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Article

# Generalized Hyers–Ulam Stability of the Additive Functional Equation

1
Department of Mathematics Education, Gongju National University of Education, Gongju 32553, Korea
2
Department of Mathematics, Kangnam University, Yongin 16979, Korea
*
Author to whom correspondence should be addressed.
Axioms 2019, 8(2), 76; https://doi.org/10.3390/axioms8020076
Received: 2 May 2019 / Revised: 20 June 2019 / Accepted: 21 June 2019 / Published: 25 June 2019
(This article belongs to the Special Issue Mathematical Analysis and Applications II)

## Abstract

:
We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation $f ( x 1 + y 1 , x 2 + y 2 , … , x n + y n ) = f ( x 1 , x 2 , … , x n ) + f ( y 1 , y 2 , … , y n ) .$ By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.
MSC:
39B82; 39B5

## 1. Introduction

In 1940, Ulam [1] gave the question concerning the stability of homomorphisms in a conference of the mathematics club of the University of Wisconsin as follows:
Let $( G , · )$ be a group, and let $( G ′ , · , d )$ be a metric group with the metric d. Given $δ > 0 ,$ does there exist $ϵ > 0$ such that if a mapping $h : G → G ′$ satisfies the inequality
$d ( h ( x y ) , h ( x ) h ( y ) ) ≤ δ$
for all $x , y ∈ G$, then there is a homomorphism $H : G → H$ with
$d ( h ( x ) , H ( x ) ) ≤ ϵ$
for all $x ∈ G$?
Next year, the Ulam’s conjecture was partially solved by Hyers [2] for the additive functional equation.
Theorem 1.
[2], Let X and Y be Banach spaces. Suppose that the mapping $f : X → Y$ satisfies the inequality
Then, there exists a unique additive mapping
$A ( x + y ) = A ( x ) + A ( y ) ,$
such that $| | f ( x ) − A ( x ) | | ≤ ε$, where the limit $A ( x ) = lim n → ∞ 2 − n f ( 2 n x ) .$
Thereafter, this phenomenon has been called the Hyers–Ulam stability.
Theorem 2.
Let X and Y be Banach spaces. Suppose that the mapping $f : X → Y$ satisfies the inequality
for all $x , y ∈ X \ { 0 }$, where θ and p are constants with $θ > 0$ and $p ≠ 1 .$ Then, there exists a unique additive mapping $T : X → Y$ such that
for all $x ∈ X \ { 0 } .$
Theorem 2 is due to Aoki [3] and Rassias [4] for $0 < p < 1$, Gajda [5] for $p > 1$, Hyers [2] for $p = 0$, and Rassias [6] for $p < 0$.
In 1994, Găvruta [7] generalized these results for additive mapping by replacing $θ ( x p + y p )$ in (1) by a general function $φ ( x , y )$, which is called the ‘generalized Hyers–Ulam stability’ in this paper.
In 2001, the term hyperstability was used for the first time probably by G. Maksa and Z. Páles in [8]. However, in 1949, it seems to have created by D. G. Bourgin [9] that the first hyperstability result concerned the ring homomorphisms.
We say that a functional equation $D ( f ) = 0$ is hyperstable if any function f satisfying the equation $D ( f ) = 0$ approximately is a true solution of $D ( f ) = 0$, which is a phenomenon called hyperstability.
The hyperstability results for the additive (Cauchy) equation were investigated by Brzdȩk [10,11].
In this paper, let V and W be vector spaces, X be a real normed space, and Y be a real Banach space. We denote the set of natural numbers by $N$ and the set of real numbers by $R$.
For a given mapping $f : V n → W$, where $V n$ denotes $V × V × ⋯ × V$, let us consider the additive functional equation
$f ( x 1 + y 1 , x 2 + y 2 , … , x n + y n ) = f ( x 1 , x 2 , … , x n ) + f ( y 1 , y 2 , … , y n ) ,$
for all $x i , y i ∈ V$ ($i = 1 , 2 , … , n$).
Each solution of the additive functional Equation (3) is called an n-variable additive mapping. A typical example for the solutions of Equation (3) is the mapping $f : R n → R l$ given by $f ( x 1 , x 2 , … , x n ) = ( ∑ i = 1 n a 1 i x i , ∑ i = 1 n a 2 i x i , … , ∑ i = 1 n a l i x i )$ with real constants $a i j$.
In this paper, we will prove the generalized Hyers–Ulam stability of the additive functional Equation (3) in the spirit of Găvruta [7], and the hyperstability of the additive functional Equation (3).

## 2. Main Results

For a given mapping $f : V n → W$, we use the following abbreviation:
$D f ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) : = f ( x 1 + y 1 , x 2 + y 2 , … , x n + y n ) − f ( x 1 , x 2 , … , x n ) − f ( y 1 , y 2 , … , y n )$
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. We need the following lemma to prove main theorems.
Lemma 1.
If a mapping $f : V n → W$ satisfies (3) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$, then f satisfies (3) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$.
Proof.
Let $x ∈ V \ { 0 }$ be a fixed element, and let $i ∈ { 1 , 2 , … , n }$. For given $x i , y i ∈ V$, let $x i ( 1 )$, $x i ( 2 )$, $y i ( 1 )$, $y i ( 2 )$ be
$x i ( 1 ) = x , x i ( 2 ) = − x , y i ( 1 ) = x , y i ( 2 ) = − x if x i = 0 and y i = 0 , x i ( 1 ) = y i , x i ( 2 ) = − y i , y i ( 1 ) = y i 2 , y i ( 2 ) = y i 2 if x i = 0 and y i ≠ 0 , x i ( 1 ) = x i 2 , x i ( 2 ) = x i 2 , y i ( 1 ) = x i , y i ( 2 ) = − x i if x i ≠ 0 and y i = 0 , x i ( 1 ) = x i 2 , x i ( 2 ) = x i 2 , y i ( 1 ) = ( k + 1 ) y i , y i ( 2 ) = − k y i if x i ≠ 0 and y i ≠ 0 ,$
where k is a fixed integer, such that $x i 2 + ( k + 1 ) y i ≠ 0 , x i 2 − k y i ≠ 0$. Then, $x i ( 1 ) , x i ( 2 ) , y i ( 1 ) , y i ( 2 ) , x i ( 1 ) + y i ( 1 ) , x i ( 2 ) + y i ( 2 ) ∈ V \ { 0 }$ and $x i ( 1 ) + y i ( 1 ) + x i ( 2 ) + y i ( 2 ) = x i + y i$ for all $i = 1 , 2 , … , n$.
Hence, the equalities $D f ( x 1 ( 1 ) , y 1 ( 1 ) , … , x n ( 1 ) , y n ( 1 ) ) = 0$, $D f ( x 1 ( 2 ) , y 1 ( 2 ) , … , x n ( 2 ) , y n ( 2 ) ) = 0$, $D f ( x 1 ( 1 ) , x 1 ( 2 ) , x 2 ( 1 ) , x 2 ( 2 ) , … , x n ( 1 ) , x n ( 2 ) ) = 0$, and $D f ( y 1 ( 1 ) , y 1 ( 2 ) , y 2 ( 1 ) , y 2 ( 2 ) , … , y n ( 1 ) , y n ( 2 ) ) = 0$ hold for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. Since the equality
$D f ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) = D f ( x 1 ( 1 ) + y 1 ( 1 ) , x 1 ( 2 ) + y 1 ( 2 ) , x 2 ( 1 ) + y 2 ( 1 ) , x 2 ( 2 ) + y 2 ( 2 ) , … , x n ( 1 ) + y n ( 1 ) , x n ( 2 ) + y n ( 2 ) ) + D f ( x 1 ( 1 ) , y 1 ( 1 ) , x 2 ( 1 ) , y 2 ( 1 ) , … , x n ( 1 ) , y n ( 1 ) ) + D f ( x 1 ( 2 ) , y 1 ( 2 ) , x 2 ( 2 ) , y 2 ( 2 ) , … , x n ( 2 ) , y n ( 2 ) ) − D f ( x 1 ( 1 ) , x 1 ( 2 ) , x 2 ( 1 ) , x 2 ( 2 ) , … , x n ( 1 ) , x n ( 2 ) ) − D f ( y 1 ( 1 ) , y 1 ( 2 ) , y 2 ( 1 ) , y 2 ( 2 ) , … , y n ( 1 ) , y n ( 2 ) )$
holds for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$, we conclude that f satisfies $D f ( x 1 , y 1 , … , x n , y n ) = 0$ for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. ☐
Thereafter, let $i ∈ { 1 , 2 , 3 , … , n }$. For a given element , we can choose a fixed element $x ′ ≠ 0$, such that $x ′ ∈ { x 1 , x 2 , … , x n }$. Moreover, let $x i ( 1 )$, $x i ( 2 ) ∈ V \ { 0 }$ be the elements defined by
$x i ( 1 ) = x i , x i ( 2 ) = x i if x i ≠ 0 , x i ( 1 ) = x ′ , x i ( 2 ) = − x ′ if x i = 0 .$
By using Lemma 1, we can prove the following set of stability theorems.
Theorem 3.
Suppose that $f : V n → Y$ is a mapping for which there exists a function $φ : ( V \ { 0 } ) 2 n → [ 0 , ∞ )$, such that
$∑ m = 0 ∞ φ ( 2 m x 1 , 2 m y 1 , 2 m x 2 , 2 m y 2 , … , 2 m x n , 2 m y n ) 2 m < ∞$
and
$∥ D f ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) ∥ ≤ φ ( x 1 , y 1 , x 2 , y 2 , … , x n , y n )$
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. Then, there exists a unique mapping $F : V n → Y$ that satisfies
$D F ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) = 0$
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$ and
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$, where the function $μ : V n → R$ is defined by
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$.
Proof.
From the inequality (6) and the equalities
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$, we have
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$. From the above inequality, we get the (following- 4 palces) inequality
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$ and all positive integers $m , m ′$. Thus, the sequence is a Cauchy sequence for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$. Since Y is a real Banach space and , we can define a mapping $F : V n → Y$ by
for all $x 1 , x 2 , … , x n ∈ V$. By putting $m = 0$ and by letting $m ′ → ∞$ in the inequalities (10), we can obtain the inequalities (8) for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$.
From the inequality (6), we can obtain
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. Since the right-hand side in the above equality tends to zero as $m → ∞$, and the equality
$D F ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) = lim m → ∞ D f ( 2 m x 1 , 2 m y 1 , 2 m x 2 , 2 m y 2 , … , 2 m x n , 2 m y n ) 2 m$
holds, then F satisfies the equality (7) for all $x 1 , y 1 , … , x n , y n ∈ V \ { 0 }$. By Lemma 1, F satisfies the equality (3) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. If $G : V n → Y$ is another n-variable additive mapping that satisfies (8), then we obtain $G ( 0 , 0 , … , 0 ) = 0 = F ( 0 , 0 … , 0 )$ and
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$ and all $k ∈ N$. Since $∑ m = k ∞ μ ( 2 m x 1 , 2 m x 2 , … , 2 m x n ) 2 m → 0$ as $k → ∞$, we have $G ( x 1 , x 2 , … , x n ) = F ( x 1 , x 2 , … , x n )$ for all $x 1 , x 2 , … , x n ∈ V$. Hence, the mapping F is the unique n-variable additive mapping, as desired. ☐
The condition $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$ used in the inequality (6) differs from the condition $( x 1 , x 2 , … , x n ) ≠ ( 0 , 0 , … , 0 )$ and $( y 1 , y 2 , … , y n ) ≠ ( 0 , 0 , … , 0 )$ handled by the other authors. If the function f satisfies the inequality (3.2) for all $( x 1 , x 2 , … , x n ) ≠ ( 0 , 0 , … , 0 )$ and $( y 1 , y 2 , … , y n ) ≠ ( 0 , 0 , … , 0 )$, then the function f satisfies the inequality (3.2) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. Therefore, the condition $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$ used in the inequality (3.2) in this paper is a generalization of the conditions used in the inequality (3.2) in the well-known pre-results ([10,11]). This condition will apply until Corollary 1.
Theorem 4.
Suppose that $f : V n → Y$ is a mapping for which there exists a function $φ : ( V \ { 0 } ) 2 n → [ 0 , ∞ )$ that satisfies
and (6) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. Then, there exists a unique mapping $F : V n → Y$ that satisfies (7) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$ and
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$, where the function $μ : V n → R$ is defined as Theorem 3.
Proof.
By choosing a fixed element $x ∈ V \ { 0 }$, we can obtain
so $f ( 0 , 0 , … , 0 ) = 0$. Since the equality (9) holds for all $( x 1 , x 2 , … , x n ) ∈ V \ { ( 0 , 0 , … , 0 ) }$, the inequality (6) implies the inequality
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$. From the above inequality, we can also obtain the inequality
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$ and all positive integers $m , m ′$. Thus, the sequences is a Cauchy sequence for all $( x 1 , … , x n ) ∈ V n \ { ( 0 , … , 0 ) }$. Since $f ( 0 , 0 , … , 0 ) = 0$ and Y is a real Banach space, we can define a mapping $F : V n → Y$ by
for all $x 1 , x 2 , … , x n ∈ V$. By putting $m = 0$ and by letting $m ′ → ∞$ in the inequality (13), we can obtain the inequality (12) for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$.
From the inequality (6), we get
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. Since the right-hand side in the above equality tends to zero as $m → ∞$, then F satisfies the equality (7) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. By Lemma 1, F satisfies the equality (3) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. If $G : V n → Y$ is another n-variable additive mapping satisfying (12), then we obtain $G ( 0 , 0 , … , 0 ) = 0 = F ( 0 , 0 , … , 0 )$ and
for all $( x 1 , x 2 , … , x n ) ∈ V n \ { ( 0 , 0 , … , 0 ) }$. Hence, the mapping F is the unique n-variable additive mapping, as desired. ☐
The following corollary follows from Theorems 3 and 4.
Corollary 1.
Let $( X , | | | · | | | )$ be a normed space, $θ > 0$, and let p be a real number with $p ≠ 1$. Suppose that $f : X n → Y$ is a mapping that satisfies
$∥ D f ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) ∥ ≤ θ ( | | | x 1 | | | p + | | | y 1 | | | p + | | | x 2 | | | p + … + | | | x n | | | p + | | | y n | | | p )$
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ X \ { 0 }$. Then, there exists a unique n-variable additive mapping $F : X n → Y$, such that
for all $( x 1 , x 2 , … , x n ) ∈ X n \ { ( 0 , 0 , … , 0 ) }$.
Proof.
Put $φ ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) : = θ ( | | | x 1 | | | p + | | | y 1 | | | p + | | | x 2 | | | p + | | | y 2 | | | p + … + | | | x n | | | p + | | | y n | | | p )$ for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ X \ { 0 }$, then $| | | x i ( 1 ) | | | , | | | x i ( 2 ) | | | ≤ max x i ≠ 0 { | | | x i | | | p : 1 ≤ i ≤ n }$ for all i from (4). Hence, due to $μ$ of Theorems 3 and 4, we obtain that
for all $( x 1 , x 2 , … , x n ) ∈ X n \ { ( 0 , 0 , … , 0 ) }$. Therefore, the inequality (15) can be obtained easily from (8) and (12) in Theorems 3 and 4. ☐
The following theorem for the hyperstability of n-variable additive functional equation follows from Corollary 1.
Theorem 5.
Let $( X , | | | · | | | )$ be a normed space and p be a real number with $p < 0$. Suppose that $f : X n → Y$ is a mapping that satisfies (14) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ X \ { 0 }$. Then, f is an n-variable additive mapping itself.
Proof.
By Corollary 1, there exists a unique n-variable additive mapping $F : X n → Y$, such that (15) for all $x 1 , x 2 , … , x n ∈ X n \ { ( 0 , 0 , … , 0 ) }$ and for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ X$.
For a given , let $x ′ ≠ 0$ be a nonzero fixed element in ${ x 1 , x 2 , … , x n }$, and let
$x i ( 3 ) = ( m + 1 ) x i , x i ( 4 ) = − m x i when x i ≠ 0 , x i ( 3 ) = m x ′ , x i ( 4 ) = − m x ′ when x i = 0 .$
Then, we can easily show that $| | | x i ( 3 ) | | | , | | | x i ( 4 ) | | | ≤ m p max x i ≠ 0 { | | | x i | | | p : 1 ≤ i ≤ n }$ for all i from (4). If $( x 1 , x 2 , … , x n ) ∈ X \ { ( 0 , 0 , … , 0 ) }$, then the equality $f ( x 1 , x 2 , … , x n ) = F ( x 1 , x 2 , … , x n )$ follows from the inequalities
as $m → ∞$. For $( x 1 , x 2 , … , x n ) = ( 0 , 0 , … , 0 )$, if we choose a fixed element of $x ∈ X \ { 0 }$, then the equality $f 0 , 0 , … , 0 ) = F 0 , 0 , … , 0 )$ follows from the inequalities
as $m → ∞$. Therefore, f is an n-variable additive mapping itself. ☐
The following example follows from Theorem 5.
Example 1.
Let $( R , | · | )$ be a normed space with absolute value $| · |$, $( R l , ∥ · ∥ )$ be a Banach space with Euclid norm $∥ · ∥$, and $p < 0$ be a real number. Suppose that $f : R n → R l$ is a continuous mapping such that
$∥ D f ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) ∥ ≤ θ ( | x 1 | p + | y 1 | p + | x 2 | p + | y 2 | p + … + | x n | p + | y n | p )$
for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ R \ { 0 }$. Then, the mapping $f : R n → R l$ given by
where$a 1 i , a 2 i , … , a l i$are real constants, indicates that
$f ( 1 , 0 , 0 , … , 0 ) = ( a 11 , a 21 , … , a l 1 ) , f ( 0 , 1 , 0 , … , 0 ) = ( a 12 , a 22 , … , a l 2 ) , ⋮ ⋮ f ( 0 , … , 0 , 0 , 1 ) = ( a 1 n , a 2 n , … , a l n ) .$
Proof.
Since $f : R n → R l$ is a continuous n-variable additive mapping by Theorem 5, then the function $f : R n → R l$ is given by (16). ☐
In the following theorems, we replace the domain $( V \ { 0 } ) 2 n$ of $φ$ and $D f$ in Theorems 3 and 4 with $V 2 n$. Then, we can improve the result inequality (8).
Theorem 6.
Suppose that $f : V n → Y$ is a mapping for which there exists a function $φ : V 2 n → [ 0 , ∞ )$ satisfying (5) and (6) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. Then, there exists a unique mapping $F : V n → Y$, such that (7) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$ and
for all $x 1 , x 2 , … , x n ∈ V$.
Proof.
The equality
for all $x 1 , x 2 , … , x n ∈ V$ and the inequality (6) imply that the inequality
for all $x 1 , x 2 , … , x n ∈ V$. From the above inequality, we can derive the inequalities
for all $x 1 , x 2 , … , x n ∈ V$ and all positive integers $m , m ′$. The remainder of the proof of this theorem developed after inequality (19) is omitted because it is similar to that of Theorem 3. ☐
Theorem 7.
Suppose that $f : V n → Y$ is a mapping for which there exists a function $φ : V 2 n → [ 0 , ∞ )$ satisfying (11) and (6) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$. Then, there exists a unique mapping $F : V n → Y$ that satisfies (7) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V$ and
for all $x 1 , x 2 , … , x n ∈ V$.
Proof.
The equality (18) for all $x 1 , x 2 , … , x n ∈ V$ and the inequality (6) imply that the inequality
for all $x 1 , x 2 , … , x n ∈ V$. From the above inequality, we can derive the inequality
for all $x 1 , x 2 , … , x n ∈ V$ and all positive integers $m , m ′$. The remainder of the proof of this theorem developed after inequality (21) is omitted because it is similar to that of Theorem 4. ☐
The following corollary follows from Theorems 6 and 7.
Corollary 2.
Let $( X , | | | · | | | )$ be a normed space and p be a nonnegative real number with $p ≠ 1$. Suppose that $f : X n → Y$ is a mapping satisfying (14) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ X$. Then, there exists a unique n-variable additive mapping $F : X n → Y$ such that
for all $x 1 , x 2 , … , x n ∈ X$.
Proof.
By putting $φ ( x 1 , y 1 , x 2 , y 2 , … , x n , y n ) : = θ ( | | | x 1 | | ∥ p + | | | y 1 | | | p + | | | x 2 | | ∥ p + | | | y 2 | | ∥ p + ⋯ + | | | x n | | | p + | | | y n | | | p )$ for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ X$, then we easily obtain (22) from (17) and (20) of Theorems 6 and 7. ☐

## 3. Conclusions

We obtained two stability results.
Theorems 3 and 4 are the generalized Hyers–Ulam stability for the additive functional Equation (3) on $V n$, which is a generalization for the stability of the Cauchy functional equation in papers of Aoki [3], Rassias [4], Gajda [5], Hyers [2], and Găvruta [7].
Theorems 6 and 7 are the hyperstablity of the additive functional Equation (3) on $V n$, which is a generalization of the Brzdȩk’s results [10,11] for the Cauchy functional equation.
If the function f satisfies the inequality (6) for all $( x 1 , x 2 , … , x n ) ≠ ( 0 , 0 , … , 0 )$ and $( y 1 , y 2 , … , y n ) ≠ ( 0 , 0 , … , 0 )$, then the function f satisfies the inequality (6) for all $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$. Therefore, the condition $x 1 , y 1 , x 2 , y 2 , … , x n , y n ∈ V \ { 0 }$ used in the inequality (3.2) of this paper is a generalization of the conditions used in the inequality (6) in well-known pre-results ([10,11]).

## Author Contributions

Conceptualization, Y.-H.L. and G.H.K.; Investigation, Y.-H.L. and G.H.K.

## Funding

This research received no external funding.

## Conflicts of Interest

The authors declare no conflict of interest.

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Lee, Y.-H.; Kim, G.H. Generalized Hyers–Ulam Stability of the Additive Functional Equation. Axioms 2019, 8, 76. https://doi.org/10.3390/axioms8020076

AMA Style

Lee Y-H, Kim GH. Generalized Hyers–Ulam Stability of the Additive Functional Equation. Axioms. 2019; 8(2):76. https://doi.org/10.3390/axioms8020076

Chicago/Turabian Style

Lee, Yang-Hi, and Gwang Hui Kim. 2019. "Generalized Hyers–Ulam Stability of the Additive Functional Equation" Axioms 8, no. 2: 76. https://doi.org/10.3390/axioms8020076

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