Hyers–Ulam Stability of Additive Functional Equation Using Direct and Fixed-Point Methods

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equation χ(􏽐lg�1 vg) � 􏽐1≤g<h<i≤lχ(vg + vh + vi) − 􏽐1≤g<h≤lχ(vg + vh) − ((l2 − 5l + 2)/2) 􏽐 l g�1(χ(vg) − χ(− vg)/2). where l is a nonnegative integer with N − 0, 1, 2, 3, 4 { } in Banach spaces.


Introduction
e problem of Ulam-Hyers stability concerns determining circumstances under which, given an approximate solution of a functional equation, one may locate an exact key that is closer to it in some sense. e investigation of stability problem for functional equations is identified to a question of Ulam [1] about the stability of group homomorphisms and affirmatively answered for Banach space by Hyers [2,3]. It was further generalized and interesting results were obtained by a number of authors [4][5][6].
In 2019, Park et al. [33] introduced additive s-functional inequality. Using the fixed-point method and direct method, he established the Hyers-Ulam stability for the abovementioned one in complex Banach spaces. Also, he examined the Hyers-Ulam stability of homomorphism and derivations in complex Banach algebras. In 2018, Almahalebi [34] investigated the quadratic functional equation in Banach spaces. And, he established the hyperstability outcome of the same equation through the fixed-point approach.
Radu [35] investigated various results about the stability problem by using the fixed-point alternative. He applied the fixed-point method to examine the stability of Cauchy functional equation and Jensen's functional equations. After his work, numerous authors used the fixed-point method to investigate several functional equations [36][37][38][39][40][41]. e functional equation is called the Cauchy additive functional equation and it is the most famous functional equation. As f(x) � cx is the solution of (1), every solution of the additive equation is called an additive function.
In this present work, we derive the solution of the generalized additive functional equation along with established Hyers-Ulam stability results by using direct and fixed-point methods for a generalized additive functional equation where l ≥ 5 is a nonnegative integer in Banach spaces.

General Solution of the Functional Equation (2)
In this section, we derive the general solution of the generalized additive functional equation (2). Here, we consider Φ and Ω be real vector spaces.
Proof. Suppose a mapping χ: Φ ⟶ Ω satisfies the functional equation (2). (2) and using the property of odd function, we have for all v ∈ Φ. Replacing v by 2v in (3), we obtain for all v ∈ Φ. Again, replacing v by 2v in (5) and using (3), we have for all v ∈ Φ. We can generalize for any nonnegative integer n and we get (2), we obtain our desired result of equation (1).

Remark 1.
Let Ω be a linear space and a function χ: Φ ⟶ Ω satisfies the functional equation (2). en, the following claims hold: In Sections 3 and 4, we take Φ be a normed space and Ω be a Banach space. For our convincing effortlessness, we describe a function Θ: Φ ⟶ Ω as for

Hyers-Ulam Stability of the Functional Equation (2): Direct Method
In this section, we investigated the Hyers-Ulam stability of the generalized additive functional equation (2) in Banach space by using the direct method.
for all v 1 , v 2 , . . . , v l ∈ Φ, then there exists a unique additive mapping Ψ: Φ ⟶ Ω satisfying equation (2) and for all v ∈ Φ. From equality (12), we get for all v ∈ Φ. Exchanging v through 2v in (13), we obtain for all v ∈ Φ. From (14), we achieve for all v ∈ Φ. Adding together (13) and (15), we get the following outcome: for all v ∈ Φ. It follows from (13), (15), and (16), and we can generalize that as follows: for all v ∈ Φ. In order to establish the convergence of the sequence χ(2 w v)/2 w , switch v through 2 s v and also divide by 2 s in (17). We conclude that, for some w, s > 0, for all v ∈ Φ. erefore, the sequence χ(2 w v)/2 w is a Cauchy. As Ω is complete, there exists Ψ: Φ ⟶ Ω so that Ψ(v) � lim w⟶∞ (χ(2 w v)/2 w ) for all v ∈ Φ. Taking the limit w ⟶ ∞ in (17), we obtain that result (11) holds for all v ∈ Φ. To prove that the function Ψ satisfies equation (2), replacing (v 1 , v 2 , . . . , v l ) by (2 w v 1 , 2 w v 2 , . . . , 2 w v l ) and also dividing by 2 w in (10), we get for all v 1 , v 2 , . . . , v l ∈ Φ. Taking the limit w ⟶ ∞ in the above inequality and using the definition of Ψ(v), we have us, the function Ψ satisfies equation (2). To prove that the function Ψ is unique, let φ: Φ ⟶ Ω be another additive mapping satisfying the functional equation (2) and (11). Hence, Hence, Ψ is unique. Now, replacing v through (v/2) in (12), we have for all v ∈ Φ. e rest of the proof is similar to that when ζ � 1. So for ζ � − 1, we can prove the results by a similar manner. Hence, the proof is completed.

Corollary 1. Let ϕ and ϑ be positive real numbers. If there exists a mapping Θ: Φ ⟶ Ω satisfying the inequality
for all v 1 , v 2 , . . . , v l ∈ Φ, then there exists a unique additive mapping Ψ: Φ ⟶ Ω such that for all v ∈ Φ.

Hyers-Ulam Stability of the Functional Equation (2): Fixed-Point Method
In this section, we examined the Hyers-Ulam stability of the generalized additive functional equation (2) in Banach space by using the fixed-point method.
Theorem 3. Let Ψ: Φ ⟶ Ω be a mapping for which there exists a mapping ξ: Φ l ⟶ [0, ∞) and and such that it satisfies the inequality has the property for all v ∈ Φ. en, there exists a unique additive mapping Ψ: Φ ⟶ Ω satisfying equation (2) and for all v ∈ Φ.