 , ∞ ) : μ F ( w 1 ) − F 1 ( w 1 ) , Λ ≥ μ K Ψ ( w 1 , w 1 , w 1 ) , Λ ν F ( w 1 ) − F 1 ( w 1 ) , Λ ≤ ν K Ψ ( w 1 , w 1 , w 1 ) , Λ .
for all w 1 ∈ W 1 and all Λ > 0 . It is easy to see that ( G , d ) is complete. Define a function H : G → G as by Theorem 2.7 of (50) and for F , F 1 ∈ G and w 1 ∈ W 1 and all Λ > 0 , we see
d ( F , F 1 ) ≤ K ⇒ μ F ( w 1 ) − F 1 ( w 1 ) , Λ ≥ μ K Ψ ( w 1 , w 1 , w 1 ) , Λ ν F ( w 1 ) − F 1 ( w 1 ) , Λ ≤ ν K Ψ ( w 1 , w 1 , w 1 ) , Λ ⇒ μ 1 τ ν F ( τ ν w 1 ) − 1 τ ν F 1 ( τ ν w 1 ) , Λ ≥ μ τ ν K Ψ ( 1 τ ν w 1 , τ ν w 1 , τ ν w 1 ) , Λ ν 1 τ ν F ( τ ν w 1 ) − 1 τ ν F 1 ( τ ν w 1 ) , Λ ≤ ν τ ν K Ψ ( 1 τ ν w 1 , τ ν w 1 , τ ν w 1 ) , Λ ⇒ μ H F ( w 1 ) − H F 1 ( w 1 ) , Λ ≥ μ L K Ψ ( w 1 , w 1 , w 1 ) , Λ ν H F ( w 1 ) − H F 1 ( w 1 ) , Λ ≤ ν L K Ψ ( w 1 , w 1 , w 1 ) , Λ ⇒ d ( H F , H F 1 ) ≤ L K ,
i.e., H is a strictly contractive mapping on G with Lipschitz constant L (see [24]).
For the case ν = 0 , it follows from (11) and with the help of (44), (50), (47), we get
μ 1 5 F ( 5 w 1 ) − F ( w 1 ) , 4 3 Λ ≥ μ ′ 1 5 Ψ A w 1 , Λ ν 1 5 F ( 5 w 1 ) − F ( w 1 ) , 4 3 Λ ≤ ν ′ 1 5 Ψ A w 1 , Λ ⇒ d ( H F , F ) ≤ L = L 1 − ν ,
for all w 1 ∈ W 1 and all Λ > 0 .
For the case ν = 1 , it follows from (17) and with the help of (44), (50), (47), we obtain
μ F ( w 1 ) − 5 F w 1 5 , 4 3 · I Λ ≥ μ ′ Ψ A w 1 5 , Λ ν F ( w 1 ) − 5 F w 1 5 , 4 3 · I Λ ≤ ν ′ Ψ A w 1 5 , Λ ⇒ d ( F , H F ) ≤ 1 = L 1 − ν ,
for all w 1 ∈ W 1 and all Λ > 0 . Combining (48) and (49), we have
d ( F , H F ) ≤ 1 = L 1 − ν .
Therefore ( F P C 1 ) of Theorem 1.3 holds. The rest of the proof follows by Theorem 1.3. Hence the proof is complete. □
Corollary 3.15. 
Suppose that an odd function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a a unique additive mapping A ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality (20) for all w 1 ∈ W 1 .

3.6. Evenness of F : Quadratic Case Stability Results : Fixed Method

Theorem 3.16. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the condition
lim ℓ → ∞ μ ′ Ψ τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 , τ ν 2 ℓ Λ = 1 lim ℓ → ∞ ν ′ Ψ τ ν ℓ w 1 , τ ν ℓ w 2 , τ ν ℓ w 3 , τ ν 2 ℓ Λ = 0 ; τ ν = 5 ; ν = 0 1 5 ; ν = 1
for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . If there exists L = L ( ν ) be function have the property
μ Ψ Q ( w 1 ) , Λ = μ Ψ Q w 1 5 , Λ ν Ψ Q ( w 1 ) , Λ = ν Ψ Q w 1 5 , Λ a n d μ 1 τ ν 2 Ψ Q τ ν w 1 , Λ = μ L Ψ Q ( w 1 ) , Λ ν 1 τ ν 2 Ψ Q τ ν w 1 , Λ = ν L Ψ Q ( w 1 ) , Λ ,
for all w 1 ∈ W 1 and all Λ > 0 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ Q ( w 1 ) − F ( w 1 ) , Λ ≥ μ ′ L 1 − ν 1 − L Ψ Q w 1 , 7 Λ 3 = μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 7 Λ 3 * μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 7 Λ 3 ν Q ( w 1 ) − F ( w 1 ) , Λ ≤ ν ′ L 1 − ν 1 − L Ψ Q w 1 , 7 Λ 3 = ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 7 Λ 3 ⋄ ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 7 Λ 3
and the mapping Q ( w 1 ) is obtained by
lim ℓ → ∞ μ 1 τ ν 2 ℓ F τ ν ℓ w 1 − Q ( w 1 ) , Λ = 1 lim ℓ → ∞ ν 1 τ ν 2 ℓ F τ ν ℓ w 1 − Q ( w 1 ) , Λ = 0
for all w 1 ∈ W 1 and all Λ > 0 .
Proof. 
Define a function H : G → G as by Theorem 2.9 of (62) and for F , F 1 ∈ G and w 1 ∈ W 1 and all Λ > 0 , we see
d ( F , F 1 ) ≤ K ⇒ μ F ( w 1 ) − F 1 ( w 1 ) , Λ ≥ μ K Ψ ( w 1 , w 1 , w 1 ) , Λ ν F ( w 1 ) − F 1 ( w 1 ) , Λ ≤ ν K Ψ ( w 1 , w 1 , w 1 ) , Λ ⇒ μ 1 τ ν 2 F ( τ ν w 1 ) − 1 τ ν 2 F 1 ( τ ν w 1 ) , Λ ≥ μ τ ν K Ψ ( τ ν w 1 , τ ν w 1 , τ ν w 1 ) , Λ ν 1 τ ν 2 F ( τ ν w 1 ) − 1 τ ν 2 F 1 ( τ ν w 1 ) , Λ ≤ ν τ ν 2 K Ψ ( τ ν w 1 , τ ν w 1 , τ ν w 1 ) , Λ ⇒ μ H F ( w 1 ) − H F 1 ( w 1 ) , Λ ≥ μ L K Ψ ( w 1 , w 1 , w 1 ) , Λ ν H F ( w 1 ) − H F 1 ( w 1 ) , Λ ≤ ν L K Ψ ( w 1 , w 1 , w 1 ) , Λ ⇒ d ( H F , H F 1 ) ≤ L K ,
i.e., H is a strictly contractive mapping on G with Lipschitz constant L (see [24]). The rest of the proof is similar to that of Theorem 3.14. Hence the proof is complete. □
Corollary 3.17. 
Suppose that an even function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality (37) for all w 1 ∈ W 1 .

3.7. Oddness and Evenness of F : Additive Quadratic Case Stability Results : Fixed Method

Theorem 3.18. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (1) where Ψ : W 1 3 → [ 0 , ∞ ) with the conditions (43) and (51) for all w 1 , w 2 , w 3 ∈ W 1 and all Λ > 0 . If there exists L = L ( ν ) be function have the properties (44) and (52) for all w 1 ∈ W 1 and all Λ > 0 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality
μ F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≥ μ ′ L 1 − ν 1 − L Ψ A w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ A − w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ Q w 1 , 7 Λ 3 * μ ′ L 1 − ν 1 − L Ψ Q − w 1 , 7 Λ 3 = μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , − w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , w 1 , 3 Λ 4 * μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 7 Λ 3 * μ ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 7 Λ 3 * μ ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , − w 1 , 7 Λ 3 * μ ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , w 1 , 7 Λ 3 ν F ( w 1 ) − A ( w 1 ) − Q ( w 1 ) , 4 Λ ≤ ν ′ L 1 − ν 1 − L Ψ A w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ A − w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ Q w 1 , 7 Λ 3 ⋄ ν ′ L 1 − ν 1 − L Ψ Q − w 1 , 7 Λ 3 = ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , − w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , w 1 , 3 Λ 4 ⋄ ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , w 1 , 7 Λ 3 ⋄ ν ′ L 1 − ν 1 − L Ψ w 1 , w 1 , − w 1 , 7 Λ 3 ⋄ ν ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , − w 1 , 7 Λ 3 ⋄ ν ′ L 1 − ν 1 − L Ψ − w 1 , − w 1 , w 1 , 7 Λ 3
and the mapping A ( w 1 ) and Q ( w 1 ) are given in (45) and (54) for all w 1 ∈ W 1 and all Λ > 0 .
Proof. 
The proof is similar ideas to that of Theorem 3.12. □
Corollary 3.19. 
Suppose that a function F : W 1 → W 2 satisfying the functional inequality (2) for all w 1 , w 2 , w 3 ∈ W 1 . Then there exists a unique additive mapping A ( w 1 ) : W 1 → W 2 and a unique quadratic mapping Q ( w 1 ) : W 1 → W 2 which satisfying (3) and the functional inequality (42) for all w 1 ∈ W 1 .

Acknowledgment

Supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. https://doi.org/10.54499/UIDB/04106/2020 and https://doi.org/10.54499/UIDP/04106/2020. References

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