# Stability for Functional Equation of Mixed Type in Non-Archimedean Normed Spaces

Authors: Ick-Soon Chang; Yang-Hi Lee
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We take into account the stability of the following functional equationf
(
x
+
a
y
)
+
f
(
x
-
a
y
)
-
2
f
(
x
)
-
a
2
f
(
y
)
-
a
2
f
(
-
y
)
=
0 in non-Archimedean normed spaces.


---

## Body

## 1. Introduction and Preliminaries

The study of stability problems has originally been formulated by Ulam [1]:under what condition does there exist a homomorphism near an approximate homomorphism? Hyers [2] had answered affirmatively the question of Ulam for Banach spaces. The theorem of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper work of Rassias [4] has had a lot of influence in the development of what is called the generalized Hyers-Ulam stability of functional equations. Thereafter, many interesting results of the generalized Hyers-Ulam stability to a number of functional equations have been investigated by a number of mathematicians; see [5–13] and references therein.Now we demonstrate some definitions used in this work.Definition 1.
A field𝕂, equipped with a function (valuation) |
·
| from 𝕂 into [
0
,
∞
), is called a non-Archimedean field if the function |
·
|
:
𝕂
→
[
0
,
∞
) satisfies the following conditions: (1)
|
r
|
=
0 if and only if r
=
0;
(2)
|
r
s
|
=
|
r
|
|
s
|;
(3)
the strong triangle inequality, namely,|
r
+
s
|
≤
max
⁡
{
|
r
|
,
|
s
|
} for all r
,
s
∈
𝕂.
Of course, it is easy to see that|
1
|
=
1
=
|
-
1
| and |
n
|
≤
1 for all nonzero integer n.Definition 2.
Let𝒳 be a vector space over the non-Archimedean field 𝕂 with a nontrivial non-Archimedean valuation |
·
|. A function ∥
·
∥
:
𝒳
→
[
0
,
∞
) is said to be anon-Archimedean norm (valuation) if it satisfies the following conditions: (1)
∥
x
∥
=
0 if and only if x
=
0;
(2)
∥
r
x
∥
=
|
r
|
∥
x
∥ for all x
∈
𝒳 and all r
∈
𝕂;
(3)
the strong triangle inequality, namely,(1)
∥
x
+
y
∥
≤
max
⁡
{
∥
x
∥
,
∥
y
∥
}
∀
x
,
y
∈
𝒳
.In this case, (
𝒳
,
∥
·
∥
) is called anon-Archimedean space. Moreover, if every Cauchy sequence is convergent, then (
𝒳
,
∥
·
∥
) is said to be a complete non-Archimedean space.
It follows from the strong triangle inequality that(2)
∥
x
n
-
x
m
∥
≤
max
⁡
{
∥
x
j
+
1
-
x
j
∥
:
m
≤
j
<
n
-
1
}
,
for all x
n
,
x
m
∈
𝒳 and all m
,
n
∈
ℕ with n
>
m. Therefore a sequence {
x
n
} is a Cauchy sequence in non-Archimedean space if and only if the sequence {
x
n
+
1
-
x
n
} converges to zero in the space.
On the other hand, Moslehian and Rassias [14] discussed the stability of the additive functional equation and the quadratic functional equation in non-Archimedean normed spaces. Quite recently, the new results on stability of functional equations in non-Archimedean metric spaces have been investigated (e.g., [6, 12, 15]).
Here and now, we consider aquadratic-additive type functional equation
(3)
f
(
x
+
a
y
)
+
f
(
x
-
a
y
)
-
2
f
(
x
)
-
a
2
f
(
y
)
-
a
2
f
(
-
y
)
=
0
,
whose solution is called a quadratic-additive mapping. Quite recently, the second author [16] investigated the Hyers-Ulam stability of the functional equation (3) on restricted domains for the case a
=
1. In particular, hyperstability of the functional equation (3) in the case when a
=
1 was investigated in [17]. The purpose of this work establishes the generalized Hyers-Ulam stability of the functional equation (3) in non-Archimedean normed spaces.


## 2. Main Results

Throughout this section, we assume that𝒳 is a non-Archimedean normed space and 𝒴 is a complete non-Archimedean space. For a given mapping f
:
E
1
→
E
2 with vector spaces E
1 and E
2, we use the abbreviations
(4)
𝒜
f
(
x
,
y
)
:
=
f
(
x
+
y
)
-
f
(
x
)
-
f
(
y
)
,
𝒬
f
(
x
,
y
)
:
=
f
(
x
+
y
)
+
f
(
x
-
y
)
-
2
f
(
x
)
-
2
f
(
y
)
,
D
f
(
x
,
y
)
:
=
f
(
x
+
a
y
)
+
f
(
x
-
a
y
)
-
2
f
(
x
)
-
a
2
f
(
y
)
-
a
2
f
(
-
y
)
,
for all x
,
y
∈
E
1, where a is a nonzero integer.Theorem 3.
LetE
1 and E
2 be vector spaces. A mapping f
:
E
1
→
E
2 satisfies equation D
f
(
x
,
y
)
=
0 for all x
,
y
∈
E
1 if and only if there exist a quadratic mapping g
:
E
1
→
E
2 and an additive mapping h
:
E
1
→
E
2 such that
(5)
f
(
x
)
=
g
(
x
)
+
h
(
x
)
,
for all x
∈
E
1.Proof.
(⇒) We decompose f into the even part and the odd part by putting
(6)
g
(
x
)
=
f
(
x
)
+
f
(
-
x
)
2
,
h
(
x
)
=
f
(
x
)
-
f
(
-
x
)
2
,
for all x
∈
E
1. Notice that f
(
0
)
=
-
D
f
(
0,0
)
/
2
a
2
=
0. From the equalities
(7)
𝒬
g
(
x
,
y
)
=
D
g
(
x
,
y
a
)
-
D
g
(
0
,
y
a
)
=
0
,
𝒜
h
(
x
,
y
)
=
D
h
(
x
+
y
2
,
x
+
y
2
a
)
-
D
h
(
x
+
y
2
,
x
-
y
2
a
)
=
0
,
for all x
,
y
∈
E
1, we conclude that g is a quadratic mapping and h is an additive mapping.
(⇐) If there exist a quadratic mapping g
:
E
1
→
E
2 and an additive mapping h
:
E
1
→
E
2 such that
(8)
f
(
x
)
=
g
(
x
)
+
h
(
x
)
,
for all x
∈
E
1, then we find that
(9)
D
f
(
x
,
y
)
=
D
g
(
x
,
y
)
+
D
h
(
x
,
y
)
=
𝒬
g
(
x
,
a
y
)
+
2
g
(
a
y
)
-
a
2
g
(
y
)
-
a
2
g
(
-
y
)
+
𝒜
h
(
x
,
x
)
-
𝒜
h
(
x
+
a
y
,
x
-
a
y
)
=
0
,
for all x
,
y
∈
E
1. We arrive at the desired conclusion.Theorem 4.
Letφ
:
𝒳
2
→
[
0
,
∞
) be a function such that
(10)
lim
⁡
n
→
∞
φ
(
2
n
x
,
2
n
y
)
|
2
|
n
=
φ
(
a
n
x
,
a
n
y
)
|
a
|
2
n
=
0
,
for all x
,
y
∈
𝒳 and, for each x
∈
𝒳, let the limit
(11)
lim
⁡
n
→
∞
max
⁡
{
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
,
φ
(
2
j
x
,
2
j
x
/
a
)
|
2
|
j
+
2
,
f
g
f
g
g
g
g
g
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
:
0
≤
j
<
n
}
,
denoted by φ
~
(
x
), exist. Suppose that f
:
𝒳
→
𝒴 is a mapping satisfying
(12)
∥
D
f
(
x
,
y
)
∥
≤
φ
(
x
,
y
)
,
for all x
,
y
∈
𝒳. Then there exists a quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(13)
∥
f
(
x
)
-
T
(
x
)
∥
≤
φ
~
(
x
)
,
for all x
∈
𝒳, where the mapping T is given by
(14)
T
(
x
)
=
lim
⁡
n
→
∞
[
f
(
a
n
x
)
+
f
(
-
a
n
x
)
2
a
2
n
+
f
(
2
n
x
)
-
f
(
-
2
n
x
)
2
n
+
1
]
,
for all x
∈
𝒳.Proof.
For a given mappingf
:
𝒳
→
𝒴 and n
∈
ℕ, let J
n
f
:
𝒳
→
𝒴 be a mapping defined by
(15)
J
n
f
(
x
)
=
f
(
a
n
x
)
+
f
(
-
a
n
x
)
2
a
2
n
+
f
(
2
n
x
)
-
f
(
-
2
n
x
)
2
n
+
1
,
for all x
∈
𝒳. Note that J
0
f
(
x
)
=
f
(
x
) and
(16)
∥
J
j
f
(
x
)
-
J
j
+
1
f
(
x
)
∥
=
∥
-
D
f
(
0
,
a
j
x
)
2
a
2
j
+
2
-
D
f
(
2
j
x
,
2
j
x
/
a
)
2
j
+
2
+
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
2
j
+
2
∥
≤
max
⁡
{
∥
D
f
(
0
,
a
j
x
)
∥
|
2
|
|
a
|
2
j
+
2
,
g
g
g
∥
D
f
(
2
j
x
,
2
j
x
/
a
)
∥
|
2
|
j
+
2
,
g
h
h
∥
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
∥
|
2
|
j
+
2
}
≤
max
⁡
{
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
,
φ
(
2
j
x
,
2
j
x
/
a
)
|
2
|
j
+
2
,
g
h
h
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
}
,
for all x
∈
𝒳 and j
≥
0. It follows from (10) and (16) that the sequence {
J
n
f
(
x
)
} is Cauchy. Since 𝒴 is complete, we conclude that {
J
n
f
(
x
)
} is convergent. So we can define T
:
𝒳
→
𝒴 by
(17)
T
(
x
)
:
=
lim
⁡
n
→
∞
J
n
f
(
x
)
,
for all x
∈
𝒳. An induction implies that
(18)
∥
J
n
f
(
x
)
-
f
(
x
)
∥
≤
max
⁡
{
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
,
φ
(
2
j
x
,
2
j
x
/
a
)
|
2
|
j
+
2
,
f
g
f
h
g
g
g
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
:
0
≤
j
<
n
}
,
for all n
∈
ℕ and all x
∈
𝒳. By taking the limit as n
→
∞ in (18) and using (10), we obtain inequality (13).
From (12), we get
(19)
∥
D
J
n
f
(
x
,
y
)
∥
=
∥
D
f
(
2
n
x
,
2
n
y
)
-
D
f
(
-
2
n
x
,
-
2
n
y
)
2
n
+
1
b
+
D
f
(
a
n
x
,
a
n
y
)
+
D
f
(
-
a
n
x
,
-
a
n
y
)
2
a
2
n
∥
≤
max
⁡
{
φ
(
2
n
x
,
2
n
y
)
|
2
|
n
+
1
,
φ
(
-
2
n
x
,
-
2
n
y
)
|
2
|
n
+
1
,
g
h
g
h
h
φ
(
a
n
x
,
a
n
y
)
|
2
|
|
a
|
2
n
,
φ
(
-
a
n
x
,
-
a
n
y
)
|
2
|
|
a
|
2
n
}
for all x
,
y
∈
𝒳. Take the limit as n
→
∞ in the above inequality and then use (10) to have D
T
(
x
,
y
)
=
0 for all x
,
y
∈
𝒳.Corollary 5.
Letr
>
2 be a real number and |
2
|, |
a
|
<
1. If a mapping f
:
𝒳
→
𝒴 satisfies the condition
(20)
∥
D
f
(
x
,
y
)
∥
≤
θ
(
∥
x
∥
r
+
∥
y
∥
r
)
,
for all x
,
y
∈
𝒳, then there exists a unique quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(21)
∥
f
(
x
)
-
T
(
x
)
∥
≤
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
|
a
|
r
,
for all x
∈
𝒳.Proof.
Letφ
(
x
,
y
)
=
θ
(
∥
x
∥
r
+
∥
y
∥
r
). Since |
2
|
r
-
1
<
1 and |
a
|
r
-
2
<
1, we know that
(22)
lim
⁡
n
→
∞
φ
(
2
n
x
,
2
n
y
)
|
2
|
n
=
φ
(
a
n
x
,
a
n
y
)
|
a
|
2
n
=
0
,
for all x
,
y
∈
𝒳. Therefore the conditions of Theorem 4 are fulfilled. In particular, it is easy to see that
(23)
φ
~
(
x
)
=
max
⁡
{
θ
∥
x
∥
r
|
2
|
|
a
|
2
,
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
|
a
|
r
}
=
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
|
a
|
r
.
Theorem 4 guarantees that there exists a quadratic-additive mapping T
:
𝒳
→
𝒴 with (21).
Now, to show uniqueness of the mappingT, let us assume that T
′ is another quadratic-additive mapping satisfying (21). Then we have
(24)
T
′
(
x
)
-
J
k
T
′
(
x
)
=
∑
j
=
0
k
-
1
‍
(
-
D
T
′
(
0
,
a
j
x
)
2
a
2
j
+
2
d
d
d
f
f
f
f
d
g
h
-
D
T
′
(
2
j
x
,
2
j
x
/
a
)
2
j
+
2
+
D
T
′
(
-
2
j
x
,
-
2
j
x
/
a
)
2
j
+
2
)
=
0
,
for any k
∈
ℕ, and thus we feel that
(25)
∥
T
(
x
)
-
T
′
(
x
)
∥
=
lim
⁡
k
→
∞
∥
J
k
T
(
x
)
-
J
k
T
′
(
x
)
∥
≤
lim
⁡
k
→
∞
max
⁡
{
∥
J
k
f
(
x
)
-
J
k
T
′
(
x
)
∥

∥
J
k
T
(
x
)
-
J
k
f
(
x
)
∥
,
g
h
h
h
h
f
h
h
h
h
∥
J
k
f
(
x
)
-
J
k
T
′
(
x
)
∥
}
≤
lim
⁡
k
→
∞
max
⁡
{
|
2
|
-
k
-
1
∥
T
(
2
k
x
)
-
f
(
2
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
k
-
1
∥
T
(
-
2
k
x
)
-
f
(
-
2
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
k
-
1
∥
f
(
2
k
x
)
-
T
′
(
2
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
k
-
1
∥
f
(
-
2
k
x
)
-
T
′
(
-
2
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
1
|
a
|
-
2
k
∥
T
(
a
k
x
)
-
f
(
a
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
1
|
a
|
-
2
k
∥
T
(
-
a
k
x
)
-
f
(
-
a
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
1
|
a
|
-
2
k
∥
f
(
a
k
x
)
-
T
′
(
a
k
x
)
∥
,
g
g
g
g
g
g
g
g
g
g
g
|
2
|
-
1
|
a
|
-
2
k
∥
f
(
-
a
k
x
)
-
T
′
(
-
a
k
x
)
∥
}
≤
lim
⁡
k
→
∞
max
⁡
{
|
a
|
k
(
r
-
2
)
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
3
|
a
|
r
,
g
h
h
h
h
h
g
g
h
h
h
|
2
|
k
(
r
-
1
)
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
3
|
a
|
r
}
=
0
,
for all x
∈
𝒳, which implies that T is unique.Theorem 6.
Letφ
:
𝒳
2
→
[
0
,
∞
) be a function such that
(26)
lim
⁡
n
→
∞
|
2
|
n
φ
(
x
2
n
,
x
2
n
)
=
lim
⁡
n
→
∞
|
a
|
2
n
φ
(
x
a
n
,
y
a
n
)
=
0
,
for all x
,
y
∈
𝒳 and, for each x
∈
𝒳, let the limit
(27)
lim
⁡
n
→
∞
max
⁡
{
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
|
a
|
2
j
|
2
|
φ
(
0
,
x
a
j
+
1
)
,
|
2
|
j
-
1
φ
(
x
2
j
+
1
,
x
2
j
+
1
a
)
,
g
h
h
h
g
h
h
|
2
|
j
-
1
φ
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
:
0
≤
j
<
n
|
a
|
2
j
|
2
|
}
,
denoted by φ
~
(
x
), exist. Suppose that f
:
𝒳
→
𝒴 is a mapping satisfying
(28)
∥
D
f
(
x
,
y
)
∥
≤
φ
(
x
,
y
)
,
for all x
,
y
∈
𝒳. Then there exists a quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(29)
∥
f
(
x
)
-
T
(
x
)
∥
≤
φ
~
(
x
)
,
for all x
∈
𝒳, where the mapping T is given by
(30)
T
(
x
)
=
lim
⁡
n
→
∞
[
a
2
n
2
(
f
(
x
a
n
)
+
f
(
-
x
a
n
)
)
g
g
h
a
2
n
2
+
2
n
-
1
(
f
(
x
2
n
)
-
f
(
-
x
2
n
)
)
]
,
for all x
∈
𝒳.Proof.
For a given mappingf
:
𝒳
→
𝒴 and n
∈
ℕ, let J
n
f
:
𝒳
→
𝒴 be a mapping defined by
(31)
J
n
f
(
x
)
=
a
2
n
2
(
f
(
x
a
n
)
+
f
(
-
x
a
n
)
)
+
2
n
-
1
(
f
(
x
2
n
)
-
f
(
-
x
2
n
)
)
,
for all x
∈
𝒳. Observe that J
0
f
(
x
)
=
f
(
x
) and
(32)
∥
J
j
f
(
x
)
-
J
j
+
1
f
(
x
)
∥
=
∥
a
2
j
2
D
f
(
0
,
x
a
j
+
1
)
h
+
2
j
-
1
D
f
(
x
2
j
+
1
,
x
2
j
+
1
a
)
h
-
2
j
-
1
D
f
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
a
2
j
2
∥
≤
max
⁡
{
|
a
|
2
j
|
2
|
∥
D
f
(
0
,
x
a
j
+
1
)
∥
,
g
h
h
h
h
|
2
|
j
-
1
∥
D
f
(
x
2
j
+
1
,
x
2
j
+
1
a
)
∥
,
g
h
h
h
h
|
2
|
j
-
1
∥
D
f
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
∥

|
a
|
2
j
|
2
|
}
≤
max
⁡
{
|
a
|
2
j
|
2
|
φ
(
0
,
x
a
j
+
1
)
,
g
h
h
h
h
|
2
|
j
-
1
φ
(
x
2
j
+
1
,
x
2
j
+
1
a
)
,
f
g
g
g
g
g
|
2
|
j
-
1
φ
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)

|
a
|
2
j
|
2
|
}
,
for all x
∈
𝒳 and j
≥
0. It follows by (26) and (32) that the sequence {
J
n
f
(
x
)
} is Cauchy. Since 𝒴 is complete, {
J
n
f
(
x
)
} converges and so we can define T
:
𝒳
→
𝒴 by
(33)
T
(
x
)
:
=
lim
⁡
n
→
∞
J
n
f
(
x
)
,
for all x
∈
𝒳. Applying the induction, we yield that
(34)
∥
J
n
f
(
x
)
-
f
(
x
)
∥
≤
max
⁡
{
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
|
a
|
2
j
|
2
|
φ
(
0
,
x
a
j
+
1
)
,
|
2
|
j
-
1
φ
(
x
2
j
+
1
,
x
2
j
+
1
a
)
,
g
h
h
h
h
h
h
|
2
|
j
-
1
φ
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
:
0
≤
j
<
n
|
a
|
2
j
|
2
|
}
,
for all n
∈
ℕ and all x
∈
𝒳. Sending the limit as n
→
∞ in (34) with (26), we arrive at inequality (29).
According to (28), we see that
(35)
∥
D
J
n
f
(
x
,
y
)
∥
=
∥
(
D
f
(
x
2
n
,
y
2
n
)
-
D
f
(
-
x
2
n
,
-
y
2
n
)
)
a
2
n
2
(
D
f
(
x
a
n
,
y
a
n
)
+
D
f
(
-
x
a
n
,
-
y
a
n
)
)
g
h
h
+
2
n
-
1
(
D
f
(
x
2
n
,
y
2
n
)
-
D
f
(
-
x
2
n
,
-
y
2
n
)
)
a
2
n
2
∥
≤
max
⁡
{
|
a
|
2
n
|
2
|
φ
(
x
a
n
,
y
a
n
)
,
|
a
|
2
n
|
2
|
φ
(
-
x
a
n
,
-
y
a
n
)
,
g
h
h
h
h
h
h
h
h
|
2
|
n
-
1
φ
(
x
2
n
,
y
2
n
)
,
|
2
|
n
-
1
φ
(
-
x
2
n
,
-
y
2
n
)

|
a
|
2
n
|
2
|
}
,
for all x
,
y
∈
𝒳. Taking the limit as n
→
∞ and using (26), we get D
T
(
x
,
y
)
=
0 for all x
,
y
∈
𝒳.Corollary 7.
Letr
<
1 be a real number and |
2
|, |
a
|
<
1. If a mapping f
:
𝒳
→
𝒴 satisfies the condition
(36)
∥
D
f
(
x
,
y
)
∥
≤
θ
(
∥
x
∥
r
+
∥
y
∥
r
)
,
for all x
,
y
∈
𝒳, then there exists a unique quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(37)
∥
f
(
x
)
-
T
(
x
)
∥
≤
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
1
+
r
|
a
|
r
,
for all x
∈
𝒳.Proof.
Letφ
(
x
,
y
)
=
θ
(
∥
x
∥
r
+
∥
y
∥
r
). Since |
a
|
1
-
r
<
1 and |
a
|
2
-
r
<
1, the mapping φ satisfies equalities (26) for all x
,
y
∈
𝒳. Moreover, we see that
(38)
φ
~
(
x
)
=
max
⁡
{
θ
∥
x
∥
r
|
2
|
|
a
|
r
,
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
1
+
r
|
a
|
r
}
=
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
1
+
r
|
a
|
r
.
Then it follows from Theorem 6 that there exists a quadratic-additive mapping T
:
𝒳
→
𝒴 satisfying (37).
In order to prove the uniqueness ofT, we assume that T
′ is another quadratic-additive mapping satisfying (37). Then
(39)
T
′
(
x
)
-
J
k
T
′
(
x
)
=
∑
j
=
0
k
-
1
‍
(
a
2
j
2
D
T
′
(
0
,
x
a
j
+
1
)
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
ghhhhh
+
2
j
-
1
D
T
′
(
x
2
j
+
1
,
x
2
j
+
1
a
)
ghhhhh
a
2
j
2
-
2
j
-
1
D
T
′
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
)
=
0
,
for any k
∈
ℕ, whence we give that
(40)
∥
T
(
x
)
-
T
′
(
x
)
∥
=
lim
⁡
k
→
∞
∥
J
k
T
(
x
)
-
J
k
T
′
(
x
)
∥
≤
lim
⁡
k
→
∞
max
⁡
{
∥
J
k
T
(
x
)
-
J
k
f
(
x
)
∥
,
∥
J
k
f
(
x
)
-
J
k
T
′
(
x
)
∥
}
≤
lim
⁡
k
→
∞
max
⁡
{
|
2
|
k
-
1
∥
(
T
-
f
)
(
x
2
k
)
∥
,
ghhhhhhhhhhhh
|
2
|
k
-
1
∥
(
T
-
f
)
(
-
x
2
k
)
∥
,
ghhhhhhhhhhhh
|
2
|
k
-
1
∥
(
f
-
T
′
)
(
x
2
k
)
∥
,
ghhhhhhhhhhhh
|
2
|
k
-
1
∥
(
f
-
T
′
)
(
-
x
2
k
)
∥
,
ghhhhhhhhhhhh
|
a
|
2
k
|
2
|
∥
(
T
-
f
)
(
x
a
k
)
∥
,
ghhhhhhhhhhhh
|
a
|
2
k
|
2
|
∥
(
T
-
f
)
(
-
x
a
k
)
∥
,
ghhhhhhhhhhhh
|
a
|
2
k
|
2
|
∥
(
f
-
T
′
)
(
x
a
k
)
∥
,
ghhhhhhhhhhhh
|
a
|
2
k
|
2
|
∥
(
f
-
T
′
)
(
-
x
a
k
)
∥
}
≤
lim
⁡
k
→
∞
max
⁡
⁡
{
|
a
|
(
2
-
r
)
k
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
+
r
|
a
|
r
,
g
h
h
h
h
h
h
h
h
h
h
h
h
|
2
|
k
(
1
-
r
)
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
+
r
|
a
|
r
}
=
0
,
for all x
∈
𝒳, which means that T is unique.Theorem 8.
Letφ
:
𝒳
2
→
[
0
,
∞
) be a function such that
(41)
lim
⁡
n
→
∞
|
2
|
n
φ
(
x
2
n
,
y
2
n
)
=
lim
⁡
n
→
∞
φ
(
a
n
x
,
a
n
y
)
|
a
|
2
n
=
0
,
for all x
,
y
∈
𝒳 and, for each x
∈
𝒳, let the limit
(42)
lim
⁡
n
→
∞
max
⁡
{
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
,
|
2
|
j
-
1
φ
(
x
2
j
+
1
,
x
2
j
+
1
a
)
,
h
j
h
j
j
j
j
j
j
|
2
|
j
-
1
φ
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
:
0
≤
j
<
n
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
}
,
denoted by φ
~
(
x
), exist. Suppose that f
:
𝒳
→
𝒴 is a mapping satisfying
(43)
∥
D
f
(
x
,
y
)
∥
≤
φ
(
x
,
y
)
,
for all x
,
y
∈
𝒳. Then there exists a quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(44)
∥
f
(
x
)
-
T
(
x
)
∥
≤
φ
~
(
x
)
,
for all x
∈
𝒳, where the mapping T is given by
(45)
T
(
x
)
=
lim
⁡
n
→
∞
[
f
(
a
n
x
)
+
f
(
-
a
n
x
)
2
a
2
n
+
f
(
a
n
x
)
+
f
(
-
a
n
x
)
2
a
2
n
2
n
-
1
(
f
(
x
2
n
)
-
f
(
-
x
2
n
)
)
]
,
for all x
∈
𝒳.Proof.
For a given mappingf
:
𝒳
→
𝒴 and n
∈
ℕ, let J
n
f
:
𝒳
→
𝒴 be a mapping defined by
(46)
J
n
f
(
x
)
=
f
(
a
n
x
)
+
f
(
-
a
n
x
)
2
a
2
n
+
2
n
-
1
(
f
(
x
2
n
)
-
f
(
-
x
2
n
)
)
,
for all x
∈
𝒳. Now we note that J
0
f
(
x
)
=
f
(
x
) and
(47)
∥
J
j
f
(
x
)
-
J
j
+
1
f
(
x
)
∥
=
∥
-
D
f
(
0
,
a
j
x
)
2
a
2
j
+
2
+
2
j
-
1
D
f
(
x
2
j
+
1
,
x
2
j
+
1
a
)
g
D
f
(
0
,
a
j
x
)
2
a
2
j
+
2
-
2
j
-
1
D
f
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
∥
≤
max
⁡
{
∥
D
f
(
0
,
a
j
x
)
∥
|
2
|
|
a
|
2
j
+
2
,
g
g
g
g
g
g
g
|
2
|
j
-
1
∥
D
f
(
x
2
j
+
1
,
x
2
j
+
1
a
)
∥
,
g
g
g
g
g
g
g
∥
D
f
(
0
,
a
j
x
)
∥
|
2
|
|
a
|
2
j
+
2
|
2
|
j
-
1
∥
D
f
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
∥
}
≤
max
⁡
{
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
,
g
h
g
g
g
g
h
|
2
|
j
-
1
φ
(
x
2
j
+
1
,
x
2
j
+
1
a
)
,
g
h
g
g
g
g
h
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
|
2
|
j
-
1
φ
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
}
,
for all x
∈
𝒳 and j
≥
0. Hence it follows by (41) and (47) that the sequence {
J
n
f
(
x
)
} is Cauchy. Since 𝒴 is complete, we conclude that {
J
n
f
(
x
)
} is convergent. Thus we can define T
:
𝒳
→
𝒴 by
(48)
T
(
x
)
:
=
lim
⁡
n
→
∞
J
n
f
(
x
)
,
for all x
∈
𝒳. Using the induction argument, we prove that
(49)
∥
J
n
f
(
x
)
-
f
(
x
)
∥
≤
max
⁡
{
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
,
|
2
|
j
-
1
φ
(
x
2
j
+
1
,
x
2
j
+
1
a
)
,
ghghhhh
φ
(
0
,
a
j
x
)
|
2
|
|
a
|
2
j
+
2
|
2
|
j
-
1
φ
(
-
x
2
j
+
1
,
-
x
2
j
+
1
a
)
:
0
≤
j
<
n
}
,
for all n
∈
ℕ and all x
∈
𝒳. Taking the limit as n
→
∞ in (49) and using (41) lead to inequality (44).
By virtue of (43), we get
(50)
∥
D
J
n
f
(
x
,
y
)
∥
=
∥
D
f
(
a
n
x
,
a
n
y
)
+
D
f
(
-
a
n
x
,
-
a
n
y
)
2
a
2
n
D
f
(
a
n
x
,
a
n
y
)
+
D
f
(
-
a
n
x
,
-
a
n
y
)
2
a
2
n
+
2
n
-
1
(
D
f
(
x
2
n
,
y
2
n
)
-
D
f
(
-
x
2
n
,
-
y
2
n
)
)
∥
≤
max
⁡
{
φ
(
a
n
x
,
a
n
y
)
|
2
|
|
a
|
2
n
,
φ
(
-
a
n
x
,
-
a
n
y
)
|
2
|
|
a
|
2
n
,
h
h
g
h
|
2
|
n
-
1
φ
(
x
2
n
,
y
2
n
)
,
ghgh
φ
(
a
n
x
,
a
n
y
)
|
2
|
|
a
|
2
n
|
2
|
n
-
1
φ
(
-
x
2
n
,
-
y
2
n
)
}
,
for all x
,
y
∈
𝒳. Send the limit as n
→
∞ and then use (41) to find D
T
(
x
,
y
)
=
0 for all x
,
y
∈
𝒳.Theorem 9.
Letφ
:
𝒳
2
→
[
0
,
∞
) be a function such that
(51)
lim
⁡
n
→
∞
φ
(
2
n
x
,
2
n
y
)
|
2
|
n
=
lim
⁡
n
→
∞
|
a
|
2
n
φ
(
x
a
n
,
y
a
n
)
=
0
,
for all x
,
y
∈
𝒳 and, for each x
∈
𝒳, let the limit
(52)
lim
⁡
n
→
∞
max
⁡
{
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
|
a
|
2
j
|
2
|
φ
(
0
,
x
a
j
+
1
)
,
f
g
f
g
g
g
h
φ
(
2
j
x
,
2
j
x
/
a
)
|
2
|
j
+
2
,
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
:
0
≤
j
<
n
|
a
|
2
j
|
2
|
}
,
denoted by φ
~
(
x
), exist. Suppose that f
:
𝒳
→
𝒴 is a mapping satisfying
(53)
∥
D
f
(
x
,
y
)
∥
≤
φ
(
x
,
y
)
,
for all x
,
y
∈
𝒳. Then there exists a quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(54)
∥
f
(
x
)
-
T
(
x
)
∥
≤
φ
~
(
x
)
,
for all x
∈
𝒳, where the mapping T is given by
(55)
T
(
x
)
=
lim
⁡
n
→
∞
[
a
2
n
2
(
f
(
x
a
n
)
+
f
(
-
x
a
n
)
)
g
g
h
+
f
(
2
n
x
)
-
f
(
-
2
n
x
)
2
n
+
1
]
,
for all x
∈
𝒳.Proof.
For a given mappingf
:
𝒳
→
𝒴 and n
∈
ℕ, let J
n
f
:
𝒳
→
𝒴 be a mapping defined by
(56)
J
n
f
(
x
)
=
a
2
n
2
(
f
(
x
a
n
)
+
f
(
-
x
a
n
)
)
+
f
(
2
n
x
)
-
f
(
-
2
n
x
)
2
n
+
1
,
for all x
∈
𝒳. We remark that J
0
f
(
x
)
=
f
(
x
) and
(57)
∥
J
j
f
(
x
)
-
J
j
+
1
f
(
x
)
∥
=
∥
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
2
j
+
2
a
2
j
2
D
f
(
0
,
x
a
j
+
1
)
-
D
f
(
2
j
x
,
2
j
x
/
a
)
2
j
+
2
+
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
2
j
+
2
a
2
j
2
∥
≤
max
⁡
{
∥
D
f
(
2
j
x
,
2
j
x
/
a
)
∥
|
2
|
j
+
2
|
a
|
2
j
|
2
|
∥
D
f
(
0
,
x
a
j
+
1
)
∥
,
g
h
h
∥
D
f
(
2
j
x
,
2
j
x
/
a
)
∥
|
2
|
j
+
2
,
g
h
h
∥
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
∥
|
2
|
j
+
2
|
a
|
2
j
|
2
|
}
≤
max
⁡
{
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
|
a
|
2
j
|
2
|
φ
(
0
,
x
a
j
+
1
)
,
g
h
h
φ
(
2
j
x
,
2
j
x
/
a
)
|
2
|
j
+
2
,
g
h
h
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
|
a
|
2
j
|
2
|
}
,
for all x
∈
𝒳 and j
≥
0. So it follows from (51) and (57) that the sequence {
J
n
f
(
x
)
} is Cauchy. Due to the completeness of 𝒴, this sequence is convergent. Let T
:
𝒳
→
𝒴 be a mapping defined by
(58)
T
(
x
)
:
=
lim
⁡
n
→
∞
J
n
f
(
x
)
,
for all x
∈
𝒳. An induction implies that
(59)
∥
J
n
f
(
x
)
-
f
(
x
)
∥
≤
max
⁡
{
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
:
0
≤
j
<
n
|
a
|
2
j
|
2
|
}
|
a
|
2
j
|
2
|
φ
(
0
,
x
a
j
+
1
)
,
g
h
h
h
h
h
φ
(
2
j
x
,
2
j
x
/
a
)
|
2
|
j
+
2
,
ghhh
φ
(
-
2
j
x
,
-
2
j
x
/
a
)
|
2
|
j
+
2
:
0
≤
j
<
n
|
a
|
2
j
|
2
|
}
,
for all n
∈
ℕ and all x
∈
𝒳. By passing the limit as n
→
∞ in (59) with (51), we obtain the relation (54).
From (53), we find that
(60)
∥
D
J
n
f
(
x
,
y
)
∥
=
∥
D
f
(
2
n
x
,
2
n
y
)
-
2
n
D
f
(
-
2
n
x
,
-
2
n
y
)
2
n
+
1
a
2
n
2
(
D
f
(
x
a
n
,
y
a
n
)
+
D
f
(
-
x
a
n
,
-
y
a
n
)
)
g
+
D
f
(
2
n
x
,
2
n
y
)
-
2
n
D
f
(
-
2
n
x
,
-
2
n
y
)
2
n
+
1
a
2
n
2
∥
≤
max
⁡
{
φ
(
-
2
n
x
,
-
2
n
y
)
|
2
|
n
+
1
|
a
|
2
n
|
2
|
φ
(
x
a
n
,
y
a
n
)
,
|
a
|
2
n
|
2
|
φ
(
-
x
a
n
,
-
y
a
n
)
,

g
h
h
φ
(
2
n
x
,
2
n
y
)
|
2
|
n
+
1
,
φ
(
-
2
n
x
,
-
2
n
y
)
|
2
|
n
+
1
|
a
|
2
n
|
2
|
}
,
for all x
,
y
∈
𝒳. Taking the limit as n
→
∞ and using (51), we arrive at D
T
(
x
,
y
)
=
0 for all x
,
y
∈
𝒳.Corollary 10.
Let|
a
|
,
|
2
|
<
1 and let r be a real number such that 1
<
r
<
2. If a mapping f
:
𝒳
→
𝒴 satisfies the condition
(61)
∥
D
f
(
x
,
y
)
∥
≤
θ
(
∥
x
∥
r
+
∥
y
∥
r
)
,
for all x
,
y
∈
𝒳, then there exists a unique quadratic-additive mapping T
:
𝒳
→
𝒴 such that
(62)
∥
f
(
x
)
-
T
(
x
)
∥
≤
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
|
a
|
r
,
for all x
∈
𝒳.Proof.
We considerφ
(
x
,
y
)
=
θ
(
∥
x
∥
r
+
∥
y
∥
r
). Based on the fact that |
a
|
2
-
r
<
1 and |
2
|
r
-
1
<
1, the mapping φ satisfies conditions (51). In fact, it is easy to see that
(63)
φ
~
(
x
)
=
max
⁡
{
θ
∥
x
∥
r
|
2
|
|
a
|
r
,
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
|
a
|
r
}
=
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
2
|
a
|
r
.
On account of Theorem 9, there is a quadratic-additive mapping T
:
𝒳
→
𝒴 satisfying (62).
To show uniqueness of the mappingT, we suppose that T
′ is another quadratic-additive mapping satisfying (62). Then
(64)
T
′
(
x
)
-
J
k
T
′
(
x
)
=
∑
j
=
0
k
-
1
‍
(
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
2
j
+
2
a
2
j
2
D
f
(
0
,
x
a
j
+
1
)
ghhhhh
-
D
f
(
2
j
x
,
2
j
x
/
a
)
2
j
+
2
+
D
f
(
-
2
j
x
,
-
2
j
x
/
a
)
2
j
+
2
)
=
0
,
for any k
∈
ℕ and so we figure out the following:
(65)
∥
T
(
x
)
-
T
′
(
x
)
∥
=
lim
⁡
k
→
∞
∥
J
k
T
(
x
)
-
J
k
T
′
(
x
)
∥
≤
lim
⁡
k
→
∞
max
⁡
{
∥
J
k
f
(
x
)
-
J
k
T
′
(
x
)
∥
∥
J
k
T
(
x
)
-
J
k
f
(
x
)
∥
,
g
h
h
h
h
h
h
h
h
∥
J
k
f
(
x
)
-
J
k
T
′
(
x
)
∥
}
≤
lim
⁡
k
→
∞
max
⁡
⁡
{
|
a
|
2
k
|
2
|
∥
(
f
-
T
′
)
(
-
x
a
k
)
∥
|
2
|
-
k
-
1
∥
T
(
2
k
x
)
-
f
(
2
k
x
)
∥
,
ghghhhhhh
|
2
|
-
k
-
1
ghghhhhhh
×
∥
T
(
-
2
k
x
)
-
f
(
-
2
k
x
)
∥
,
ghghhhhhh
|
2
|
-
k
-
1
∥
f
(
2
k
x
)
-
T
′
(
2
k
x
)
∥
,
ghghhhhhh
|
2
|
-
k
-
1
ghghhhhhh
×
∥
f
(
-
2
k
x
)
-
T
′
(
-
2
k
x
)
∥
,
ghghhhhhh
|
a
|
2
k
|
2
|
∥
(
T
-
f
)
(
x
a
k
)
∥
,
ghghhhhhh
|
a
|
2
k
|
2
|
∥
(
T
-
f
)
(
-
x
a
k
)
∥
,
ghhghhhhh
|
a
|
2
k
|
2
|
∥
(
f
-
T
′
)
(
x
a
k
)
∥
,
ghghhhhhh
|
a
|
2
k
|
2
|
∥
(
f
-
T
′
)
(
-
x
a
k
)
∥
}
≤
lim
⁡
k
→
∞
max
⁡
⁡
{
|
a
|
k
(
2
-
r
)
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
3
|
a
|
r
,
ghghhhhhh
|
2
|
k
(
r
-
1
)
(
|
a
|
r
+
1
)
θ
∥
x
∥
r
|
2
|
3
|
a
|
r
}
=
0
,
for all x
∈
𝒳. This implies that T is unique.The problem is whether or not the previous corollaries hold the cases whenr
=
1 or r
=
2.



---

*Source: 919646-2014-02-25.xml*