# A Functional equation related to inner product spaces in non-archimedean normed spaces

## Abstract

In this paper, we prove the Hyers-Ulam stability of a functional equation related to inner product spaces in non-Archimedean normed spaces.

2010 Mathematics Subject Classification: Primary 46S10; 39B52; 47S10; 26E30; 12J25.

## 1. Introduction and preliminaries

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive mappings and by Rassias [4] for approximate linear mappings by allowing the difference Cauchy equation || f(x 1 + x 2) - f(x 1) - f(x 2) || to be controlled by ε (|| x 1 || p + || x 2 || p ). Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. In 1994, a generalization of the Rassias' theorem was obtained by Gǎvruta [5], who replaced ε (|| x 1 || p + || x 2 || p ) by a general control function φ(x 1, x 2).

Quadratic functional equations were used to characterize inner product spaces [6]. A square norm on an inner product space satisfies the parallelogram equality ||x 1 + x 2||2 + ||x 1 - x 2||2 = 2(||x 1||2 + ||x 1||2). The functional equation

$f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y )$
(1.1)

is related to a symmetric bi-additive mapping [7, 8]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.1) is said to be a quadratic mapping.

It was shown by Rassias [9] that the norm defined over a real vector space X is induced by an inner product if and only if for a fixed integer n ≥ 2

$∑ i = 1 n x i - 1 n ∑ j = 1 n x j 2 = ∑ i = 1 n ∥ x i ∥ 2 - n 1 n ∑ i = 1 n x i 2$

for all x 1,, x n X.

Let $K$ be a field. A non-Archimedean absolute value on $K$ is a function $||:K→ℝ$ such that for any $a,b∈K$, we have

1. (i)

|a| ≥ 0 and equality holds if and only if a = 0,

2. (ii)

|ab| = |a||b|,

3. (iii)

|a + b| ≤ max{|a|, |b|}.

The condition (iii) is called the strict triangle inequality. By (ii), we have |1| = | - 1| = 1. Thus, by induction, it follows from (iii) that |n| ≤ 1 for each integer n. We always assume in addition that | | is non-trivial, i.e., that there is an $a 0 ∈K$ such that |a 0| ≠ 0, 1.

Let X be a linear space over a scalar field $K$ with a non-Archimedean non-trivial valuation |·|. A function || · || : X is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(NA1) ||x|| = 0 if and only if x = 0;

(NA2) ||rx|| = |r|||x|| for all $r∈K$ and x X;

(NA3) the strong triangle inequality (ultrametric); namely,

$∥ x + y ∥ ≤ max { ∥ x ∥ , ∥ y ∥ } ( x , y ∈ X ) .$

Then (X, || · ||) is called a non-Archimedean space.

Thanks to the inequality

$∥ x m - x l ∥ ≤ max { ∥ x j + 1 - x j ∥ : l ≤ j ≤ m - 1 } ( m > l )$

a sequence {x m } is Cauchy in X if and only if {x m+1- x m } converges to zero in a non-Archimedean space. By a complete non-Archimedean space, we mean a non-Archimedean space in which every Cauchy sequence is convergent.

In 1897, Hensel [10] introduced a normed space which does not have the Archimedean property.

During the last three decades, the theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings and superstrings [11]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition [1216].

The main objective of this paper is to prove the Hyers-Ulam stability of the following functional equation related to inner product spaces

$∑ i = 1 n f x i - 1 n ∑ j = 1 n x j = ∑ i = 1 n f ( x i ) - n f 1 n ∑ i = 1 n x i$
(1.2)

(n , n ≥ 2) in non-Archimedean normed spaces. Interesting new results concerning functional equations related to inner product spaces have recently been obtained by Najati and Rassias [17] as well as for the fuzzy stability of a functional equation related to inner product spaces by Park [18] and Eshaghi Gordji and Khodaei [19]. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians (see [2049]).

## 2. Hyers-Ulam stability in non-Archimedean spaces

In the rest of this paper, unless otherwise explicitly stated, we will assume that G is an additive group and that X is a complete non-Archimedean space. For convenience, we use the following abbreviation for a given mapping f : GX:

$Δ f ( x 1 , … , x n ) = ∑ i = 1 n f x i - 1 n ∑ j = 1 n x j - ∑ i = 1 n f ( x i ) + n f 1 n ∑ i = 1 n x i$

for all x 1,, x n G, where n ≥ 2 is a fixed integer.

Lemma 2.1. [17]. Let V 1 and V 2 be real vector spaces. If an odd mapping f : V 1V 2 satisfies the functional equation (1.2), then f is additive.

In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean spaces for an odd case.

Theorem 2.2. Let φ : G n → [0, ∞) be a function such that

$lim m → ∞ φ ( 2 m x 1 ,2 m x 2 , … ,2 m x n ) ∣ 2 ∣ m = 0 = lim m → ∞ 1 ∣ 2 ∣ m Φ ( 2 m − 1 x )$
(2.1)

for all x, x 1, x 2,, x n G, and

$φ ˜ a ( x ) = lim m → ∞ max { 1 ∣ 2 ∣ k Φ ( 2 k x ) : 0 ≤ k < m }$
(2.2)

exists for all x G, where

$Φ ( x ) : = max { φ ( 2 x ,0, …, 0 ), 1 ∣ 2 ∣ max { n φ ( x , x ,0, …, 0 ), φ ( x , − x , … , − x ), φ ( − x , x , … , x ) } }$
(2.3)

for all x G. Suppose that an odd mapping f : GX satisfies the inequality

$∥ Δ f ( x 1 , … , x n ) ∥ ≤ φ ( x 1 , x 2 , … , x n )$
(2.4)

for all x 1, x 2,, x n G. Then there exists an additive mapping A : GX such that

$∥ f ( x ) - A ( x ) ∥ ≤ 1 | 2 | φ ̃ a ( x )$
(2.5)

for all x G, and if

$lim ℓ → ∞ lim m → ∞ max { 1 ∣ 2 ∣ k Φ ( 2 k x ) : ℓ ≤ k < m + ℓ } = 0$
(2.6)

then A is a unique additive mapping satisfying (2.5).

Proof. Letting x 1 = nx 1, $x i =n x 1 ′ ( i = 2 , … , n )$ in (2.4) and using the oddness of f, we obtain that

$∥ n f ( x 1 + ( n - 1 ) x ′ 1 ) + f ( ( n - 1 ) ( x 1 - x ′ 1 ) ) - ( n - 1 ) f ( x 1 - x ′ 1 ) - f ( n x 1 ) - ( n - 1 ) f ( n x ′ 1 ) ∥ ≤ φ ( n x 1 , n x ′ 1 , … , n x ′ 1 )$
(2.7)

for all $x 1 , x 1 ′ ∈G$. Interchanging x 1 with $x 1 ′$ in (2.7) and using the oddness of f, we get

$∥ n f ( ( n - 1 ) x 1 + x ′ 1 ) - f ( ( n - 1 ) ( x 1 - x ′ 1 ) ) + ( n - 1 ) f ( x 1 - x ′ 1 ) - ( n - 1 ) f ( n x 1 ) - f ( n x ′ 1 ) ∥ ≤ φ ( n x ′ 1 , n x 1 , … , n x 1 )$
(2.8)

for all $x 1 , x ′ 1 ∈ G$. It follows from (2.7) and (2.8) that

$∥ n f ( x 1 + ( n - 1 ) x ′ 1 ) - n f ( ( n - 1 ) x 1 + x ′ 1 ) + 2 f ( ( n - 1 ) ( x 1 - x ′ 1 ) ) - 2 ( n - 1 ) f ( x 1 - x ′ 1 ) + ( n - 2 ) f ( n x 1 ) - ( n - 2 ) f ( n x ′ 1 ) ∥ ≤ max { φ ( n x 1 , n x ′ 1 , … , n x ′ 1 ) , φ ( n x ′ 1 , n x 1 , … , n x 1 ) }$
(2.9)

for all $x 1 , x ′ 1 ∈ G$. Setting x 1 = nx 1, $x 2 =-n x 1 ′$, x i = 0 (i = 3,..., n) in (2.4) and using the oddness of f, we get

$∥ f ( ( n - 1 ) x 1 + x ′ 1 ) - f ( x 1 + ( n - 1 ) x ′ 1 ) + 2 f ( x 1 - x ′ 1 ) - f ( n x 1 ) + f ( n x ′ 1 ) ∥ ≤ φ ( n x 1 , - n x ′ 1 , 0 , … , 0 )$
(2.10)

for all $x 1 , x ′ 1 ∈ G$. It follows from (2.9) and (2.10) that

$∥ f ( ( n − 1 ) ( x 1 − x ′ 1 ) ) + f ( x 1 − x ′ 1 ) − f ( n x 1 ) + f ( n x ′ 1 ) ∥ ≤ 1 ∣ 2 ∣ max { n φ ( n x 1 , − n x ′ 1 ,0, …, 0 ), φ ( n x 1 , n x ′ 1 , … , n x ′ 1 ), φ ( n x ′ 1 , n x 1 , … , n x 1 ) }$
(2.11)

for all $x 1 , x ′ 1 ∈ G$. Putting $x 1 =n ( x 1 - x 1 ′ )$, x i = 0 (i = 2,..., n) in (2.4), we obtain

$∥ f ( n ( x 1 - x 1 ′ ) ) - f ( ( n - 1 ) ( x 1 - x 1 ′ ) ) - f ( ( x 1 - x 1 ′ ) ) ∥ ≤ φ ( n ( x 1 - x 1 ′ ) , 0 , … , 0 )$
(2.12)

for all $x 1 , x ′ 1 ∈ G$. It follows from (2.11) and (2.12) that

(2.13)

for all $x 1 , x ′ 1 ∈ G$. Replacing x 1 and $x 1 ′$ by $x n$ and $- x n$ in (2.13), respectively, we obtain

$∥ f ( 2 x ) − 2 f ( x ) ∥ ≤ max { φ ( 2 x ,0, …, 0 ), 1 ∣ 2 ∣ max { n φ ( x , x ,0, …, 0 ), φ ( x , − x , … , − x ), φ ( − x , x , … , x ) } }$

for all x G. Hence,

$f ( 2 x ) 2 - f ( x ) ≤ 1 | 2 | Φ ( x )$
(2.14)

for all x G. Replacing x by 2m-1 x in (2.14), we have

$‖ f ( 2 m − 1 x ) 2 m − 1 − f ( 2 m x ) 2 m ‖ ≤ 1 ∣ 2 ∣ m Φ ( 2 m − 1 x )$
(2.15)

for all x G. It follows from (2.1) and (2.15) that the sequence ${ f ( 2 m x ) 2 m }$ is Cauchy. Since X is complete, we conclude that ${ f ( 2 m x ) 2 m }$ is convergent. So one can define the mapping A : GX by $A ( x ) : = lim m → ∞ f ( 2 m x ) 2 m$ for all x G. It follows from (2.14) and (2.15) that

$‖ f ( x ) − f ( 2 m x ) 2 m ‖ ≤ 1 ∣ 2 ∣ max { 1 ∣ 2 ∣ k Φ ( 2 k x ) : 0 ≤ k < m }$
(2.16)

for all m and all x G. By taking m to approach infinity in (2.16) and using (2.2), one gets (2.5). By (2.1) and (2.4), we obtain

$∥ Δ A ( x 1 , x 2 , … , x n ) ∥ = lim m → ∞ 1 | 2 | m ∥ Δ f ( 2 m x 1 , 2 m x 2 , … , 2 m x n ) ∥ ≤ lim m → ∞ 1 | 2 | m φ ( 2 m x 1 , 2 m x 2 , … , 2 m x n ) = 0$

for all x 1, x 2,, x n G. Thus, the mapping A satisfies (1.2). By Lemma 2.1, A is additive.

If A' is another additive mapping satisfying (2.5), then

$∥ A ( x ) − A ′ ( x ) ∥ = lim ℓ → ∞ ∣ 2 ∣ − ℓ ∥ A ( 2 ℓ x ) − A ′ ( 2 ℓ x ) ∥ ≤ lim ℓ → ∞ ∣ 2 ∣ − ℓ max { ∥ A ( 2 ℓ x ) − f ( 2 ℓ x ) ∥ , ∥ f ( 2 ℓ x ) − Q ′ ( 2 ℓ x ) ∥ } ≤ 1 ∣ 2 ∣ lim ℓ → ∞ lim m → ∞ max { 1 ∣ 2 ∣ k φ ˜ ( 2 k x ) : ℓ ≤ k < m + ℓ } = 0$

for all x G, Thus A = A'. □

Corollary 2.3. Let ρ : [0, ∞) → [0, ∞) be a function satisfying

1. (i)

ρ (|2|t) ≤ ρ(|2|)ρ(t) for all t ≥ 0,

2. (ii)

ρ(|2|) < |2|.

Let ε > 0 and let G be a normed space. Suppose that an odd mapping f : GX satisfies the inequality

$∥ Δ f ( x 1 , … , x n ) ∥ ≤ ε ∑ i = 1 n ρ ( ∥ x i ∥ )$

for all x 1,, x n G. Then there exists a unique additive mapping A : GX such that

$∥ f ( x ) - A ( x ) ∥ ≤ 2 n | 2 | 2 ε ρ ( ∥ x ∥ )$

for all x G.

Proof. Defining φ : G n → [0, ∞) by $φ ( x 1 , … , x n ) :=ε ∑ i = 1 n ρ ( ∥ x i ∥ )$, we have

$lim m → ∞ 1 | 2 | m φ ( 2 m x 1 , … , 2 m x n ) ≤ lim m → ∞ ρ ( | 2 | ) | 2 | m φ ( x 1 , … , x n ) = 0$

for all x 1,, x n G. So we have

$φ ˜ a ( x ) : = lim m → ∞ max { 1 ∣ 2 ∣ k Φ ( 2 k x ) : 0 ≤ k < m } = Φ ( x )$

and

$lim ℓ → ∞ lim m → ∞ max { 1 ∣ 2 ∣ k Φ ( 2 k x ) : ℓ ≤ k < m + ℓ } = lim ℓ → ∞ 1 ∣ 2 ∣ ℓ Φ ( 2 ℓ x ) = 0$

for all x G. It follows from (2.3) that

$Φ ( x ) = max ε ρ ( ∥ 2 x ∥ ) , 1 | 2 | max { 2 n ε ρ ( ∥ x ∥ ) , n ε ρ ( ∥ x ∥ ) , n ε ρ ( ∥ x ∥ ) } = max ε ρ ( ∥ 2 x ∥ ) , 1 | 2 | max { 2 n ε ρ ( ∥ x ∥ ) , n ε ρ ( ∥ x ∥ ) } = max ε ρ ( ∥ 2 x ∥ ) , 1 | 2 | 2 n ε ρ ( ∥ x ∥ ) = 2 n | 2 | ε ρ ( ∥ x ∥ ) .$

Applying Theorem 2.2, we conclude that

$∥ f ( x ) - A ( x ) ∥ ≤ 1 | 2 | φ ̃ a ( x ) = 1 | 2 | Φ ( x ) = 2 n | 2 | 2 ε ρ ( ∥ x ∥ )$

for all x G. □

Lemma 2.4. [17]. Let V 1 and V 2 be real vector spaces. If an even mapping f : V 1V 2 satisfies the functional equation (1.2), then f is quadratic.

In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean spaces for an even case.

Theorem 2.5. Let φ : G n → [0, ∞) be a function such that

$lim m → ∞ φ ( 2 m x 1 ,2 m x 2 , … ,2 m x n ) ∣ 2 ∣ 2 m = 0 = lim m → ∞ 1 ∣ 2 ∣ 2 m φ ˜ ( 2 m − 1 x )$
(2.17)

for all x, x 1, x 2,, x n G, and

$φ ̃ q ( x ) = lim m → ∞ max 1 | 2 | 2 k φ ̃ ( 2 k x ) : 0 ≤ k < m$
(2.18)

exists for all x G, where

(2.19)

and

$Ψ ( x ) : = 1 | 2 | max { n φ ( n x , 0 , … , 0 ) , φ ( n x , 0 , … , 0 ) , φ ( 0 , n x , … , n x ) }$
(2.20)

for all x G. Suppose that an even mapping f : GX with f(0) = 0 satisfies the inequality (2.4) for all x 1, x 2,, x n G. Then there exists a quadratic mapping Q : GX such that

$∥ f ( x ) - Q ( x ) ∥ ≤ 1 | 2 | 2 φ ̃ q ( x )$
(2.21)

for all x G, and if

$lim ℓ → ∞ lim m → ∞ max 1 | 2 | 2 k φ ̃ ( 2 k x ) : ℓ ≤ k < m + ℓ = 0$
(2.22)

then Q is a unique quadratic mapping satisfying (2.21).

Proof. Letting x 1 = nx 1, x i = nx 2 (i = 2,, n) in (2.4) and using the evenness of f, we obtain

$∥ n f ( x 1 + ( n - 1 ) x 2 ) + f ( ( n - 1 ) ( x 1 - x 2 ) ) + ( n - 1 ) f ( x 1 - x 2 ) - f ( n x 1 ) - ( n - 1 ) f ( n x 2 ) ∥ ≤ φ ( n x 1 , n x 2 , … , n x 2 )$
(2.23)

for all x 1, x 2 G. Interchanging x 1 with x 2 in (2.23) and using the evenness of f, we obtain

$∥ n f ( ( n - 1 ) x 1 + x 2 ) + f ( ( n - 1 ) ( x 1 - x 2 ) ) + ( n - 1 ) f ( x 1 - x 2 ) - ( n - 1 ) f ( n x 1 ) - f ( n x 2 ) ∥ ≤ φ ( n x 2 , n x 1 , … , n x 1 )$
(2.24)

for all x 1, x 2 G. It follows from (2.23) and (2.24) that

$∥ n f ( ( n - 1 ) x 1 + x 2 ) + n f ( x 1 + ( n - 1 ) x 2 ) + 2 f ( ( n - 1 ) ( x 1 - x 2 ) ) + 2 ( n - 1 ) f ( x 1 - x 2 ) - n f ( n x 1 ) - n f ( n x 2 ) ∥ ≤ max { φ ( n x 1 , n x 2 , … , n x 2 ) , φ ( n x 2 , n x 1 , … , n x 1 ) }$
(2.25)

for all x 1, x 2 G. Setting x 1 = nx 1, x 2 = -nx 2, x i = 0 (i = 3,, n) in (2.4) and using the evenness of f, we obtain

$∥ f ( ( n - 1 ) x 1 + x 2 ) + f ( x 1 + ( n - 1 ) x 2 ) + 2 ( n - 1 ) f ( x 1 - x 2 ) - f ( n x 1 ) - f ( n x 2 ) ∥ ≤ φ ( n x 1 , - n x 2 , 0 , … , 0 )$
(2.26)

for all x 1, x 2 G. So we obtain from (2.25) and (2.26) that

$∥ f ( ( n - 1 ) ( x 1 - x 2 ) ) - ( n - 1 ) 2 f ( x 1 - x 2 ) ∥ ≤ 1 | 2 | max { n φ ( n x 1 , - n x 2 , 0 , … , 0 ) , φ ( n x 1 , n x 2 , … , n x 2 ) , φ ( n x 2 , n x 1 , … , n x 1 ) }$
(2.27)

for all x 1, x 2 G. Setting x 1 = x, x 2 = 0 in (2.27), we obtain

$∥ f ( ( n - 1 ) x ) - ( n - 1 ) 2 f ( x ) ∥ ≤ 1 | 2 | max { n φ ( n x , 0 , … , 0 ) , φ ( n x , 0 , … , 0 ) , φ ( 0 , n x , … , n x ) }$
(2.28)

for all x G. Putting x 1 = nx, x i = 0 (i = 2,, n) in (2.4), one obtains

$∥ f ( n x ) - f ( ( n - 1 ) x ) - ( 2 n - 1 ) f ( x ) ∥ ≤ φ ( n x , 0 , … , 0 )$
(2.29)

for all x G. It follows from (2.28) and (2.29) that

(2.30)

for all x G. Letting x 2 = - (n - 1) x 1 and replacing x 1 by $x n$ in (2.26), we get

$∥ f ( ( n - 1 ) x ) - f ( ( n - 2 ) x ) - ( 2 n - 3 ) f ( x ) ∥ ≤ φ ( x , ( n - 1 ) x , 0 , … , 0 )$
(2.31)

for all x G. It follows from (2.28) and (2.31) that

(2.32)

for all x G. It follows from (2.30) and (2.32) that

$∥ f ( n x ) - f ( ( n - 2 ) x ) - 4 ( n - 1 ) f ( x ) ∥ ≤ max { φ ( n x , 0 , … , 0 ) , φ ( x , ( n - 1 ) x , 0 , … , 0 ) , Ψ ( x ) }$
(2.33)

for all x G. Setting x 1 = x 2 = n x , x i = 0 (i = 3,, n) in (2.4), we obtain

$∥ f ( ( n - 2 ) x ) + ( n - 1 ) f ( 2 x ) - f ( n x ) ∥ ≤ 1 | 2 | φ ( n x , n x , 0 , … , 0 )$
(2.34)

for all x G. It follows from (2.33) and (2.34) that

(2.35)

for all x G. Thus,

$f ( x ) - f ( 2 x ) 2 2 ≤ 1 | 2 | 2 φ ̃ ( x )$
(2.36)

for all x G. Replacing x by 2 m - 1 x in (2.36), we have

$f ( 2 m - 1 x ) 2 2 ( m - 1 ) - f ( 2 m x ) 2 2 m ≤ 1 | 2 | 2 m φ ̃ ( 2 m - 1 x )$
(2.37)

for all x G. It follows from (2.17) and (2.37) that the sequence ${ f ( 2 m x ) 2 2 m }$ is Cauchy. Since X is complete, we conclude that ${ f ( 2 m x ) 2 2 m }$ is convergent. So one can define the mapping Q : GX by $Q ( x ) := lim m → ∞ f ( 2 m x ) 2 2 m$ for all x G. By using induction, it follows from (2.36) and (2.37) that

$f ( x ) - f ( 2 m x ) 2 2 m ≤ 1 | 2 | 2 max 1 | 2 | 2 k φ ̃ ( 2 k x ) : 0 ≤ k < m$
(2.38)

for all n and all x G. By taking m to approach infinity in (2.38) and using (2.18), one gets (2.21).

The rest of proof is similar to proof of Theorem 2.2. □

Corollary 2.6. Let η : [0, ∞) → [0, ∞) be a function satisfying

1. (i)

η(|l|t) ≤ η(|l|)η(t) for all t ≥ 0,

2. (ii)

η(|l|) < |l|2 for l {2, n - 1, n}.

Let ε > 0 and let G be a normed space. Suppose that an even mapping f : GX with f(0) = 0 satisfies the inequality

$∥ Δ f ( x 1 , … , x n ) ∥ ≤ ε ∑ i = 1 n η ( ∥ x i ∥ )$

for all x 1,, x n G. Then there exists a unique quadratic mapping Q : GX such that

$∥ f ( x ) - Q ( x ) ∥ ≤ 2 | 2 | 2 ε η ( ∥ x ∥ ) , i f n = 2 ; n | 2 | 3 | n - 1 | ε η ( ∥ n x ∥ ) , i f n > 2 ,$

for all x G.

Proof. Defining φ : G n → [0, ∞) by $φ ( x 1 , … , x n ) :=ε ∑ i = 1 n η ( ∥ x i ∥ )$, we have

$lim m → ∞ 1 | 2 | 2 m φ ( 2 m x 1 , … , 2 m x n ) ≤ lim m → ∞ η ( | 2 | ) | 2 | 2 m φ ( x 1 , … , x n ) = 0$

for all x 1,, x n G. We have

$φ ̃ q ( x ) : = lim m → ∞ max 1 | 2 | 2 k φ ̃ ( 2 k x ) : 0 ≤ k < m = φ ̃ ( x )$

and

$lim ℓ → ∞ lim m → ∞ max 1 | 2 | 2 k φ ̃ ( 2 k x ) : ℓ ≤ k < m + ℓ = lim ℓ → ∞ 1 | 2 | 2 ℓ φ ̃ ( 2 ℓ x ) = 0$

for all x G. It follows from (2.20) that

$Ψ ( x ) = 1 | 2 | max { n ε η ( ∥ n x ∥ ) , ε η ( ∥ n x ∥ ) , ( n - 1 ) ε η ( ∥ n x ∥ ) } = 1 | 2 | max { n ε η ( ∥ n x ∥ ) , ( n - 1 ) ε η ( ∥ n x ∥ ) } = n | 2 | ε η ( ∥ n x ∥ )$

Hence, by using (2.19), we obtain

for all x G.

Applying Theorem 2.5, we conclude the required result. □

Lemma 2.7. [17]. Let V 1 and V 2 be real vector spaces. A mapping f : V 1V 2 satisfies (1.2) if and only if there exist a symmetric bi-additive mapping B : V 1 × V 1V 2 and an additive mapping A : V 1V 2 such that f(x) = B(x, x) + A(x) for all x V 1.

Now, we prove the main theorem concerning the Hyers-Ulam stability problem for the functional equation (1.2) in non-Archimedean spaces.

Theorem 2.8. Let φ : G n → [0, ∞) be a function satisfying (2.1) and (2.17) for all x, x 1, x 2,, x n G, and $φ ̃ a ( x )$ and $φ ̃ q ( x )$ exist for all x G, where $φ ̃ a ( x )$ and $φ ̃ q ( x )$ are defined as in Theorems 2.2 and 2.5. Suppose that a mapping f : GX with f(0) = 0 satisfies the inequality (2.4) for all x 1, x 2,, x n G. Then there exist an additive mapping A : GX and a quadratic mapping Q : GX such that

$∥ f ( x ) - A ( x ) - Q ( x ) ∥ ≤ 1 | 2 | 2 max φ ̃ a ( x ) , φ ̃ a ( - x ) , 1 | 2 | φ ̃ q ( x ) , 1 | 2 | φ ̃ q ( - x )$
(2.39)

for all x G. If

$lim ℓ → ∞ lim m → ∞ max 1 | 2 | k Φ ( 2 k x ) : ℓ ≤ k < m + ℓ = 0 = lim ℓ → ∞ lim m → ∞ max 1 | 2 | 2 k φ ̃ ( 2 k x ) : ℓ ≤ k < m + ℓ$

then A is a unique additive mapping and Q is a unique quadratic mapping satisfying (2.39).

Proof. Let $f e ( x ) = 1 2 ( f ( x ) + f ( - x ) )$ for all x G. Then

$∥ Δ f e ( x 1 , … , x n ) ∥ = 1 2 ( Δ f ( x 1 , … , x n ) + Δ f ( - x 1 , … , - x n ) ) ≤ 1 | 2 | max { φ ( x 1 , … , x n ) , φ ( - x 1 , … , - x n ) }$

for all x 1, x 2,, x n G. By Theorem 2.5, there exists a quadratic mapping Q : GX such that

$∥ f e ( x ) - Q ( x ) ∥ ≤ 1 | 2 | 3 max { φ ̃ q ( x ) , φ ̃ q ( - x ) }$
(2.40)

for all x G. Also, let $f o ( x ) = 1 2 ( f ( x ) - f ( - x ) )$ for all x G. By Theorem 2.2, there exists an additive mapping A : GX such that

$∥ f o ( x ) - A ( x ) ∥ ≤ 1 | 2 | 2 max { φ ̃ a ( x ) , φ ̃ a ( - x ) }$
(2.41)

for all x G. Hence (2.39) follows from (2.40) and (2.41).

The rest of proof is trivial. □

Corollary 2.9. Let γ : [0, ∞) → [0, ∞) be a function satisfying

(i) γ(|l|t) ≤ γ(|l|) γ(t) for all t ≥ 0,

(ii) γ(|l|) < |l|2 for l {2, n - 1, n}.

Let ε > 0, G a normed space and let f : GX satisfy

$∥ Δ f ( x 1 , … , x n ) ∥ ≤ ε ∑ i = 1 n γ ( ∥ x i ∥ )$

for all x 1,, x n G and f (0) = 0. Then there exist a unique additive mapping A : GX and a unique quadratic mapping Q : GX such that

$∥ f ( x ) - A ( x ) - Q ( x ) ∥ ≤ 2 n | 2 | 3 ε γ ( ∥ x ∥ )$

for all x G.

Proof. The result follows by Corollaries 2.6 and 2.3. □

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## Acknowledgements

Dong Yun Shin was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Gordji, M.E., Khodabakhsh, R., Khodaei, H. et al. A Functional equation related to inner product spaces in non-archimedean normed spaces. Adv Differ Equ 2011, 37 (2011). https://doi.org/10.1186/1687-1847-2011-37