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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 such that for any a,bK, 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 rK 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 lim m max { 1 2 k Φ ( 2 k x ) : k < m + } = 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

f ( n ( x 1 - x 1 ) ) - f ( n x 1 ) + f ( n x 1 ) max φ ( n ( x 1 - x 1 ) , 0 , , 0 ) , n | 2 | φ ( n x 1 , - n x 1 , 0 , , 0 ) , 1 | 2 | max { φ ( n x 1 , n x 1 , , n x 1 ) , φ ( n x 1 , n x 1 , , n x 1 ) }
(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 2 A ( 2 x ) A ( 2 x ) lim 2 max { A ( 2 x ) f ( 2 x ) , f ( 2 x ) Q ( 2 x ) } 1 2 lim lim m max { 1 2 k φ ˜ ( 2 k x ) : k < m + } = 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 lim m max { 1 2 k Φ ( 2 k x ) : k < m + } = lim 1 2 Φ ( 2 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

φ ̃ ( x ) : = 1 | n - 1 | max 1 | 2 | φ ( n x , n x , 0 , , 0 ) , φ ( n x , 0 , , 0 ) , φ ( x , ( n - 1 ) x , 0 , , 0 ) , Ψ ( x )
(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 lim m max 1 | 2 | 2 k φ ̃ ( 2 k x ) : k < m + = 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

f ( n x ) - n 2 f ( x ) max φ ( n x , 0 , , 0 ) , n | 2 | φ ( n x , 0 , , 0 ) , 1 | 2 | φ ( n x , 0 , , 0 ) , 1 | 2 | φ ( 0 , n x , , n x )
(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

f ( ( n - 2 ) x ) - ( n - 2 ) 2 f ( x ) max φ ( x , ( n - 1 ) x , 0 , , 0 ) , n | 2 | φ ( n x , 0 , , 0 ) , 1 | 2 | φ ( n x , 0 , , 0 ) , 1 | 2 | φ ( 0 , n x , , n x )
(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

f ( 2 x ) - 4 f ( x ) 1 | n - 1 | max 1 | 2 | φ ( n x , n x , 0 , , 0 ) , φ ( n x , 0 , , 0 ) , φ ( x , ( n - 1 ) x , 0 , , 0 ) , Ψ ( x )
(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 lim m max 1 | 2 | 2 k φ ̃ ( 2 k x ) : k < m + = lim 1 | 2 | 2 φ ̃ ( 2 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

φ ̃ ( x ) = 1 | n - 1 | max 2 | 2 | ε η ( n x ) , ε η ( n x ) , n | 2 | ε η ( n x ) , ε ( η ( x ) + η ( ( n - 1 ) x ) ) = 2 ε η ( x ) , i f n = 2 ; n | 2 | | n - 1 | ε η ( n x ) , i f n > 2 ,

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 lim m max 1 | 2 | k Φ ( 2 k x ) : k < m + = 0 = lim lim m max 1 | 2 | 2 k φ ̃ ( 2 k x ) : k < m +

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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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

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Keywords

  • non-Archimedean spaces
  • additive and quadratic functional equation
  • Hyers-Ulam stability
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