Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

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Analyzing the Stability of Euler-Lagrange Additive Functional 

Equations in Banach Spaces: A Fixed Point and Direct Method 

Perspective 

 

P. Agilan 𝟏∗, M.Seetharaman 𝟐, M. Sophia 𝟑, Vaddi Seshagiri Rao 𝟒 

 1 Department of Mathematics, St.Joseph’s College of Engineering, OMR, Chennai - 600 119, Tamil Nadu, India. E-

mail: agilram@gmail.com 

 2Department of Mathematics, Roever Engineering College, Elambalur, Perambalur - 621 220, Tamil Nadu, India. E-

mail: saram1183@gmail.com. 

 3Department of Mathematics, SIMATS Engineering, Saveetha Nagar, Thandalam, Kanchipuram-Chennai Rd, Chennai- 

602105, Tamil Nadu, India. E-mail: sophia.raj2005@gmail.com. 

 4 Department of Mechanical Engineering, St.Joseph’s College of Engineering, OMR, Chennai - 600 119, Tamil Nadu, 

India. E-mail: raosvaddi@gmail.com. 

 ∗Corresponding author: agilram@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: This study explores the Ulam-Hyers stability of the Euler-Lagrange additive 

functional equation in the framework of Banach spaces. By employing both direct and fixed 

point methods, we establish new stability results that contribute to the understanding of 

functional equations in normed spaces. The direct method provides explicit estimates for 

stability bounds, while the fixed point approach ensures the existence and uniqueness of 

stable solutions under suitable conditions. Our findings offer a deeper insight into the 

interplay between functional equations and stability theory, with potential applications in 

analysis and mathematical modeling.  

Keywords: Euler - Lagrange functional equations, generalized Ulam - Hyers stability, 

Banach Space, fixed point.  

 

1.   Introduction 

One of the most influential stability notions was introduced by Ulam in 1940, when he posed a 

fundamental question regarding the stability of group homomorphisms[1]. This question was later 

answered affirmatively by Hyers in 1941[2], leading to what is now known as Ulam-Hyers stability. 

Since then, this stability concept has been extensively studied and extended to various mathematical 

settings, including functional equations, differential equations, and difference equations. Various types 

of functional equations, such as Cauchy’s functional equation, Jensen’s equation, and the Euler-

Lagrange additive functional equation, have been studied under Ulam-Hyers stability [3, 4, 5]. In many 

cases, researchers employ direct analytical methods, fixed point theorems, and iterative techniques to 

establish stability results. Banach space settings provide a rich framework for these studies, as normed 

spaces allow rigorous error estimations and convergence analysis.[6, 7, 8].  

 Differential equations are essential in modeling physical phenomena, engineering systems, and 

biological processes. Stability analysis is a key aspect of differential equation theory, ensuring that 



Communications on Applied Nonlinear Analysis 

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small perturbations in initial conditions or parameters do not lead to drastic changes in solutions [9]. 

The concept of Ulam-Hyers stability has been extended to differential equations, particularly to 

ordinary differential equations (ODEs) and partial differential equations (PDEs).For first-order and 

higher-order differential equations[10, 11], stability is often analyzed using fixed point methods, 

integral inequalities, and operator theory. The stability of linear and nonlinear differential equations 

has been extensively studied, leading to significant results that ensure the robustness of solutions under 

small perturbations. The application of fixed point theorems such as Banach’s contraction principle 

has been particularly useful in proving the existence and stability of solutions. In partial differential 

equations, researchers have investigated Ulam-Hyers stability in settings involving boundary value 

problems, heat equations, and wave equations. Such studies are crucial in mathematical physics, where 

slight variations in initial or boundary conditions should not lead to unbounded or highly unstable 

behaviors.  

 Difference equations serve as discrete counterparts to differential equations and have applications in 

numerical analysis, dynamical systems, and computer science. Ulam-Hyers stability in difference 

equations ensures that small deviations in the system’s structure do not lead to significant instabilities 

in discrete-time solutions. The stability analysis of difference equations has been widely explored using 

discrete fixed point methods, iterative approximation techniques, and functional inequalities. Linear 

and nonlinear difference equations, recurrence relations, and fractional difference equations have all 

been studied under the Ulam-Hyers stability framework[12, 13]. These results are particularly useful 

in numerical methods for solving differential equations, where approximations and discretization 

errors must be controlled. Recently Agilan et.al exploring the stability results in various additive 

functional equation through various normed spaces such as [14, 15, 16, 17, 18, 19, 20, 21].  

 In this paper, we introduce new kind of functional Equation and investigate the Ulam-Hyers stability 

of the Euler-Lagrange additive functional equation in Banach spaces using direct and fixed point 

methods.  

(𝑠2 + 2𝑠)ℎ(𝑝𝑥 + 𝑞𝑦) + (1 − 2𝑠)ℎ(𝑝𝑦 + 𝑞𝑧) + 2𝑠ℎ(𝑝𝑧 + 𝑞𝑥) − 2𝑠𝑝ℎ(𝑥 − 𝑦) 

               −2𝑠𝑞ℎ(𝑦 − 𝑧) = (𝑠2𝑝 + 2𝑠𝑞)ℎ(𝑥) + (𝑝 + 𝑠2𝑞)ℎ(𝑦) + (2𝑠𝑝 + 𝑞)ℎ(𝑧) (1) 

 with (𝑠2 + 2𝑠), (1 − 2𝑠), 𝑠𝑝, 𝑠𝑞 ≠ 0 and    𝑝, 𝑞, 𝑠 ∈ ℝ  

We aim to:1.Establish new stability results for the Euler-Lagrange equation under different conditions. 

     2.Compare the effectiveness of direct analytical techniques and fixed point methods. 

    3.Extend existing stability results to broader classes of functional equations. 

By doing so, we contribute to the ongoing research in stability theory, offering insights into its 

applications in various mathematical and applied fields.  

 Hereafter through out this paper, let us consider 𝑋 and 𝑌 to be a normed linear space and a Banach 

space, respectively.  

𝐻(𝑥, 𝑦, 𝑧) = (𝑠2 + 2𝑠)ℎ(𝑝𝑥 + 𝑞𝑦) + (1 − 2𝑠)ℎ(𝑝𝑦 + 𝑞𝑧) + 2𝑠ℎ(𝑝𝑧 + 𝑞𝑥) − 2𝑠𝑝ℎ(𝑥 − 𝑦)

− 2𝑠𝑞ℎ(𝑦 − 𝑧) − (𝑠2𝑝 + 2𝑠𝑞)ℎ(𝑥) − (𝑝 + 𝑠2𝑞)ℎ(𝑦) − (2𝑠𝑝 + 𝑞)ℎ(𝑧) 



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 where (𝑠2 + 2𝑠), (1 − 2𝑠), 𝑠𝑝, 𝑠𝑞 ≠ 0 and    𝑝, 𝑞, 𝑠 ∈ ℝ for all 𝑥, 𝑦, 𝑧 ∈ 𝑋.   

2  STABILITY RESULTS FOR THE FUNCTIONAL EQUATION (1) 

Theorem 2.1  Let j ∈ {−1,1} and Θ: X3 → [0,∞) be a function such that  

 lim
n→∞

Θ(𝒬njx,𝒬njy,𝒬njz)

𝒬nj
= 0 (1) 

 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Let ℎ: 𝑋 → 𝑌 be a function satisfying the inequality  

 ‖𝐻(𝑥, 𝑦, 𝑧)‖ ≤ Θ(𝑥, 𝑦, 𝑧) (2) 

 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then there exists a unique additive mapping 𝐴: 𝑋 → 𝑌 and satisfying the functional 

equation (1) such that  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
1

𝒬(𝑠2+2𝑠+1)
∑∞
𝑘=

1−𝑗

2

Θ(𝒬𝑘𝑗𝑥,𝒬𝑘𝑗𝑥,𝒬𝑘𝑗𝑥)

𝒬𝑘𝑗
 (3) 

 for all 𝑥 ∈ 𝑋. The mapping 𝐴(𝑥) is defined by  

 𝐴(𝑥) = lim
𝑛→∞

ℎ(𝒬𝑛𝑗𝑥)

𝒬𝑛𝑗
 (4) 

 for all 𝑥 ∈ 𝑋. Here 𝒬 = (𝑝 + 𝑞) .  

Proof. Assume 𝑗 = 1. Replacing (𝑥, 𝑦, 𝑧) by (𝑥, 𝑥, 𝑥) in (2), we get  

 ‖(𝑠2 + 2𝑠 + 1)ℎ(𝒬𝑥) − 𝒬(𝑠2 + 2𝑠 + 1)ℎ(𝑥)‖ ≤ Θ(𝑥, 𝑥, 𝑥) (5) 

 for all 𝑥 ∈ 𝑋. The above inequality can written as  

 ‖𝒬(𝑠2 + 2𝑠 + 1)ℎ(𝑥) − (𝑠2 + 2𝑠 + 1)ℎ(𝒬𝑥)‖ ≤ Θ(𝑥, 𝑥, 𝑥) (6) 

 for all 𝑥 ∈ 𝑋. bothside divide by (𝑠2 + 2𝑠 + 1)𝒬 in (6), we have  

 ‖ℎ(𝑥) −
ℎ(𝒬𝑥)

𝒬
‖ ≤

Θ(𝑥,𝑥,𝑥)

𝒬(𝑠2+2𝑠+1)
 (7) 

 for all 𝑥 ∈ 𝑋. Now replacing 𝑥 by 𝒬𝑥 and dividing by 𝒬 in (7), we get  

 ‖
ℎ(𝒬𝑥)

𝒬
−

ℎ(𝒬2𝑥)

𝒬2
‖ ≤

Θ(𝒬𝑥,𝒬𝑥,𝒬𝑥)

𝒬2(𝑠2+2𝑠+1)
 (8) 

 for all 𝑥 ∈ 𝑋. From (7) and (8), we obtain  

 ‖ℎ(𝑥) −
ℎ(𝒬2𝑥)

𝒬2
‖ ≤ ‖ℎ(𝑥) −

ℎ(𝒬𝑥)

𝒬
‖ + ‖

ℎ(𝒬𝑥)

𝒬
−
ℎ(𝒬2𝑥)

𝒬2
‖ 

 ≤
1

𝒬(𝑠2+2𝑠+1)
[Θ(𝑥, 𝑥, 𝑥) +

Θ(𝒬𝑥,𝒬𝑥,𝒬𝑥)

𝒬
] (9) 

 for all 𝑥 ∈ 𝑋. In general for any positive integer 𝑛 , we get  

 ‖ℎ(𝑥) −
ℎ(𝒬𝑛𝑥)

𝒬𝑛
‖ ≤

1

𝒬(𝑠2+2𝑠+1)
∑𝑛−1𝑘=0

Θ(𝒬𝑘𝑥,𝒬𝑘𝑥,𝒬𝑘𝑥)

𝒬𝑘
 (10) 

 ≤
1

𝒬(𝑠2+2𝑠+1)
∑∞𝑘=0

Θ(𝒬𝑘𝑥,𝒬𝑘𝑥,𝒬𝑘𝑥)

𝒬𝑘
 



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 for all 𝑥 ∈ 𝑋. In order to prove the convergence of the sequence  

 {
ℎ(𝒬𝑛𝑥)

𝒬𝑛
}, 

replace 𝑥 by 𝒬𝑚𝑥 and dividing by 𝒬𝑚 in (10), for any 𝑚,𝑛 > 0 , we deduce  

 ‖
ℎ(𝒬𝑚𝑥)

𝒬𝑚
−
ℎ(𝒬𝑛+𝑚𝑥)

𝒬(𝑛+𝑚)
‖ =

1

𝒬𝑚
  ‖ℎ(𝒬𝑚𝑥) −

ℎ(𝒬𝑛⋅𝒬𝑚𝑥)

𝒬𝑛
‖ 

 ≤
1

𝒬(𝑠2+2𝑠+1)
∑𝑛−1𝑘=0

Θ(𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥)

𝒬𝑘+𝑚
 

 ≤
1

𝒬(𝑠2+2𝑠+1)
∑∞𝑘=0

Θ(𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥)

𝒬𝑘+𝑚
 

 → 0     𝑎𝑠   𝑚 → ∞ 

 for all 𝑥 ∈ 𝑋. Hence the sequence {
ℎ(𝒬𝑛𝑥)

𝒬𝑛
} is Cauchy sequence. Since 𝑌 is complete, there exists a 

mapping 𝐴:𝑋 → 𝑌 such that  

 𝐴(𝑥) = lim
𝑛→∞

 
ℎ(𝒬𝑛𝑥)

𝒬𝑛
    ∀   𝑥 ∈ 𝑋. 

Letting 𝑛 → ∞ in (10) we see that (3) holds for all 𝑥 ∈ 𝑋. To prove that 𝐴 satisfies (1), replacing 

(𝑥, 𝑦, 𝑧) by (𝒬𝑛𝑥, 𝒬𝑛𝑦, 𝒬𝑛𝑧) and dividing by 𝒬𝑛 in (2), we obtain  

 
1

𝒬𝑛
‖𝐻(𝒬𝑛𝑥, 𝒬𝑛𝑦, 𝒬𝑛𝑧)‖ ≤

1

𝒬𝑛
Θ(𝒬𝑛𝑥, 𝒬𝑛𝑦, 𝒬𝑛𝑧) 

 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Letting 𝑛 → ∞ in the above inequality and using the definition of 𝐴(𝑥), we see that  

 (𝑠2 + 2𝑠)𝐴(𝑝𝑥 + 𝑞𝑦) + (1 − 2𝑠)𝐴(𝑝𝑦 + 𝑞𝑧) + 2𝑠𝐴(𝑝𝑧 + 𝑞𝑥) − 2𝑠𝑝𝐴(𝑥 − 𝑦) 

                 −2𝑠𝑞𝐴(𝑦 − 𝑧) = (𝑠2𝑝 + 2𝑠𝑞)𝐴(𝑥) + (𝑝 + 𝑠2𝑞)𝐴(𝑦) + (2𝑠𝑝 + 𝑞)𝐴(𝑧) 

 Hence 𝐴 satisfies (1) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. To prove 𝐴 is unique, we let 𝐵(𝑥) be another mapping 

satisfying (1) and (3), then  

 ‖𝐴(𝑥) − 𝐵(𝑥)‖ =
1

𝒬𝑛
‖𝐴(𝒬𝑛𝑥) − 𝐵(𝒬𝑛𝑥)‖ 

 ≤
1

𝒬𝑛
{‖𝐴(𝒬𝑛𝑥) − ℎ(𝒬𝑛𝑥)‖ + ‖ℎ(𝒬𝑛𝑥) − 𝐵(𝒬𝑛𝑥)‖} 

 ≤
2

(𝑠2+2𝑠+1)
∑∞𝑘=0

Θ(𝒬𝑘+𝑛𝑥,𝒬𝑘+𝑛𝑥,𝒬𝑘+𝑛𝑥)

𝒬(𝑘+𝑛)
 

 → 0     𝑎𝑠   𝑛 → ∞ 

 for all 𝑥 ∈ 𝑋. Hence 𝐴 is unique.  

Corollary 2.2 Let Θ and s be nonnegative real numbers. Let a function h: X → Y satisfies the inequality  

‖𝐻(𝑥, 𝑦, 𝑧)‖ ≤

{
 

 
Θ,
Θ{||𝑥||𝑠 + ||𝑦||𝑠 + ||𝑧||𝑠},  𝑠 ≠ 1; 

Θ||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠,  3𝑠 ≠ 1; 

Θ{||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠 + {||𝑥||3𝑠 + ||𝑦||3𝑠 + ||𝑧||3𝑠}},  3𝑠 ≠ 1; 

 (11) 



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 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then there exists a unique additive function 𝐴: 𝑋 → 𝑌 such that  

‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤

{
  
 

  
 

Θ

(𝑠2+2𝑠+1)|𝒬−1|
,

3 Θ||𝑥||𝑠

(𝑠2+2𝑠+1)|𝒬−𝒬𝑠|
,

Θ||𝑥||3𝑠

(𝑠2+2𝑠+1)|𝒬−𝒬3𝑠|
,

4 Θ||𝑥||3𝑠

(𝑠2+2𝑠+1)|𝒬−𝒬3𝑠|

 (12) 

 for all 𝑥 ∈  𝑋.  

3  STABILITY RESULTS:FIXED POINT METHOD 

Theorem 3.1 [22](The alternative of fixed point)  Suppose that for a complete generalized metric space 

(X, d) and a strictly contractive mapping T: X → X with Lipschitz constant L. Then, for each given 

element x ∈ X, either 

(B1)   d(T
nx, Tn+1x) = ∞    ∀  n ≥ 0,  

(B2) there exists a natural number n0 such that: 

(i) d(Tnx, Tn+1x) < ∞ for all n ≥ n0 ; 

(ii)The sequence (Tnx) is convergent to a fixed point y∗ of T; 

(iii) y∗ is the unique fixed point of T in the set Y = {y ∈ X: d(Tn0x, y) < ∞}; 

(iv) d(y∗, y) ≤
1

1−L
  d(y, Ty) for all y ∈ Y.  

Theorem 3.2  Let h: V → B be a mapping for which there exists functions α, β, γ: V3 → [0,∞) with the 

condition  

 lim
k→∞

 
α(μi

kx,μi
ky,μi

kz)

μi
k = 0, (13) 

 where  

 μi = {
𝒬,  i = 0 ,
1

𝒬
,  i = 1  

 satisfying the functional inequality  

 ‖H(x, y, z)‖ ≤ α(x, y, z) (14) 

 for all x, y, z ∈ V. If there exists an L = L(i) < 1 such that the function  

 x → γ(x) =
1

(s2+2s+1)
Θ (

x

𝒬
), 

one has the property  

 γ(x) = L μi γ (
x

μi
) (15) 



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 for all x ∈ V. Then there exists a unique additive function A: V → B satisfying the functional equation 

(1) and  

 ‖h(x) − A(x)‖ ≤
L1−i

1−L
γ(x) (16) 

 holds for all x ∈ X.  

  

Proof. Consider the set 𝑋 = {𝑝/𝑝: 𝑉 → 𝐵, 𝑝(0) = 0} and introduce the generalized metric on 𝑋,  

 𝑑(𝑝, 𝑞) = inf{𝐾 ∈ (0,∞): ∥ 𝑝(𝑥) − 𝑞(𝑥) ∥≤ 𝐾𝛾(𝑥), 𝑥 ∈ 𝑉}. 

It is easy to see that (𝑋, 𝑑) is complete.  

 Define 𝑇: 𝑋 → 𝑋 by  

 𝑇𝑝(𝑥) =
1

𝜇𝑖
𝑝(𝜇𝑖𝑥), ∀     𝑥 ∈ 𝑉. 

Now 𝑝, 𝑞 ∈ 𝑋,  

 𝑑(𝑝, 𝑞) ≤ 𝐾 ⇒∥ 𝑝(𝑥) − 𝑞(𝑥) ∥≤ 𝐾𝛾(𝑥), 𝑥 ∈ 𝑉. 

 ⇒ ‖
1

𝜇𝑖
𝑝(𝜇𝑖𝑥) −

1

𝜇𝑖
𝑞(𝜇𝑖𝑥)‖ ≤

1

𝜇𝑖
𝐾𝛾(𝜇𝑖𝑥), 𝑥 ∈ 𝑉, 

 ⇒ ‖
1

𝜇𝑖
𝑝(𝜇𝑖𝑥) −

1

𝜇𝑖
𝑞(𝜇𝑖𝑥)‖ ≤ 𝐿𝐾𝛾(𝑥), 𝑥 ∈ 𝑉, 

 ⇒∥ 𝑇𝑝(𝑥) − 𝑇𝑞(𝑥) ∥≤ 𝐿𝐾𝛾(𝑥), 𝑥 ∈ 𝑉, 

 ⇒ 𝑑(𝑇𝑝, 𝑇𝑞) ≤ 𝐿𝐾. 

 This implies  

 𝑑(𝑇𝑝, 𝑇𝑞) ≤ 𝐿𝑑(𝑝, 𝑞), 

for all 𝑝, 𝑞 ∈ 𝑋 . i.e., 𝑇 is a strictly contractive mapping on 𝑋 with Lipschitz constant 𝐿.  

 From (7), we have  

 ‖ℎ(𝑥) −
ℎ(𝒬𝑥)

𝒬
‖ ≤

Θ(𝑥,𝑥,𝑥)

𝒬(𝑠2+2𝑠+1)
 (17) 

 where  𝛽(𝑥) =
Θ(𝑥,𝑥,𝑥)

𝒬(𝑠2+2𝑠+1)
 

for all 𝑥 ∈ 𝑉. Using (15) for the case 𝑖 = 0, it reduces to  

‖
1

𝒬
ℎ(𝒬𝑥) − ℎ(𝑥)‖ ≤

1

𝒬
𝛾(𝑥) 

for all 𝑥 ∈ 𝑉.  

 𝑖. 𝑒. ,        𝑑(𝑇ℎ, ℎ) ≤
1

𝒬
= 𝐿 = 𝐿1−0 = 𝐿1−𝑖 < ∞. 

Again replacing 𝑥 =
𝑥

𝒬
 in (17), we get  



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 ‖ℎ(𝑥) − 𝒬𝑓 (
𝑥

𝒬
)‖ ≤

1

(𝑠2+2𝑠+1)
Θ(

𝑥

𝒬
). 

 for all 𝑥 ∈ 𝑉. Using (15) for the case 𝑖 = 1, it reduces to  

 ‖ℎ(𝑥) − 𝒬𝑓 (
𝑥

𝒬
)‖ ≤ 𝛾(𝑥) 

for all 𝑥 ∈ 𝑉.  

 𝑖. 𝑒. ,        𝑑(ℎ, 𝑇ℎ) ≤ 1 = 𝐿0 = 𝐿1−1 = 𝐿1−𝑖 < ∞. 

 

 In the above cases, we arrive  

 𝑑(ℎ, 𝑇ℎ) ≤ 𝐿1−𝑖 . 

Therefore (𝐵2(𝑖)) holds.  

 By (𝐵2(𝑖𝑖)), it follows that there exists a fixed point 𝐴 of 𝑇 in 𝑋 such that  

 𝐴(𝑥) = lim
𝑘→∞

 
ℎ(𝜇𝑖

𝑘𝑥)

𝜇𝑖
𝑘 ,        ∀  𝑥 ∈ 𝑉. (18) 

  

 Claim that 𝐴: 𝑉 → 𝐵 is additive. Replacing (𝑥, 𝑦, 𝑧) by (𝜇𝑖
𝑘𝑥, 𝜇𝑖

𝑘𝑦, 𝜇𝑖
𝑘𝑧) in (14) and dividing by 𝜇𝑖

𝑘, it 

follows from (13) and (18), 𝐴 satisfies (1) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋.  

 By (𝐵2(𝑖𝑖𝑖)), 𝐴 is the unique fixed point of 𝑇 in the set 𝑌 = {ℎ ∈ 𝑋: 𝑑(𝑇ℎ, 𝐴) < ∞}, using the fixed 

point alternative result 𝐴 is the unique function such that  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤ 𝐾𝛾(𝑥) 

for all 𝑥 ∈ 𝑉 and 𝐾 > 0. Finally by (𝐵2(𝑖𝑣)), we obtain  

 𝑑(ℎ, 𝐴) ≤
1

1−𝐿
𝑑(𝑔, 𝑇𝑔) 

implying  

 𝑑(ℎ, 𝐴) ≤
𝐿1−𝑖

1−𝐿
. 

Hence we conclude that  

 ∥ ℎ(𝑥) − 𝐴(𝑥) ∥≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥). 

for all 𝑥 ∈ 𝑉.  

 Corollary 3.3  Let h: V → B be a mapping and there exist real numbers Θ and s such that  

 ‖H(x, y, z)‖ ≤

{
 

 
Θ,
Θ{||x||s + ||y||s + ||z||s},  s ≠ 1; 

Θ||x||s||y||s||z||s,  3s ≠ 1; 

Θ{||x||s||y||s||z||s + {||x||3s + ||y||3s + ||z||3s}},  3s ≠ 1; 

 (19) 



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 for all x, y, z ∈ X, then there exists a unique additive function A: V → B such that  

 ‖h(x) − A(x)‖ ≤

{
  
 

  
 

Θ

(s2+2s+1)|𝒬−1|
,

3 Θ||x||s

(s2+2s+1)|𝒬−𝒬s|
,

Θ||x||3s

(s2+2s+1)|𝒬−𝒬3s|
,

4 Θ||x||3s

(s2+2s+1)|𝒬−𝒬3s|

 (20) 

 for all x ∈ X.  

 Proof. Let us set  

 𝛼(𝑥, 𝑦, 𝑧) = {

Θ,
Θ{||𝑥||𝑠 + ||𝑦||𝑠 + ||𝑧||𝑠},
Θ  ||𝑥||𝑠 ||𝑦||𝑠 ||𝑧||𝑠 ,

Θ{||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠 + (||𝑥||3𝑠 + ||𝑦||3𝑠 + ||𝑧||3𝑠)}

 

 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. Now  

 
𝛼(𝜇𝑖

𝑘𝑥,𝜇𝑖
𝑘𝑦,𝜇𝑖

𝑘𝑧)

𝜇𝑖
𝑘 =

{
  
 

  
 
Θ

𝜇𝑖
𝑘 ,

Θ

𝜇𝑖
𝑘 {||𝜇𝑖

𝑘𝑥||𝑠 + ||𝜇𝑖
𝑘𝑦||𝑠 + ||𝜇𝑖

𝑘𝑧||𝑠},

Θ

𝜇𝑖
𝑛 ||𝜇𝑖

𝑛𝑥||𝑠 ||𝜇𝑖
𝑛𝑦||𝑠  ||𝜇𝑖

𝑛𝑧||𝑠,

Θ

𝜇𝑖
𝑘 {||𝜇𝑖

𝑘𝑥||𝑠 ||𝜇𝑖
𝑘𝑦||𝑠 ||𝜇𝑖

𝑘𝑧||𝑠{||𝜇𝑖
𝑘𝑥||3𝑠 + ||𝜇𝑖

𝑘𝑦||3𝑠 + ||𝜇𝑖
𝑘𝑧||3𝑠}}

 

 = {

→ 0 𝑎𝑠  𝑘 → ∞,
→ 0 𝑎𝑠  𝑘 → ∞,
→ 0 𝑎𝑠  𝑘 → ∞,
→ 0 𝑎𝑠  𝑘 → ∞.

 

 i.e., (13) is holds. But, we have  

 𝛾(𝑥) =
1

(𝑠2+2𝑠+1)
[Θ (

𝑥

𝒬
,
𝑥

𝒬
,
𝑥

𝒬
)]. 

 Hence  

 𝛾(𝑥) =
1

(𝑠2+2𝑠+1)
[Θ (

𝑥

𝒬
,
𝑥

𝒬
,
𝑥

𝒬
)] =

{
  
 

  
 

Θ

(𝑠2+2𝑠+1)𝒬
,

3Θ

(𝑠2+2𝑠+1)𝒬𝑠
||𝑥||𝑠,

Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠,

4Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠.

 

 Also,  



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1

𝜇𝑖
𝛾(𝜇𝑖𝑥) =

{
  
 

  
 

Θ

𝜇𝑖⋅(𝑠
2+2𝑠+1)𝒬

,

3Θ

𝜇𝑖⋅(𝑠
2+2𝑠+1)𝒬𝑠

||𝜇𝑖𝑥||
𝑠 ,

Θ

𝜇𝑖⋅(𝑠
2+2𝑠+1)𝒬3𝑠

||𝜇𝑖𝑥||
3𝑠,

Θ

𝜇𝑖⋅(𝑠
2+2𝑠+1)𝒬3𝑠

||𝜇𝑖𝑥||
3𝑠.

 

 =

{
  
 

  
 𝜇𝑖

−1 Θ

(𝑠2+2𝑠+1)
,

𝜇𝑖
𝑠−1 3Θ

(𝑠2+2𝑠+1)𝒬𝑠
||𝑥||𝑠,

𝜇𝑖
3𝑠−1 Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠,

𝜇𝑖
3𝑠−1 4Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠.

 

 =

{
 
 

 
 𝜇𝑖

−1𝛾(𝑥),

𝜇𝑖
𝑠−1𝛾(𝑥),

𝜇𝑖
3𝑠−1𝛾(𝑥),

𝜇𝑖
3𝑠−1𝛾(𝑥).

 

 we prove the following cases for conditions using (16) 

Case:1  𝐿 = 𝒬−1 if 𝑖 = 0  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(𝒬−1)
1−0

1−(𝒬)−1
⋅

Θ

(𝑠2+2𝑠+1)𝒬
=

Θ

(𝑠2+2𝑠+1)(𝒬−1)
. 

 Case:2  𝐿 = 𝒬 if 𝑖 = 1  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(𝒬)1−1

1−𝒬
⋅

Θ

(𝑠2+2𝑠+1)𝒬
=

Θ

(𝑠2+2𝑠+1)(1−𝒬)
. 

 Case:1  𝐿 = 𝒬𝑠−1 if 𝑖 = 0  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(𝒬𝑠−1)
1−0

1−𝒬𝑠−1
3Θ

(𝑠2+2𝑠+1)𝒬𝑠
||𝑥||𝑠 

 =
𝒬𝑠

𝒬−𝒬𝑠
3Θ

(𝑠2+2𝑠+1)𝒬𝑠
||𝑥||𝑠 

 =
3Θ||𝑥||𝑠

(𝑠2+2𝑠+1)(𝒬−𝒬𝑠)
. 

 Case:2  𝐿 =
1

𝒬𝑠−1
 if 𝑖 = 1  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(
1

𝒬𝑠−1
)
1−1

1−
1

(𝑠2+2𝑠+1)𝒬𝑠−1

3Θ

(𝑠2+2𝑠+1)𝒬𝑠
||𝑥||𝑠 

 =
𝒬𝑠

𝒬𝑠−𝒬

3Θ

(𝑠2+2𝑠+1)𝒬𝑠
||𝑥||𝑠 

 =
3Θ||𝑥||𝑠

(𝑠2+2𝑠+1)(𝒬𝑠−𝒬)
. 



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 Case:1  𝐿 = 𝒬3𝑠−1 if 𝑖 = 0  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(𝒬3𝑠−1)
1−0

1−𝒬3𝑠−1
Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
𝒬3𝑠

𝒬−𝒬3𝑠
Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
Θ||𝑥||3𝑠

(𝑠2+2𝑠+1)(𝒬−𝒬3𝑠)
. 

 Case:2  𝐿 =
1

𝒬3𝑠−1
 if 𝑖 = 1  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(
1

𝒬3𝑠−1
)
1−1

1−
1

𝒬3𝑠−1

Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
𝒬3𝑠

𝒬3𝑠−𝒬

Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
Θ||𝑥||3𝑠

(𝑠2+2𝑠+1)(𝒬3𝑠−𝒬)
. 

 Case:1  𝐿 = 𝒬3𝑠−1 if 𝑖 = 0  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(𝒬3𝑠−1)
1−0

1−𝒬3𝑠−1
4Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
𝒬3𝑠

𝒬−𝒬3𝑠
4Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
4Θ||𝑥||3𝑠

(𝑠2+2𝑠+1)(𝒬−𝒬3𝑠)
. 

 Case:2  𝐿 =
1

𝒬3𝑠−1
 if 𝑖 = 1  

 ‖ℎ(𝑥) − 𝐴(𝑥)‖ ≤
𝐿1−𝑖

1−𝐿
𝛾(𝑥) =

(
1

𝒬3𝑠−1
)
1−1

1−
1

𝒬3𝑠−1

4Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
𝒬3𝑠

𝒬3𝑠−𝒬

4Θ

(𝑠2+2𝑠+1)𝒬3𝑠
||𝑥||3𝑠 

 =
4Θ||𝑥||3𝑠

(𝑠2+2𝑠+1)(𝒬3𝑠−𝒬)
. 

4  Conclusion 

 The research article titled "Analyzing the Stability of Euler-Lagrange Additive Functional Equations 

in Banach Spaces: A Fixed Point and Direct Method Perspective" investigates the stability of specific 

functional equations within Banach spaces. The study employs both fixed point and direct methods to 

establish conditions under which these equations exhibit stability. The findings contribute to a deeper 

understanding of the behavior of Euler-Lagrange additive functional equations in the context of 

Banach spaces, offering valuable insights for further research in this area.  

Conflict of interest. The authors declare that they have no competing interests. 



Communications on Applied Nonlinear Analysis 

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