Open Access

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

  • Madjid Eshaghi Gordji1, 2,
  • Razieh Khodabakhsh1, 2,
  • Hamid Khodaei1, 2,
  • Choonkil Park3 and
  • Dong Yun shin4Email author
Advances in Difference Equations20112011:37

https://doi.org/10.1186/1687-1847-2011-37

Received: 29 March 2011

Accepted: 29 September 2011

Published: 29 September 2011

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.

Keywords

non-Archimedean spacesadditive and quadratic functional equationHyers-Ulam stability

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 , 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 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 2 m-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. □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Semnan University
(2)
Center of Excellence in Nonlinear Analysis and Applications (Cenaa), Semnan University
(3)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(4)
Department of Mathematics, University of Seoul

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