# Approximate Homomorphisms and Derivations on Normed Lie Triple Systems

Authors: Hark-Mahn Kim; Juri Lee
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We investigate the generalized Hyers-Ulam stability of homomorphisms and derivations on normed Lie triple systems for the following generalized Cauchy-Jensen additive equationr
0
f
(
(
s
∑
j
=
1
p
x
j
+
t
∑
j
=
1
d
y
j
)
/
r
0
)
=
s
∑
j
=
1
p
‍
f
(
x
j
)
+
t
∑
j
=
1
d
‍
f
(
y
j
), where r
0
,
s
, and
t are nonzero real numbers. As a results, we generalize some stability results concerning this equation.


---

## Body

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1], concerning the stability of group homomorphisms.Let(
G
,
∘
) be a group and let (
H
,
⋆
,
d
) be a metric group with the metric d
(
·
,
·
). Given ε
>
0, there exist a scalar δ
=
δ
(
ε
)
>
0 such that if a mapping f
:
G
→
H satisfies the inequality d
(
f
(
x
∘
y
)
,
f
(
x
)
⋆
f
(
y
)
)
<
δ for all x
,
y
∈
G, then a homomorphism F
:
G
→
H exists with d
(
f
(
x
)
,
F
(
x
)
)
<
ε for all x
∈
G?Hyers [2] gave the first affirmative partial answer to the question of Ulam for additive mappings on Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. In 1990, Rassias [5] during 27th international symposium on functional equations asked the question whether such a theorem can also be proved for p
≥
1. In 1991, Gajda [6] following the same approach as in Rassias [5] gave an affirmative solution to this question for p
>
1. It was proved by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove Rassias’ type theorem when p
=
1.During the last three decades, several stability problems of functional equations have been investigated by a number of mathematicians; compare or confer [8–12] and references therein.Now, we introduce the concept of normed Lie triple systems. First of all, ternary algebraic operations were considered in 19th century by several mathematicians such as Cayley [13] who introduced the notion of cubic matrix which in turn was generalized by Kapranov et al. [14] in 1990. There are some applications, although they are still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation. The comments on physical applications of ternary structures can be found in [15–22].A normed (Banach) Lie triple system is a normed (Banach) space(
A
;
∥

·

∥
) with a trilinear mapping (
x
,
y
,
z
)
↦
[
x
,
y
,
z
] from A
×
A
×
A to A satisfying the following axioms:(i)
[
x
,
y
,
z
]
=
-
[
y
,
x
,
z
],
(ii)
[
x
,
y
,
z
]
+
[
y
,
z
,
x
]
+
[
z
,
x
,
y
]
=
0,
(iii)
[
u
,
v
,
[
x
,
y
,
z
]
]
=
[
[
u
,
v
,
x
]
,
y
,
z
]
+
[
x
,
[
u
,
v
,
y
]
,
z
]
+
[
x
,
y
,
[
u
,
v
,
z
]
],
(iv)
∥
[
x
,
y
,
z
]
∥
≤
∥
x
∥
∥
y
∥
∥
z
∥,for allu
,
v
,
x
,
y
,
z
∈
A. The concept of Lie triple system was first introduced by Lister [23](see also [24, 25]). Let A and B be normed Lie triple systems. A ℂ-linear mapping H
:
A
→
B is said to be a Lie triple homomorphism if H
(
[
x
,
y
,
z
]
)
=
[
H
(
x
)
,
H
(
y
)
,
H
(
z
)
] for all x
,
y
,
z
∈
A. A ℂ-linear mapping D
:
A
→
A is called a Lie triple derivation if D
(
[
x
,
y
,
z
]
)
=
[
D
(
x
)
,
y
,
z
]
+
[
x
,
D
(
y
)
,
z
]
+
[
x
,
y
,
D
(
z
)
] for all x
,
y
,
z
∈
A. The third identity asserts that the mapping D
u
,
v
:
x
↦
[
u
,
v
,
x
] is a (inner) derivation on A [25].Clearly, every Lie algebra is at the same time a Lie triple system via[
x
,
y
,
z
]
:
=
[
[
x
,
y
]
,
z
], and our definition of a homomorphism (derivation) coincides with that of prehomomorphism (prederivation) on a Lie algebra. Also, if U is an involutive automorphism of a Lie algebra (
L
,
[
,
]
), then the eigenspace E
-
1
(
U
) is a Lie triple system; see [25]. We remark that Lie triple systems are important since they give the structure of the tangent space of a symmetric space. Confer [25, 26] for reference. Also some applications of Lie triple systems can be found in Nambuòs approach by modifying the Heisenberg equation of motion [27].Now, one of the interesting functional equations is the following general Cauchy-Jensen additive equation:(1)
r
0
f
(
s
∑
j
=
1
p
x
j
+
t
∑
j
=
1
d
y
j
r
0
)
=
s
∑
j
=
1
p
f
(
x
j
)
+
t
∑
j
=
1
d
f
(
y
j
)
which is a generalization of Cauchy or Jensen additive equations, where r
0
,
s, and t are nonzero real numbers and f is a mapping between linear spaces. It is easy to see that a mapping f satisfies the above Cauchy-Jensen additive equation if and only if f
(
x
)
-
f
(
0
) is additive, where f
(
0
)
=
0 if r
0
≠
s
p
+
t
d. In this paper, we refine the generalized Hyers-Ulam stability results of [25] for Lie triple homomorphisms and Lie triple derivations on Lie triple systems associated with the general Cauchy-Jensen additive equation (1) and then we apply our results to study stability theorems of Lie triple homomorphisms and Lie triple derivations associated with Cauchy-Jensen additive equation (1) on normed Lie triple systems, which can be regarded as ternary structures. The reader may be referred to [28–30] for some other related stability results on derivations.Throughout this paper, suppose thatA is a normed Lie triple system with norm ∥

·

∥
A and that B is a Banach Lie triple system with norm ∥

·

∥
B.LetB
A be the set of all mappings from A to B, let ℒ
(
A
,
B
) be the set of all ℂ-linear mappings from A to B, let ℋ
(
A
,
B
) be the set of all Lie triple homomorphisms from A to B, let 𝒟
(
A
,
A
) be the set of all Lie triple derivations on A, and let ℝ
+ be the set of all nonnegative reals.


## 2. Stability of Homomorphisms on Normed Lie Triple Systems

In the section, we prove the stability of homomorphisms on normed Lie triple systems associated with the Cauchy-Jensen additive equation. Forf
∈
B
A, we define an operator J
λ
f
:
A
p
+
d
→
B by
(2)
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
≔
r
0
f
(
s
∑
j
=
1
p
λ
x
j
+
t
∑
j
=
1
d
λ
y
j
r
0
)
-
s
∑
j
=
1
p
λ
f
(
x
j
)
-
t
∑
j
=
1
d
λ
f
(
y
j
)
for all λ
∈
S and all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A, where S is a nontrivial connected subset of 𝕋
=
{
z
∈
ℂ
:
|
z
|
=
1
} with 1
∈
S and S
∖
{
1
}
≠
∅.Lemma 1.
Letf
∈
B
A with f
(
0
)
=
0. Then
(3)
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
=
0
for all λ
∈
S and all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A if and only if f
∈
ℒ
(
A
,
B
).Proof.
LetJ
λ
f
=
0 for all λ
∈
S. Then we note that r
0
f
(
(
s
/
r
0
)
x
)
=
s
f
(
x
) and r
0
f
(
(
t
/
r
0
)
y
)
=
t
f
(
y
) for all x
,
y
∈
A. Thus, r
0
f
(
(
λ
s
x
+
λ
t
y
)
/
r
0
)
=
s
λ
f
(
x
)
+
t
λ
f
(
y
)
=
λ
r
0
f
(
s
x
/
r
0
)
+
λ
r
0
f
(
t
y
/
r
0
) for all λ
∈
S. So f
(
λ
x
+
λ
y
)
=
λ
f
(
x
)
+
λ
f
(
y
) for all λ
∈
S, and then f is an additive mapping and f
(
λ
x
)
=
λ
f
(
x
) for all λ
∈
S and all x
∈
A. Hence the assumptions of Lemma 1 in [31] are fulfilled, for instance, in view of the comments in the papers [32, 33], and so we can deduce the lemma. The converse is trivial.We first consider stability theorem for approximate Lie triple homomorphisms of the functional equationJ
λ
f
=
0 with action of λ
∈
S.Theorem 2.
Suppose thatf
∈
B
A with f
(
0
)
=
0 satisfies
(4)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
φ
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
,
(5)
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
f
(
y
)
,
f
(
z
)
]
∥
B
≤
ψ
(
x
,
y
,
z
)
for all λ
∈
S and all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
A. If φ
:
A
p
+
d
→
ℝ
+ and ψ
:
A
3
→
ℝ
+ are functions such that
(6)
φ
~
(
x
)
≔
∑
n
=
0
∞
1
|
α
|
n
φ
(
0
,
…
,
0
︷
p
,
α
n
x
,
…
,
α
n
x
︷
d
)
<
∞
,
(
φ
~
(
x
)
≔
∑
n
=
1
∞
|
α
|
n
φ
(
0
,
…
,
0
︷
p
,
x
α
n
,
…
,
x
α
n
︷
d
)
<
∞
,
r
e
s
p
.
)
,
(7)
lim
⁡
n
→
∞
1
|
α
|
n
φ
(
α
n
x
1
,
…
,
α
n
x
p
,
α
n
y
1
,
…
,
α
n
y
d
)
=
0
,
(
lim
⁡
n
→
∞
|
α
|
n
φ
(
x
1
α
n
,
…
,
x
p
α
n
,
y
1
α
n
,
…
,
y
d
α
n
)
=
0
,
r
e
s
p
.
)
,
(8)
lim
⁡
n
→
∞
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
=
0
,
(
lim
⁡
n
→
∞
|
α
|
3
n
ψ
(
x
α
n
,
y
α
n
,
z
α
n
)
=
0
,
r
e
s
p
.
)
for all x
,
y
,
z
,
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A, where α
=
t
d
/
r
0, then there exists a unique h
∈
ℋ
(
A
,
B
) such that
(9)
∥
f
(
x
)
-
h
(
x
)
∥
B
≤
1
|
α
|
|
r
0
|
φ
~
(
x
)
=
1
|
t
|
d
φ
~
(
x
)
for all x
∈
A.Proof.
Lettingλ
=
1, x
1
=
⋯
=
x
p
=
0, and y
1
=
⋯
=
y
d
=
x in (4) and dividing by |
t
|
d, we get
(10)
∥
f
(
x
)
-
f
(
α
x
)
α
∥
B
≤
φ
(
0
,
…
,
0
,
x
,
…
,
x
)
|
t
|
d
=
φ
(
0
,
…
,
0
,
x
,
…
,
x
)
|
α
|
|
r
0
|
for all x
∈
A. Replacing x by α
n
x in (10) and dividing both sides of (10) by |
α
|
n, we get
(11)
∥
f
(
α
n
+
1
x
)
α
n
+
1
-
f
(
α
n
x
)
α
n
∥
B
≤
φ
(
0
,
…
,
0
,
α
n
x
,
…
,
α
n
x
)
|
α
|
n
+
1
|
r
0
|
for all x
∈
A and all nonnegative integers n. Therefore, one can use induction to show that
(12)
∥
f
(
α
n
x
)
α
n
-
f
(
α
m
x
)
α
m
∥
B
≤
∑
k
=
m
n
-
1
φ
(
0
,
…
,
0
,
α
k
x
,
…
,
α
k
x
)
|
α
|
k
+
1
|
r
0
|
for all nonnegative integers n
>
m and all x
∈
A. It follows from the convergence of the series (6) that the sequence {
f
(
α
n
x
)
/
α
n
} is a Cauchy sequence in B. Thus, we may define a mapping h
:
A
→
B by
(13)
h
(
x
)
≔
lim
⁡
n
→
∞
f
(
α
n
x
)
α
n
for all x
∈
A. Letting m
=
0 in (12) and passing n to infinity, we lead to the approximation (9). It follows from (4) that
(14)
∥
r
0
h
(
s
∑
j
=
1
p
λ
x
j
+
t
∑
j
=
1
d
λ
y
j
r
0
)

-

s
∑
j
=
1
p
λ
h
(
x
j
)

-

t
∑
j
=
1
d
λ
h
(
y
j
)
∥
B
=
lim
⁡
n
→
∞
1
|
α
|
n
∥
r
0
f
(
α
n
(
s
∑
j
=
1
p
λ
x
j
+
t
∑
j
=
1
d
λ
y
j
r
0
)
)
-
s
∑
j
=
1
p
λ
f
(
α
n
x
j
)
-
t
∑
j
=
1
d
λ
f
(
α
n
y
j
)
∥
B
≤
lim
⁡
n
→
∞
1
|
α
|
n
φ
(
α
n
x
1
,
…
,
α
n
x
p
,
α
n
y
1
,
…
,
α
n
y
d
)
=
0
for all λ
∈
S and x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A. Hence
(15)
r
0
h
(
s
∑
j
=
1
p
λ
x
j
+
t
∑
j
=
1
d
λ
y
j
r
0
)
=
s
∑
j
=
1
p
λ
h
(
x
j
)
+
t
∑
j
=
1
d
λ
h
(
y
j
)
for all λ
∈
S and x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A. So, h
∈
ℒ
(
A
,
B
) by Lemma 1.
It follows from (5) that
(16)
∥
h
(
[
x
,
y
,
z
]
)
-
[
h
(
x
)
,
h
(
y
)
,
h
(
z
)
]
∥
B
=
lim
⁡
n
→
∞
1
|
α
|
3
n
∥
f
(
[
α
n
x
,
α
n
y
,
α
n
z
]
)
-
[
f
(
α
n
x
)
,
f
(
α
n
y
)
,
f
(
α
n
z
)
]
∥
B
≤
lim
⁡
n
→
∞
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
=
0
for all x
,
y
,
z
∈
A. So, h
∈
ℋ
(
A
,
B
).
To prove the uniqueness of such homomorphismh subject to (9), we assume that there exists another Lie triple homomorphism h
′
:
A
→
B satisfying (9). Obviously, we have h
′
(
α
n
x
)
=
α
n
h
′
(
x
) and h
(
α
n
x
)
=
α
n
h
(
x
) for all x
∈
A. By the triangle inequality, (9), and above equality, we have
(17)
∥
h
(
x
)
-
h
′
(
x
)
∥
B
=
|
α
|
-
n
∥
h
(
α
n
x
)
-
h
′
(
α
n
x
)
∥
B
≤
|
α
|
-
n
∥
h
(
α
n
x
)
-
f
(
α
n
x
)
∥
B
+
|
α
|
-
n
∥
f
(
α
n
x
)
-
h
′
(
α
n
x
)
∥
B
≤
2
φ
~
(
α
n
x
)
|
t
|
|
d
|
|
α
|
n
for all n
∈
ℕ. By letting n
→
∞, we get h
(
x
)
=
h
′
(
x
) for any x
∈
A. This completes the proof.Corollary 3.
Letθ be a positive real number, let |
t
d
/
r
0
|
≠
1, and let r be a real number such that either 0
<
r
<
1 or r
>
1. Suppose that f
∈
B
A with f
(
0
)
=
0 satisfies
(18)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
θ
(
∑
j
=
1
p
∥
x
j
∥
A
r
+
∑
j
=
1
d
∥
y
j
∥
A
r
)
,
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
f
(
y
)
,
f
(
z
)
]
∥
B
≤
θ
(
∥
x
∥
A
3
r
+
∥
y
∥
A
3
r
+
∥
z
∥
A
3
r
)
,
for all λ
∈
S and all x
,
y
,
z
,
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A. Then there exists a unique h
∈
ℋ
(
A
,
B
) such that(19)
∥
f
(
x
)
-
h
(
x
)
∥
B
≤
{
|
t
d
/
r
0
|
θ
|
t
|
(
|
t
d
/
r
0
|
-
|
t
d
/
r
0
|
r
)
∥
x
∥
A
r
,
i
f
0
<
r
<
1
,
|
t
d
r
0
|
>
1
(
r
>
1
,
|
t
d
r
0
|
<
1
)
(
|
t
d
/
r
0
|
θ
|
t
|
(
|
t
d
/
r
0
|
r
-
|
t
d
/
r
0
|
)
∥
x
∥
A
r
,
i
f
r
>
1
,
|
t
d
r
0
|
>
1
(
0
<
r
<
1
,
|
t
d
r
0
|
<
1
)
,
r
e
s
p
.
)for all x
∈
A.Proof.
The proof follows from Theorem2 by taking
(20)
φ
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
:
=
θ
(
∑
j
=
1
p
‍
∥
x
j
∥
A
r
+
∑
j
=
1
d
‍
∥
y
j
∥
A
r
)
,
ψ
(
x
,
y
,
z
)
:
=
θ
(
∥
x
∥
A
3
r
+
∥
y
∥
A
3
r
+
∥
z
∥
A
3
r
)
for all x
,
y
,
z
,
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A.Recently, the investigation on the hyperstability of the Cauchy and Jensen functional equations has been established [34, 35] and some information on it can be found in [36, 37]. Now, we present the following hyperstability result associated with Lie triple homomorphisms between normed Lie triple systems.From Theorems 2–5 in the paper [34], we can easily derive the following.Theorem 4.
Assume thatf
:
A
→
B satisfies the inequality
(21)
∥
f
(
x
+
y
2
)
-
f
(
x
)
+
f
(
y
)
2
∥
B
≤
θ
∥
x
∥
A
p
∥
y
∥
A
q
,
f
f
f
f
f
f
f
f
f
f
f
x
,
y
∈
A
∖
{
0
}
,
x
+
y
≠
0
,
for some θ
≥
0, and p
,
q
∈
ℝ, p
+
q
≠
0,1. Then
(22)
f
(
x
+
y
2
)
=
f
(
x
)
+
f
(
y
)
2
,
x
,
y
∈
A
∖
{
0
}
,
x
+
y
≠
0
.Corollary 5.
Letf
∈
B
A with f
(
0
)
=
0. Assume that there exist θ
≥
0 and p
,
q
∈
ℝ with p
>
0, p
+
q
≠
0,1 such that
(23)
∥
2
f
(
λ
x
+
λ
y
2
)
-
λ
f
(
x
)
-
λ
f
(
y
)
∥
B
≤
θ
∥
x
∥
A
p
∥
y
∥
A
q
for all λ
∈
S and all x
,
y
∈
A, y
≠
0, and that a function ψ
:
A
3
→
ℝ
+ satisfies the functional inequality (5) and (8) for all x
,
y
,
z
∈
A and some α
∈
ℂ
∖
{
0
}. Then f
∈
ℋ
(
A
,
B
).Proof.
By (23) with λ
=
1, condition (21) holds. Hence Theorem 4 implies that condition (22) is fulfilled. Next, with x
=
0 in (23), we get
(24)
f
(
λ
v
2
)
=
λ
f
(
v
)
2
,
v
∈
A
,
λ
∈
S
,
which implies that
(25)
f
(
2
v
)
=
2
f
(
v
)
,
f
(
λ
v
)
=
λ
f
(
v
)
,
v
∈
A
,
λ
∈
S
.
Hence, by (22),
(26)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
,
x
,
y
∈
A
∖
{
0
}
,
x
+
y
≠
0
,
and next
(27)
f
(
u
)
=
f
(
2
u
-
u
)
=
f
(
2
u
)
+
f
(
-
u
)
=
2
f
(
u
)
+
f
(
-
u
)
,
u
∈
A
∖
{
0
}
,
which yields f
(
-
u
)
=
-
f
(
u
) for u
∈
A. This and (26) show that f is additive. Consequently, (26) and Lemma 1 in [31] jointly with the remarks in the papers [32, 33] imply the ℂ-linearity of f.
Further, it follows from (5) and (8) that, for all x
,
y
,
z
∈
A and all n
∈
ℕ,
(28)
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
f
(
y
)
,
f
(
z
)
]
∥
B
=
1
|
α
|
3
n
∥
f
(
[
α
n
x
,
α
n
y
,
α
n
z
]
)
-
[
f
(
α
n
x
)
,
f
(
α
n
y
)
,
f
(
α
n
z
)
]
∥
B
≤
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
,
which tends to zero as n
→
∞. So, f
∈
ℋ
(
A
,
B
).Corollary 6.
Letf
∈
B
A with f
(
0
)
=
0 and a
∈
ℂ
∖
{
0
}. Assume that there exist θ
≥
0 and p
<
0 such that
(29)
∥
f
(
λ
a
x
+
λ
a
y
)
-
λ
a
f
(
x
)
-
λ
a
f
(
y
)
∥
B
≤
θ
(
∥
x
∥
A
p
+
∥
y
∥
A
p
)
for all λ
∈
S and all x
,
y
∈
A
∖
{
0
} and that a function ψ
:
A
3
→
ℝ
+ satisfies the functional inequality (5) and (8) for all x
,
y
,
z
∈
A. Then f
∈
ℋ
(
A
,
B
).Proof.
It follows from Theorem 2 in [38] (see also, Theorem 1.2 in [35]) and (29) that
(30)
f
(
λ
a
x
+
λ
a
y
)
=
λ
a
f
(
x
)
+
λ
a
f
(
y
)
for all x
,
y
∈
A
∖
{
0
} and all λ
∈
S. Thus,
(31)
0
=
f
(
0
)
=
f
(
a
x
-
a
x
)
=
a
f
(
x
)
+
a
f
(
-
x
)
,
x
∈
A
∖
{
0
}
,
which implies f
(
-
x
)
=
-
f
(
x
) for all x
∈
A. Now, using the equality (30), we get
(32)
f
(
a
2
x
)
=
f
(
a
2
x
-
a
2
y
+
a
2
y
)
=
a
f
(
a
x
-
a
y
)
+
a
f
(
a
y
)
=
a
2
f
(
x
)
+
a
2
f
(
-
y
)
+
a
f
(
a
y
)
=
a
2
f
(
x
)
-
a
2
f
(
y
)
+
a
f
(
a
y
)
for any x
,
y
∈
A
∖
{
0
} with x
≠
y. Thus, there is c
∈
B defined as c
:
=
f
(
a
2
x
)
-
a
2
f
(
x
)
=
a
f
(
a
y
)
-
a
2
f
(
y
) for any x
,
y
∈
A
∖
{
0
}, which yields c
/
a
=
f
(
a
2
x
)
-
a
f
(
a
x
)
,
x
∈
A
∖
{
0
} and c
:
=
f
(
a
2
x
)
-
a
2
f
(
x
)
=
f
(
a
2
x
)
-
a
f
(
a
x
)
+
a
f
(
a
x
)
-
a
2
f
(
x
)
=
c
/
a
+
c for any x
∈
A
∖
{
0
}. Consequently, c
=
0 and so f
(
a
x
)
=
a
f
(
x
) for any x
∈
A. Therefore, we have
(33)
f
(
a
x
+
a
y
)
=
a
f
(
x
)
+
a
f
(
y
)
=
f
(
a
x
)
+
f
(
a
y
)
for all x
,
y
∈
A
∖
{
0
}. Since f
(
0
)
=
0, we conclude that f is additive on A, and so we lead to the ℂ-linearity of f by applying Lemma 1. The remaining proof is the same as the corresponding proof of Corollary 5.Corollary 7.
Letf
∈
B
A with f
(
0
)
=
0 and a
,
b
∈
ℂ
∖
{
0
}. Assume that there exist θ
,
p
,
q
∈
ℝ
+ with p
>
0
,
p
+
q
≠
1 such that
(34)
∥
f
(
λ
a
x
+
λ
b
y
)
-
λ
a
f
(
x
)
-
λ
b
f
(
y
)
∥
B
≤
θ
∥
x
∥
A
p
∥
y
∥
A
q
for all λ
∈
S and all x
,
y
∈
A and that a function ψ
:
A
3
→
ℝ
+ satisfies the functional inequality (5) and (8) for all x
,
y
,
z
∈
A. Then f
∈
ℋ
(
A
,
B
).Proof.
It follows from Theorem 20 in [36] and (34) that
(35)
f
(
λ
a
x
+
λ
b
y
)
=
λ
a
f
(
x
)
+
λ
b
f
(
y
)
for all x
,
y
∈
A and all λ
∈
S. Thus, for all x
,
y
∈
A and all λ
∈
S, one has f
(
λ
a
x
)
=
λ
a
f
(
x
), which implies that f is additive on A, and so we lead to the ℂ-linearity of f by applying Lemma 1. The remaining proof is the same as the corresponding proof of Corollary 5.Next, we consider stability theorem for approximate Lie triple homomorphisms of the functional equationJ
λ
f
=
0 with action of two scalars λ
=
1 and λ
=
i
∈
ℂ.Theorem 8.
Suppose thatf
∈
B
A with f
(
0
)
=
0 satisfies
(36)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
φ
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
,
(37)
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
f
(
y
)
,
f
(
z
)
]
∥
B
≤
ψ
(
x
,
y
,
z
)
for λ
=
1 and λ
=
i
∈
ℂ and for all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
A. If φ
:
A
p
+
d
→
ℝ
+ and ψ
:
A
3
→
ℝ
+ are functions such that
(38)
φ
~
(
x
)
≔
∑
n
=
1
∞
|
α
|
n
φ
(
0
,
…
,
0
︷
p
,
x
α
n
,
…
,
x
α
n
︷
d
)
<
∞
,
(
φ
~
(
x
)
≔
∑
n
=
0
∞
‍
1
|
α
|
n
φ
(
0
,
…
,
0
︷
p
,
α
n
x
,
…
,
α
n
x
︷
d
)
<
∞
,
r
e
s
p
.
)
,
lim
⁡
n
→
∞
|
α
|
n
φ
(
x
1
α
n
,
…
,
x
p
α
n
,
y
1
α
n
,
…
,
y
d
α
n
)
=
0
,
(
lim
⁡
n
→
∞
1
|
α
|
n
φ
(
α
n
x
1
,
…
,
α
n
x
p
,
α
n
y
1
,
…
,
α
n
y
d
)
=
0
,
r
e
s
p
.
)
,
lim
⁡
n
→
∞
|
α
|
3
n
ψ
(
x
α
n
,
y
α
n
,
z
α
n
)
=
0
,
(
lim
⁡
n
→
∞
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
=
0
,
r
e
s
p
.
)
for all x
,
y
,
z
,
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
A, where α
=
t
d
/
r
0, and the mapping t
→
f
(
t
x
) is continuous on ℝ for each fixed x
∈
A, then there exists a unique h
∈
ℋ
(
A
,
B
) such that
(39)
∥
f
(
x
)
-
h
(
x
)
∥
B
≤
1
|
α
|
|
r
0
|
φ
~
(
x
)
=
1
|
t
|
d
φ
~
(
x
)
for all x
∈
A.Proof.
Applying the same argument as in the proof of Theorem2, one can deduce the existence of a unique additive mapping h
:
A
→
B given by
(40)
h
(
x
)
:
=
lim
⁡
n
→
∞
α
n
f
(
x
α
n
)
satisfying the required inequality (39). By the same reasoning as in the proof of the theorem of [4], the additive mapping h is ℝ-linear.
Lettingλ
=
i, x
1
=
⋯
=
x
p
=
0, and y
1
=
⋯
=
y
d
=
x in (36), we have
(41)
∥
f
(
i
x
)
-
i
α
f
(
x
α
)
∥
B
≤
φ
(
0
,
…
,
0
,
x
/
α
,
…
,
x
/
α
)
|
r
0
|
for all x
∈
A. Then, it follows from the last inequality that
(42)
∥
α
n
f
(
i
x
α
n
)
-
i
α
n
+
1
f
(
x
α
n
+
1
)
∥
B
≤
|
α
|
n
+
1
φ
(
0
,
…
,
0
,
x
/
α
n
+
1
,
…
,
x
/
α
n
+
1
)
|
α
|
|
r
0
|
for all n
∈
ℕ and all x
∈
A. The right hand side tends to zero as n
→
∞, and so we obtain
(43)
h
(
i
x
)
=
lim
⁡
n
→
∞
α
n
f
(
i
x
α
n
)
=
lim
⁡
n
→
∞
i
α
n
+
1
f
(
x
α
n
+
1
)
=
i
h
(
x
)
for all x
∈
A. Therefore, for each μ
=
λ
1
+
i
λ
2
∈
ℂ
(
λ
1
,
λ
2
∈
ℝ
),
(44)
h
(
μ
x
)
=
h
(
λ
1
x
+
i
λ
2
x
)
=
λ
1
h
(
x
)
+
λ
2
h
(
i
x
)
=
λ
1
h
(
x
)
+
i
λ
2
h
(
x
)
=
(
λ
1
+
i
λ
2
)
h
(
x
)
=
μ
h
(
x
)
for all x
∈
A, so that h
∈
ℒ
(
A
,
B
). It follows from (37) that
(45)
∥
h
(
[
x
,
y
,
z
]
)
-
[
h
(
x
)
,
h
(
y
)
,
h
(
z
)
]
∥
B
=
lim
⁡
n
→
∞
|
α
|
3
n
∥
f
(
[
x
α
n
,
y
α
n
,
z
α
n
]
)
-
[
f
(
x
α
n
)
,
f
(
y
α
n
)
,
f
(
z
α
n
)
]
∥
B
≤
lim
⁡
n
→
∞
|
α
|
3
n
ψ
(
x
α
n
,
y
α
n
,
z
α
n
)
=
0
for all x
,
y
,
z
∈
A. So, h
∈
ℋ
(
A
,
B
).
The remainder of proof is similarly verified by the corresponding proof of Theorem2.Corollary 9.
Letθ be a positive real number, let |
t
d
/
r
0
|
≠
1, and let r be a real number such that either 0
<
r
<
1 or r
>
1. Suppose that f
∈
B
A with f
(
0
)
=
0 satisfies
(46)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
θ
(
∑
j
=
1
p
∥
x
j
∥
A
r
+
∑
j
=
1
d
∥
y
j
∥
A
r
)
,
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
f
(
y
)
,
f
(
z
)
]
∥
B
≤
θ
(
∥
x
∥
A
3
r
+
∥
y
∥
A
3
r
+
∥
z
∥
A
3
r
)
,
for all λ
=
1 and λ
=
i
∈
ℂ and for all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
A. If the mapping t
→
f
(
t
x
) is continuous on ℝ for each fixed x
∈
A, then there exists a unique h
∈
ℋ
(
A
,
B
) such that(47)
∥
f
(
x
)
-
h
(
x
)
∥
B
≤
{
|
t
d
/
r
0
|
θ
|
t
|
(
|
t
d
/
r
0
|
r
-
|
t
d
/
r
0
|
)
∥
x
∥
A
r
,
i
f
r
>
1
,
|
t
d
r
0
|
>
1
(
0
<
r
<
1
,
|
t
d
r
0
|
<
1
)
(
|
t
d
/
r
0
|
θ
|
t
|
(
|
t
d
/
r
0
|
-
|
t
d
/
r
0
|
r
)
∥
x
∥
A
r
,
i
f
0
<
r
<
1
,
|
t
d
r
0
|
>
1
(
r
>
1
,
|
t
d
r
0
|
<
1
)
,
r
e
s
p
.
)for all x
∈
A.


## 3. Stability of Derivations on Normed Lie Triple Systems

In this section, we prove the stability of derivations on normed Lie triple systems associated with the Cauchy-Jensen additive equation. Now, we consider stability theorem for approximate Lie triple derivations of the functional equationJ
λ
f
=
0 with action of λ
∈
S.Theorem 10.
Suppose thatf
∈
B
B with f
(
0
)
=
0 satisfies
(48)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
φ
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
,
(49)
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
y
,
z
]
-
[
x
,
f
(
y
)
,
z
]
-
[
x
,
y
,
f
(
z
)
]
∥
B
≤
ψ
(
x
,
y
,
z
)
for all λ
∈
S and all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
B. If φ
:
B
p
+
d
→
ℝ
+ and ψ
:
B
3
→
ℝ
+ are functions such that
(50)
φ
~
(
x
)
:
=
∑
n
=
0
∞
1
|
α
|
n
φ
(
α
n
x
,
…
,
α
n
x
)
<
∞
,
(
φ
~
(
x
)
:
=
∑
n
=
1
∞
|
α
|
n
φ
(
x
α
n
,
…
,
x
α
n
)
<
∞
,
r
e
s
p
.
)
,
(51)
lim
⁡
n
→
∞
1
|
α
|
n
φ
(
α
n
x
1
,
…
,
α
n
x
p
,
α
n
y
1
,
…
,
α
n
y
d
)
=
0
,
(
lim
⁡
n
→
∞
|
α
|
n
φ
(
x
1
α
n
,
…
,
x
p
α
n
,
y
1
α
n
,
…
,
y
d
α
n
)
=
0
,
r
e
s
p
.
)
,
(52)
lim
⁡
n
→
∞
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
=
0
,
(
lim
⁡
n
→
∞
|
α
|
3
n
ψ
(
x
α
n
,
y
α
n
,
z
α
n
)
=
0
,
r
e
s
p
.
)
for all x
,
y
,
z
,
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
B, where α
=
(
s
p
+
t
d
)
/
r
0, then there exists a unique D
∈
𝒟
(
B
,
B
), defined as D
(
x
)
=
lim
⁡
n
→
∞
⁡
(
f
(
α
n
x
)
/
α
n
)
(
D
(
x
)
=
lim
⁡
n
→
∞
⁡
α
n
f
(
x
/
α
n
)
,
r
e
s
p
.
), such that
(53)
∥
f
(
x
)
-
D
(
x
)
∥
B
≤
1
|
α
|
|
r
0
|
φ
~
(
x
)
=
1
|
s
p
+
t
d
|
φ
~
(
x
)
for all x
∈
B.Proof.
Puttingλ
=
1 and x
1
=
⋯
=
x
p
=
y
1
,
⋯
=
y
d
=
x in (48), we obtain
(54)
∥
r
0
f
(
s
p
+
t
d
r
0
x
)
-
(
s
p
+
t
d
)
f
(
x
)
∥
b
≤
φ
(
x
,
…
,
x
)
for all x
∈
B. Let α
:
=
(
s
p
+
t
d
)
/
r
0; therefore
(55)
∥
f
(
x
)
-
f
(
α
x
)
α
∥
B
≤
φ
(
x
,
…
,
x
)
|
α
|
|
r
0
|
=
1
|
s
p
+
t
d
|
φ
(
x
,
…
,
x
)
for all x
∈
B. Replacing x by α
n
x in (55) and dividing both sides of (55) by |
α
|
n, we get
(56)
∥
f
(
α
n
+
1
x
)
α
n
+
1
-
f
(
α
n
x
)
α
n
∥
B
≤
φ
(
α
n
x
,
…
,
α
n
x
)
|
α
|
n
+
1
|
r
0
|
for all x
∈
B and all nonnegative integers n. Thus, one has
(57)
∥
f
(
α
n
x
)
α
n
-
f
(
α
m
x
)
α
m
∥
B
≤
∑
k
=
m
n
-
1
‍
φ
(
α
k
x
,
…
,
α
k
x
)
|
α
|
k
+
1
|
r
0
|
for all nonnegative integers n
>
m and x
∈
B. It follows from the convergence of the series (50) that the sequence {
f
(
α
n
x
)
/
α
n
} is a Cauchy sequence. From the completeness of B, this sequence converges in B. So we can define a mapping D
:
B
→
B by D
(
x
)
:
=
lim
⁡
n
→
∞
⁡
(
f
(
α
n
x
)
/
α
n
) for all x
∈
B.
Now, by consideringm
=
0 and taking the limit as n
→
∞ in (57), we obtain the estimation (53). It follows from (49) that
(58)
∥
D
(
[
x
,
y
,
z
]
)
-
[
D
(
x
)
,
y
,
z
]
-
[
x
,
D
(
y
)
,
z
]
-
[
x
,
y
,
D
(
z
)
]
∥
B
=
lim
⁡
n
→
∞
1
|
α
|
3
n
∥
f
(
α
3
n
[
x
,
y
,
z
]
)
-
[
f
(
α
n
x
)
,
α
n
y
,
α
n
z
]
nnnnnnnnnnnn
-
[
α
n
x
,
f
(
α
n
y
)
,
α
n
z
]
nnnnnnnnnnnn
-
[
α
n
x
,
α
n
y
,
f
(
α
n
z
)
]
f
(
α
3
n
[
x
,
y
,
z
]
)
∥
B
≤
lim
⁡
n
→
∞
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
=
0
for all x
,
y
,
z
∈
B. So, D
∈
𝒟
(
B
,
B
).
The remaining proof is similar to that of Theorem2.Corollary 11.
Assume that there exist real numbersθ
≥
0, θ
′
≥
0 and that r
′, s
′, and t
′ are positive real numbers such that r
′
+
s
′
+
t
′
≠
3. Let |
(
s
p
+
t
d
)
/
r
0
|
≠
1 and r
+
s
≠
1. Suppose f
∈
B
B with f
(
0
)
=
0 satisfies
(59)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
θ
∏
i
=
1
p
∥
x
i
∥
B
r
i
∏
j
=
1
d
∥
y
j
∥
B
s
j
,
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
y
,
z
]
-
[
x
,
f
(
y
)
,
z
]
-
[
x
,
y
,
f
(
z
)
]
∥
B
≤
θ
′
∥
x
∥
B
r
′
∥
y
∥
B
s
′
∥
z
∥
B
t
′
for all λ
∈
S and all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
B. Then there exists a unique D
∈
𝒟
(
B
,
B
) such that(60)
∥
f
(
x
)
-
D
(
x
)
∥
B
≤
{
θ
∥
x
∥
B
r
+
s
|
r
0
|
(
|
(
s
p
+
t
d
)
/
r
0
|
-
|
(
s
p
+
t
d
)
/
r
0
|
r
+
s
)
,
i
f
|
s
p
+
t
d
r
0
|
<
1
,
r
+
s
>
1
,
r
′
+
s
′
+
t
′
>
3
(
|
s
p
+
t
d
r
0
|
>
1
,
r
+
s
<
1
,
r
′
+
s
′
+
t
′
<
3
)
(
θ
∥
x
∥
B
r
+
s
|
r
0
|
(
|
(
s
p
+
t
d
)
/
r
0
|
r
+
s
-
|
(
s
p
+
t
d
)
/
r
0
|
)
,
i
f
|
s
p
+
t
d
r
0
|
<
1
,
r
+
s
<
1
,
r
′
+
s
′
+
t
′
<
3
(
|
s
p
+
t
d
r
0
|
>
1
,
r
+
s
>
1
,
r
′
+
s
′
+
t
′
>
3
)
,
r
e
s
p
.
)for all x
∈
B, where r
:
=
∑
i
=
1
p
r
i, s
:
=
∑
j
=
1
d
s
j.Now, applying the above main theorem, we present the following hyperstability result associated with Lie triple derivations on Banach Lie triple systems.Corollary 12.
Letf
∈
B
B with f
(
0
)
=
0. Assume that there exist θ
≥
0 and p
,
q
∈
ℝ with p
>
0, p
+
q
≠
0,1 such that
(61)
∥
2
f
(
λ
x
+
λ
y
2
)
-
λ
f
(
x
)
-
λ
f
(
y
)
∥
B
≤
θ
∥
x
∥
A
p
∥
y
∥
A
q
for all λ
∈
S and all x
,
y
∈
B
,
y
≠
0 and that a function ψ
:
A
3
→
ℝ
+ satisfies the functional inequality (49) and (52) for all x
,
y
,
z
∈
A and some α
∈
ℂ
∖
{
0
}. Then f
∈
𝒟
(
B
,
B
).Proof.
By the same reason as in the proof of Corollary5, we lead to the ℂ-linearity of f. Further, it follows from (49) and (52) that for all x
,
y
,
z
∈
B and all n
∈
ℕ
(62)
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
y
,
z
]
-
[
x
,
f
(
y
)
,
z
]
-
[
x
,
y
,
f
(
z
)
]
∥
B
=
1
|
α
|
3
n
∥
f
(
α
3
n
[
x
,
y
,
z
]
)
-
[
f
(
α
n
x
)
,
α
n
y
,
α
n
z
]
nnnnnnnnn
-
[
α
n
x
,
f
(
α
n
y
)
,
α
n
z
]
-
[
α
n
x
,
α
n
y
,
f
(
α
n
z
)
]
f
(
α
3
n
[
x
,
y
,
z
]
)
∥
B
≤
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
which tends to zero as n
→
∞. So, f
∈
𝒟
(
B
,
B
).Corollary 13.
Letf
∈
B
B with f
(
0
)
=
0 and a
∈
ℂ
∖
{
0
}. Assume that there exist θ
≥
0 and p
<
0 such that
(63)
∥
f
(
λ
a
x
+
λ
a
y
)
-
λ
a
f
(
x
)
-
λ
a
f
(
y
)
∥
B
≤
θ
(
∥
x
∥
B
p
+
∥
y
∥
B
p
)
for all λ
∈
S and all x
,
y
∈
B
∖
{
0
} and that a function ψ
:
B
3
→
ℝ
+ satisfies the functional inequality (49) and (52) for all x
,
y
,
z
∈
A. Then f
∈
𝒟
(
B
,
B
).Corollary 14.
Letf
∈
B
B with f
(
0
)
=
0 and a
,
b
∈
ℂ
∖
{
0
}. Assume that there exist θ
,
p
,
q
∈
ℝ
+ with p
>
0
,
p
+
q
≠
1 such that
(64)
∥
f
(
λ
a
x
+
λ
b
y
)
-
λ
a
f
(
x
)
-
λ
b
f
(
y
)
∥
B
≤
θ
∥
x
∥
A
p
∥
y
∥
A
q
for all λ
∈
S and all x
,
y
∈
B and that a function ψ
:
B
3
→
ℝ
+ satisfies the functional inequality (49) and (52) for all x
,
y
,
z
∈
A. Then f
∈
𝒟
(
B
,
B
).Next, we consider stability theorem for approximate Lie triple derivation of the functional equationJ
λ
f
=
0 with action of λ
=
1 and λ
=
i
∈
ℂ.Theorem 15.
Suppose thatf
∈
B
B with f
(
0
)
=
0 satisfies
(65)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
φ
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
,
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
y
,
z
]
-
[
x
,
f
(
y
)
,
z
]
-
[
x
,
y
,
f
(
z
)
]
∥
B
≤
ψ
(
x
,
y
,
z
)
for λ
=
1 and λ
=
i
∈
ℂ and for all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
B. If φ
:
B
p
+
d
→
ℝ
+ and ψ
:
B
3
→
ℝ
+ are functions such that
(66)
φ
~
(
x
)
≔
∑
n
=
1
∞
‍
|
α
|
n
φ
(
x
α
n
,
…
,
x
α
n
)
<
∞
,
(
φ
~
(
x
)
:
=
∑
n
=
0
∞
‍
1
|
α
|
n
φ
(
α
n
x
,
…
,
α
n
x
)
<
∞
,
r
e
s
p
.
)
,
lim
⁡
n
→
∞
|
α
|
n
φ
(
x
1
α
n
,
…
,
x
p
α
n
,
y
1
α
n
,
…
,
y
d
α
n
)
=
0
,
(
lim
⁡
n
→
∞
1
|
α
|
n
φ
(
α
n
x
1
,
…
,
α
n
x
p
,
α
n
y
1
,
…
,
α
n
y
d
)
=
0
,
r
e
s
p
.
)
,
lim
⁡
n
→
∞
|
α
|
3
n
ψ
(
x
α
n
,
y
α
n
,
z
α
n
)
=
0
,
(
lim
⁡
n
→
∞
1
|
α
|
3
n
ψ
(
α
n
x
,
α
n
y
,
α
n
z
)
=
0
,
r
e
s
p
.
)
for all x
,
y
,
z
,
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
∈
B, where α
=
(
s
p
+
t
d
)
/
r
0, and the mapping t
→
f
(
t
x
) is continuous on ℝ for each fixed x
∈
B, then there exists a unique D
∈
𝒟
(
B
,
B
), defined as D
(
x
)
=
lim
⁡
n
→
∞
⁡
α
n
f
(
x
/
α
n
)
(
D
(
x
)
=
lim
⁡
n
→
∞
⁡
(
f
(
α
n
x
)
/
α
n
)
,
resp
.
), such that
(67)
∥
f
(
x
)
-
D
(
x
)
∥
B
≤
1
|
s
p
+
t
d
|
φ
~
(
x
)
for all x
∈
B.Proof.
The proof is similar to that of Theorem10.Corollary 16.
Assume that there exist real numbersθ
≥
0, θ
′
≥
0 and that r
′, s
′, and t
′ are positive real numbers such that r
′
+
s
′
+
t
′
≠
3. Let |
(
s
p
+
t
d
)
/
r
0
|
≠
1 and r
+
s
≠
1. Suppose that f
∈
B
B with f
(
0
)
=
0 satisfies(68)
∥
J
λ
f
(
x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
)
∥
B
≤
θ
∏
i
=
1
p
∥
x
i
∥
B
r
i
∏
j
=
1
d
∥
y
j
∥
B
s
j
,
∥
f
(
[
x
,
y
,
z
]
)
-
[
f
(
x
)
,
y
,
z
]
-
[
x
,
f
(
y
)
,
z
]
-
[
x
,
y
,
f
(
z
)
]
∥
B
≤
θ
∥
x
∥
B
r
′
∥
y
∥
B
s
′
∥
z
∥
B
t
′
for λ
=
1 and λ
=
i
∈
ℂ and for all x
1
,
…
,
x
p
,
y
1
,
…
,
y
d
,
x
,
y
,
z
∈
B. If the mapping t
→
f
(
t
x
) is continuous on ℝ for each fixed x
∈
B, then there exists a unique D
∈
𝒟
(
B
,
B
) such that(69)
∥
f
(
x
)
-
D
(
x
)
∥
B
≤
{
θ
∥
x
∥
B
r
+
s
|
r
0
|
(
|
(
s
p
+
t
d
)
/
r
0
|
r
+
s
-
|
(
s
p
+
t
d
)
/
r
0
|
)
,
i
f
|
s
p
+
t
d
r
0
|
<
1
,
r
+
s
<
1
,
r
′
+
s
′
+
t
′
<
3
(
|
s
p
+
t
d
r
0
|
>
1
,
r
+
s
>
1
,
r
′
+
s
′
+
t
′
>
3
)
(
θ
∥
x
∥
B
r
+
s
|
r
0
|
(
|
(
s
p
+
t
d
)
/
r
0
|
-
|
(
s
p
+
t
d
)
/
r
0
|
r
+
s
)
,
i
f
|
s
p
+
t
d
r
0
|
<
1
,
r
+
s
>
1
,
r
′
+
s
′
+
t
′
>
3
(
|
s
p
+
t
d
r
0
|
>
1
,
r
+
s
<
1
,
r
′
+
s
′
+
t
′
<
3
)
,
r
e
s
p
.
)for all x
∈
B, where r
:
=
∑
i
=
1
p
r
i, s
:
=
∑
j
=
1
d
s
j.



---

*Source: 754291-2014-05-07.xml*