Open Access

On the stability of pexider functional equation in non-archimedean spaces

Journal of Inequalities and Applications20112011:17

DOI: 10.1186/1029-242X-2011-17

Received: 7 January 2011

Accepted: 24 June 2011

Published: 24 June 2011

Abstract

In this paper, the Hyers-Ulam stability of the Pexider functional equation

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equa_HTML.gif

in a non-Archimedean space is investigated, where σ is an involution in the domain of the given mapping f.

MSC 2010:26E30, 39B52, 39B72, 46S10

Keywords

Hyers-Ulam stability of functional equation Non-Archimedean space Quadratic Cauchy and Pexider functional equations

1.Introduction

The stability problem for functional equations first was planed in 1940 by Ulam [1]:

Let G1 be group and G2 be a metric group with the metric d(·,·). Does, for any ε > 0, there exists δ > 0 such that, for any mapping f : G1G2 which satisfies d(f(xy), f(x)f(y)) ≤ δ for all x, y G1, there exists a homomorphism h : G1G2 so that, for any x G1, we have d(f (x), h(x)) ≤ ε?

In 1941, Hyers [2] answered to the Ulam's question when G1 and G2 are Banach spaces. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influences in the development of the Hyers-Ulam-Rassias stability of functional equations (for more details, see [5] where a discussion on definitions of the Hyers-Ulam stability is provided by Moszner, also [612]).

In this paper, we give a modification of the approach of Belaid et al. [13] in non-Archimedean spaces. Recently, Ciepliński [14] studied and proved stability of multi-additive mappings in non-Archimedean normed spaces, also see [1522].

Definition 1.1. The function | · | : K is called a non-Archimedean valuation or absolute value over the field K if it satisfies following conditions: for any a, b K,
  1. (1)

    |a| ≥ 0;

     
  2. (2)

    |a| = 0 if and only if a = 0;

     
  3. (3)

    |ab| = |a| |b|

     
  4. (4)

    |a + b| ≤ max{|a|, |b|};

     
  5. (5)

    there exists a member a 0 K such that |a 0| ≠ 0, 1.

     

A field K with a non-Archimedean valuation is called a non-Archimedean field.

Corollary 1.2. |-1| = |1| = 1 and so, for any a K, we have |-a| = |a|. Also, if |a| < |b| for any a, b K, then |a + b| = |b|.

In a non-Archimedean field, the triangle inequality is satisfied and so a metric is defined. But an interesting inequality changes the usual Archimedean sense of the absolute value. For any n , we have |n · 1| ≤ . Thus, for any a K, n and nonzero divisor k of n, the following inequalities hold:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ1_HTML.gif
(1.1)
Definition 1.3. Let V be a vector space over a non-Archimedean field K. A non-Archimedean norm over V is a function || · || : V → R satisfying the following conditions: for any α K and u, v V,
  1. (1)

    ||u|| = 0 if and only if u = 0;

     
  2. (2)

    ||αu|| = |α| ||u||;

     
  3. (3)

    ||u + v|| ≤ max{||u||, ||v||}.

     

Since 0 = ||0|| = ||v - v|| ≤ max{||v||, ||-v||} = ||v|| for any v V, we have ||v|| ≥ 0. Any vector space V with a non-Archimedean norm || · || : V is called a non-Archimedean space. If the metric d(u, v) = ||u - v|| is induced by a non-Archimedean norm || · || : V on a vector space V which is complete, then (V, || · ||) is called a complete non-Archimedean space.

Proposition 1.4. ([23]) A sequence https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq1_HTML.gif in a non-Archimedean space is a Cauchy sequence if and only if the sequence https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq2_HTML.gif converges to zero.

Since any non-Archimedean norm satisfies the triangle inequality, any non-Archimedean norm is a continuous function from its domain to real numbers.

Proposition 1.5. Let V be a normed space and E be a non-Archimedean space. Let f : VE be a function, continuous at 0 V such that, for any × V, f(2x) = 2f(x) (for example, additive functions). Then, f = 0.

Proof. Since f(0) = 0, for any ε > 0, there exists δ > 0 that, for any x V with ||x|| ≤ δ,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equb_HTML.gif
and, for any x V, there exists n that https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq3_HTML.gif and hence
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equc_HTML.gif

Since this inequality holds for all ε > 0, it follows that, for any x V, f(x) = 0. This completes the proof.

The preceding fact is a special case of a general result for non-Archimedean spaces, that is, every continuous function from a connected space to a non-Archimedean space is constant. This is a consequence of totally disconnectedness of every non-Archimedean space (see [23]).

2. Stability of quadratic and Cauchy functional equations

Throughout this section, we assume that V1 is a normed space and V2 is a complete non-Archimedean space. Let σ : V1V1 be a continuous involution (i.e., σ (x + y) = σ (x) + σ (y) and σ (σ (x)) = x) and φ : V1 × V1 be a function with
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ2_HTML.gif
(2.1)
and define a function ϕ : V1 × V1 by
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ3_HTML.gif
(2.2)
which easily implies
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ4_HTML.gif
(2.3)
Theorem 2.1. Suppose that φ satisfies the condition 2.1 and let ϕ is defined by Equation 2.2. If f : V1V2satisfies the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ5_HTML.gif
(2.4)
for all x, y V1, then there exists a unique solution q : V1V2of the functional equation
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ6_HTML.gif
(2.5)
such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ7_HTML.gif
(2.6)

for all x V1.

Proof. Replacing x and y in Equation 2.4 with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif , respectively, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ8_HTML.gif
(2.7)
Replacing x and y in Equation 2.4 with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq4_HTML.gif , respectively, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ9_HTML.gif
(2.8)
Also, replacing both of x, y in Equation 2.4 with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equd_HTML.gif
and so, for any n , we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ10_HTML.gif
(2.9)
Similarly, replacing both of x, y in Equation 2.4 with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq4_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ11_HTML.gif
(2.10)
Replacing x in Equation 2.7 with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Eque_HTML.gif
for all x V1 and so, by assumption Equation 2.1,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equf_HTML.gif
Thus, f(0) = 0 and the inequality Equation 2.10 reduces to
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equg_HTML.gif
and so,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ12_HTML.gif
(2.11)
For any n , define
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equh_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equi_HTML.gif
Then,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ13_HTML.gif
(2.12)

for all x, y V1.

From Equations (2.9) and (2.11), we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equj_HTML.gif

and so Proposition 1.4 and the hypothesis Equation 2.1 imply that https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq6_HTML.gif is a Cauchy sequence. Since V2 is complete, the sequence https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq7_HTML.gif converges to a point of V2 which defines a mapping q : V1V2.

Now, we prove
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ14_HTML.gif
(2.13)
for all n . Since Equation 2.7 implies
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equk_HTML.gif
Assume that ||f(x) -q n (x)|| ≤ ϕ n (x, x) holds for some n . Then, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equl_HTML.gif

Therefore, by induction on n, Equation 2.13 follows from Equation 2.12. Taking the limit of both sides of Equation 2.13, we prove that q satisfies Equation 2.6.

For any n and x, y V1, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equm_HTML.gif
and so, by the continuity of non-Archimedean norm and taking the limit of both sides of the above inequality, we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equn_HTML.gif

Thus, q is a solution of the Equation 2.5 which satisfies Equation 2.6.

Then, by replacing x, y with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq5_HTML.gif in Equation 2.5, we obtain the following identities: for any solution g : V1V2 of the Equation (2.5),
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equo_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ15_HTML.gif
(2.14)
By induction on n, one can show that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ16_HTML.gif
(2.15)
and
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ17_HTML.gif
(2.16)

for all n .

Now, suppose that q' : V1V2 is another solution of 2.5 that satisfies the Equation 2.6. It follows from Equations 2.14 to 2.16 that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equp_HTML.gif
Therefore, since
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equq_HTML.gif

we have q(x) = q'(x) for all x V1. This completes the proof.

In the proof of the next theorem, we need a result concerning the Cauchy functional equation
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ18_HTML.gif
(2.17)

which has been established in [20].

Theorem 2.2. ([20]) Suppose that φ(x, y) satisfies the condition 2.1 and, for a mapping f : V1V2,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ19_HTML.gif
(2.18)
for all x, y V1. Then, there exists a unique solution q : V1V2of the Equation 2.17 such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ20_HTML.gif
(2.19)
for all x V1, where
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equr_HTML.gif

for all x, y V1

3. Stability of the Pexider functional equation

In this section, we assume that V1 is a normed space and V2 is a complete non-Archimedean space. For any mapping f : V1V2, we define two mappings F e and F o as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equs_HTML.gif
and also define F(x) = f(x) -f(0). Then, we have obviously
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ21_HTML.gif
(3.1)
Theorem 3.1. Let σ : V1V1be a continuous involution and the mappings f i : V1V2for i = 1, 2, 3, 4 and δ > 0, satisfy
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ22_HTML.gif
(3.2)
for all x, y V1, then there exists a unique solution q : V1V2of the Equation 2.5 and a mapping v : V1V2which satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equt_HTML.gif
for all x, y V1and exists two additive mappings https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq8_HTML.gif such that https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq9_HTML.gif for i= 1, 2 and, for all x V1,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ23_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ24_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ25_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ26_HTML.gif
(3.6)
Proof. It follows from (3.2) that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equu_HTML.gif
and so, for all x, y V1,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equv_HTML.gif
then,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ27_HTML.gif
(3.7)
Similarly, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ28_HTML.gif
(3.8)

for all x, y V1.

Now, first by putting y = 0 in Equation 3.7 and applying Equation 3.2 and second by putting x = 0 in Equation 3.7 and applying Equation 3.2 once again, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ29_HTML.gif
(3.9)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ30_HTML.gif
(3.10)
for all x, y V1 and so these inequalities with Equation 3.7 imply
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ31_HTML.gif
(3.11)
Replacing y with σ(y) in Equation 3.11, we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ32_HTML.gif
(3.12)
It follows from Equations 3.1, 3.11 and 3.12 that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equw_HTML.gif
By Theorem 2.1 of [24], there exists a unique solution q : V1V2 of the functional Equation 2.5 such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ33_HTML.gif
(3.13)

for all x V1.

As a result of the inequalities Equations 3.11 and 3.12, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ34_HTML.gif
(3.14)
It is easily seen that the mapping v : V1V2 defined by
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equx_HTML.gif
is a solution of the functional equation
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equy_HTML.gif

for all x, y V1.

Replacing both of x, y in Equation 3.14 with https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq10_HTML.gif , We get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ35_HTML.gif
(3.15)
for all x V1. Now, Equations 3.13 and 3.15 imply
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ36_HTML.gif
(3.16)
and
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ37_HTML.gif
(3.17)
Similarly, it follows from the inequalities Equations 3.7, 3.10 and 3.13 that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ38_HTML.gif
(3.18)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ39_HTML.gif
(3.19)
Since Equation 3.8 implies
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ40_HTML.gif
(3.20)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ41_HTML.gif
(3.21)
for all x, y V1, we have
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ42_HTML.gif
(3.22)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ43_HTML.gif
(3.23)
for all x V1. Now, from Equations 3.8 and 3.20, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ44_HTML.gif
(3.24)
and so, by interchanging role of x, y in the preceding inequality,
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ45_HTML.gif
(3.25)
for all x, y V1. Since y + σ (x) = σ (x + σ (y), it follows from Equations 3.1, 3.24 and 3.25 that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ46_HTML.gif
(3.26)
By Theorem 2.2, there exists a unique additive mapping https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq11_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ47_HTML.gif
(3.27)
Since
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equz_HTML.gif

for all x V1, we deduce https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq12_HTML.gif for all x V1.

By a similar deduction, Equations 3.8 and 3.21 imply that there exists a unique additive mapping https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq13_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ48_HTML.gif
(3.28)
Moreover, we have https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_IEq14_HTML.gif for all x V1. Thus, by Equations 3.16, 3.22, 3.27 and 3.28, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2011-17/MediaObjects/13660_2011_Article_11_Equ49_HTML.gif
(3.29)

This proves Equation 3.3. Similarly, one can prove Equations 3.4 to 3.6.

Declarations

Acknowledgements

The authors would like to thank the referee and area editor Professor Ondrĕj Došlý for giving useful suggestions and comments for the improvement of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University (iau)
(2)
Department of Mathematics and Computer Science, Amirkabir University of Technology

References

  1. Ulam SM: Problems in Modern Mathematics, Chapter IV, Science Editions. Wiley, New York; 1960.
  2. Hyers DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView Article
  3. Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Jpn 1950, 2: 64–66. 10.2969/jmsj/00210064View Article
  4. Rassias THM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View Article
  5. Moszner Z: On the stability of functional equations. Aequationes Math 2009, 77: 33–88. 10.1007/s00010-008-2945-7MathSciNetView Article
  6. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ; 2002.
  7. Hyers DH, Isac G, Rassias THM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View Article
  8. Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.
  9. Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000, 62: 23–130. 10.1023/A:1006499223572MathSciNetView Article
  10. Rassias THM: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 2003.View Article
  11. Ciepliński K: Generalized stability of multi-additive mappings. Appl Math Lett 2010,23(10):1291–1294. 10.1016/j.aml.2010.06.015MathSciNetView Article
  12. Ciepliński K: Stability of the multi-Jensen equation. J Math Anal Appl 2010,363(1):249–254. 10.1016/j.jmaa.2009.08.021MathSciNetView Article
  13. Bouikhalene B, Elqorachi E, Rassias THM: On the Hyers-Ulam stability of approximately Pexider mappings. Math Inequal Appl 2008, 11: 805–818.MathSciNet
  14. Ciepliński K: Stability of multi-additive mappings in non-Archimedean normed spaces. J Math Anal Appl 2011, 373: 376–383. 10.1016/j.jmaa.2010.07.048MathSciNetView Article
  15. Cho YJ, Park C, Saadati R: Functional inequalities in non-Archimedean in Banach spaces. Appl Math Lett 2010, 60: 1994–2002.MathSciNet
  16. Mirmostafaee AK: Stability of quartic mappings in non-Archimedean normed spaces. Kyungpook Math J 2009, 49: 289–297.MathSciNetView Article
  17. Moslehian MS, Sadeghi GH: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal 2008, 69: 3405–3408. 10.1016/j.na.2007.09.023MathSciNetView Article
  18. Moslehian MS, Sadeghi GH: Stability of two types of cubic functional equations in non-Archimedean spaces. Real Anal Exch 2008, 33: 375–384.MathSciNet
  19. Moslehian MS, Rassias THM: Stability of functional equations in non-archimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView Article
  20. Moslehian MS, Rassias THM: Stability of functional equations in non-Archimedean spaces. Appl Anal Discrete Math 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView Article
  21. Najati A, Moradlou F: Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces. Tamsui Oxf J Math Sci 2008, 24: 367–380.MathSciNet
  22. Saadati R, Cho YJ, Vahidi J: The stability of the quartic functional equation in various spaces. Comput Math Appl 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034MathSciNetView Article
  23. Schneider P: Non-Archimedean Functional Analysis. Springer, New York; 2002.View Article
  24. Bouikhalene B, Elqorachi E, Rassias THM: On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution. Nonlinear Funct Anal Appl 2007,12(2):247–262.MathSciNet

Copyright

© Saadati et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.