Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 926390, 23 pages
doi:10.1155/2012/926390
Research Article
On the Hyers-Ulam Stability of a
General Mixed Additive and Cubic Functional
Equation in n-Banach Spaces
Tian Zhou Xu1 and John Michael Rassias2
1
2
School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Pedagogical Department E.E., Section of Mathematics and Informatics, National and Kapodistrian
University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece
Correspondence should be addressed to Tian Zhou Xu, xutianzhou@bit.edu.cn
Received 7 January 2012; Accepted 15 February 2012
Academic Editor: Krzysztof Cieplinski
Copyright q 2012 T. Z. Xu and J. M. Rassias. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The objective of the present paper is to determine the generalized Hyers-Ulam stability of the
mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition,
we show under some suitable conditions that an approximately mixed additive-cubic function can
be approximated by a mixed additive and cubic mapping.
1. Introduction and Preliminaries
A basic question in the theory of functional equations is as follows: when is it true that
a function, which approximately satisfies a functional equation, must be close to an exact
solution of the equation?
If the problem accepts a unique solution, we say the equation is stable see 1. The
study of stability problems for functional equations is related to a question of Ulam 2
concerning the stability of group homomorphisms and affirmatively answered for Banach
spaces by Hyers 3. The result of Hyers was generalized by Aoki 4 for approximate
additive mappings and by Rassias 5 for approximate linear mappings by allowing the
Cauchy difference operator CDfx, y fx y − fx fy to be controlled by
xp yp . In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa 6,
who replaced xp yp by a general control function ϕx, y. On the other hand, several
further interesting discussions, modifications, extensions, and generalizations of the original
problem of Ulam have been proposed see, e.g. 7–12 and the references therein.
Recently, Park 9 investigated the approximate additive mappings, approximate
Jensen mappings, and approximate quadratic mappings in 2-Banach spaces and proved the
2
Abstract and Applied Analysis
generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional
equation, and the quadratic functional equation in 2-Banach spaces. This is the first result for
the stability problem of functional equations in 2-Banach spaces.
In 11, 12, we introduced the following mixed additive-cubic functional equation for
fixed integers k with k / 0, ±1:
f kx y f kx − y kf x y kf x − y 2fkx − 2kfx,
1.1
with f0 0, and investigated the generalized Hyers-Ulam stability of 1.1 in quasi-Banach
spaces and non-Archimedean fuzzy normed spaces, respectively.
In this paper, we investigate, approximate mixed additive-cubic mappings in n-Banach
spaces. That is, we prove the generalized Hyers-Ulam stability of a general mixed additivecubic equation 1.1 in n-Banach spaces by the direct method.
The concept of 2-normed spaces was initially developed by Gähler 13, 14 in the
middle of 1960s, while that of n-normed spaces can be found in 15, 16. Since then, many
others have studied this concept and obtained various results; see for instance 15, 17–19.
We recall some basic facts concerning n-normed spaces and some preliminary results.
Definition 1.1. Let n ∈ N, and let X be a real linear space with dim X ≥ n and ·, . . . , · : X n →
R a function satisfying the following properties:
N1 x1 , x2 , . . . , xn 0 if and only if x1 , x2 , . . . , xn are linearly dependent,
N2 x1 , x2 , . . . , xn is invariant under permutation,
N3 αx1 , x2 , . . . , xn |α|x1 , x2 , . . . , xn ,
N4 x y, x2 , . . . , xn ≤ x, x2 , . . . , xn y, x2 , . . . , xn for all α ∈ R and x, y, x1 , x2 , . . . , xn ∈ X. Then the function ·, . . . , · is called an n-norm on X
and the pair X, ·, . . . , · is called an n-normed space.
Example 1.2. For x1 , x2 , . . . , xn ∈ Rn , the Euclidean n-norm x1 , x2 , . . . , xn E is defined by
⎞
⎛
x11 · · · x1n ⎜
⎟
⎜
.
.
.
. . .. ⎟
x1 , x2 , . . . , xn E det xij abs ⎜ ..
⎟,
⎠
⎝
xn1 · · · xnn 1.2
where xi xi1 , . . . , xin ∈ Rn for each i 1, 2, . . . , n.
Example 1.3. The standard n-norm on X, a real inner product space of dimension dim X ≥ n,
is as follows:
x1 , x1 · · · x1 , xn 1/2
,
.
.
..
..
..
x1 , x2 , . . . , xn S .
xn , x1 · · · xn , xn 1.3
where ·, · denotes the inner product on X. If X Rn , then this n-norm is exactly the same as
the Euclidean n-norm x1 , x2 , . . . , xn E mentioned earlier. For n 1, this n-norm is the usual
norm x1 x1 , x1 1/2 .
Abstract and Applied Analysis
3
Definition 1.4. A sequence {xk } in an n-normed space X is said to converge to some x ∈ X in
the n-norm if
lim xk − x, y2 , . . . , yn 0,
k→∞
1.4
for every y2 , . . . , yn ∈ X.
Definition 1.5. A sequence {xk } in an n-normed space X is said to be a Cauchy sequence with
respect to the n-norm if
lim xk − xl , y2 , . . . , yn 0,
k,l → ∞
1.5
for every y2 , . . . , yn ∈ X. If every Cauchy sequence in X converges to some x ∈ X, then X is
said to be complete with respect to the n-norm. Any complete n-normed space is said to be
an n-Banach space.
Now we state the following results as lemma see 9 for the details.
Lemma 1.6. Let X be an n-normed space. Then,
1 For xi ∈ Xi 1, . . . , n and γ, a real number,
x1 , . . . , xi , . . . , xj , . . . , xn x1 , . . . , xi , . . . , xj γxi , . . . , xn 1.6
for all 1 ≤ i / j ≤ n,
2 |x, y2 , . . . , yn − y, y2 , . . . , yn | ≤ x − y, y2 , . . . , yn for all x, y, y2 , . . . , yn ∈ X,
3 if x, y2 , . . . , yn 0 for all y2 , . . . , yn ∈ X, then x 0,
4 for a convergent sequence {xj } in X,
lim xj , y2 , . . . , yn lim xj , y2 , . . . , yn j →∞
j →∞
1.7
for all y2 , . . . , yn ∈ X.
2. Approximate Mixed Additive-Cubic Mappings
In this section, we investigate the generalized Hyers-Ulam stability of the generalized mixed
additive-cubic functional equation in n-Banach spaces. Let X be a linear space and Y an nBanach space. For convenience, we use the following abbreviation for a given mapping f :
X → Y:
Df x, y : f kx y f kx − y − kf x y − kf x − y − 2fkx 2kfx
for all x, y ∈ X.
2.1
4
Abstract and Applied Analysis
Theorem 2.1. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
∞
1 j
ϕ 2 x, 2j y, u2 , . . . , un < ∞,
j
j0 2
2.2
Dfx, y, u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.3
for all x, y, u2 , . . . , un ∈ X. Then, there is a unique additive mapping A : X → Y such that
∞
1 j
f2x − 8fx − Ax, u2 , . . . , un ≤
ϕ
2
x,
u
,
.
.
.
,
u
2
n
Y
j1
j0 2
2.4
for all x, u2 , . . . , un ∈ X, where
ϕx,
u2 , . . . , un 1
: 3
|k| 1 ϕx, 2k 1x, u2 , . . . , un ϕx, 2k − 1x, u2 , . . . , un |k − k|
ϕ3x, x, u2 , . . . , un 8k2 1 ϕx, x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un ϕx, kx, u2 , . . . , un k2 ϕ2x, 2x, u2 , . . . , un ϕ2x, 2kx, u2 , . . . , un 2ϕx, k 1x, u2 , . . . , un 2ϕx, k − 1x, u2 , . . . , un 2ϕ2x, x, u2 , . . . , un x kx
,
, u2 , . . . , un
2ϕ2x, kx, u2 , . . . , un 8ϕ
2 2
x 2k − 1x
x 2k 1x
8|k|ϕ
,
, u2 , . . . , un 8|k|ϕ
,
, u2 , . . . , un
2
2
2
2
x 3kx
|k| 1
8ϕ
,
, u2 , . . . , un ϕ0, k 1x, u2 , . . . , un 2 2
|k − 1|
8k2 1
2
ϕ0, k − 1x, u2 , . . . , un ϕ0, x, u2 , . . . , un |k − 1|
|k − 1|
k2
|k|
ϕ0, 3k − 1x, u2 , . . . , un ϕ0, 2k − 1x, u2 , . . . , un |k − 1|
|k − 1|
k2 |k| − 1
ϕ0, 2kx, u2 , . . . , un |k − 1|
8|k|
8|k|
3k − 1x
k 1x
, u2 , . . . , un , u2 , . . . , un
ϕ 0,
ϕ 0,
2
2
|k − 1|
|k − 1|
8k2 2|k| − 8
ϕ0, kx, u2 , . . . , un .
|k − 1|
2.5
Abstract and Applied Analysis
5
Proof. Letting x 0 in 2.3, we get
fy f−y, u2 , . . . , un ≤
Y
1
ϕ 0, y, u2 , . . . , un
|k − 1|
2.6
for all y, u2 , . . . , un ∈ X. Putting y x in 2.3, we have
fk 1x fk − 1x − kf2x − 2fkx 2kfx, u2 , . . . , un ≤ ϕx, x, u2 , . . . , un Y
2.7
for all x, u2 , . . . , un ∈ X. Thus
f2k 1x f2k − 1x − kf4x − 2f2kx 2kf2x, u2 , . . . , un Y
≤ ϕ2x, 2x, u2 , . . . , un 2.8
for all x, u2 , . . . , un ∈ X. Letting y kx in 2.3, we get
f2kx − kfk 1x − kf−k − 1x − 2fkx 2kfx, u2 , . . . , un ≤ ϕx, kx, u2 , . . . , un Y
2.9
for all x, u2 , . . . , un ∈ X. Letting y k 1x in 2.3, we have
f2k 1x f−x − kfk 2x − kf−kx − 2fkx 2kfx, u2 , . . . , un Y
≤ ϕx, k 1x, u2 , . . . , un 2.10
for all x, u2 , . . . , un ∈ X. Letting y k − 1x in 2.3, we have
f2k − 1x − k 2fkx − kf−k − 2x 2k 1fx, u2 , . . . , un Y
≤ ϕx, k − 1x, u2 , . . . , un 2.11
for all x, u2 , . . . , un ∈ X. Replacing x and y by 2x and x in 2.3, respectively, we get
f2k 1x f2k − 1x − 2f2kx − kf3x 2kf2x − kfx, u2 , . . . , un Y
≤ ϕ2x, x, u2 , . . . , un 2.12
for all x, u2 , . . . , un ∈ X. Replacing x and y by 3x and x in 2.3, respectively, we get
f3k 1x f3k − 1x − 2f3kx − kf4x − kf2x 2kf3x, u2 , . . . , un Y
≤ ϕ3x, x, u2 , . . . , un 2.13
6
Abstract and Applied Analysis
for all x, u2 , . . . , un ∈ X. Replacing x and y by 2x and kx in 2.3, respectively, we have
f3kx fkx − kfk 2x − kf−k − 2x − 2f2kx 2kf2x, u2 , . . . , un Y
≤ ϕ2x, kx, u2 , . . . , un 2.14
for all x, u2 , . . . , un ∈ X. Setting y 2k 1x in 2.3, we have
f3k 1x f−k 1x − kf2k 1x − kf−2kx − 2fkx 2kfx, u2 , . . . , un Y
≤ ϕx, 2k 1x, u2 , . . . , un 2.15
for all x, u2 , . . . , un ∈ X. Letting y 2k − 1x in 2.3, we have
f3k − 1x f−k − 1x − kf−2k − 1x − kf2kx − 2fkx 2kfx, u2 , . . . , un Y
≤ ϕx, 2k − 1x, u2 , . . . , un 2.16
for all x, u2 , . . . , un ∈ X. Letting y 3kx in 2.3, we have
f4kx f−2kx − kf3k 1x − kf−3k − 1x − 2fkx 2kfx, u2 , . . . , un Y
≤ ϕx, 3kx, u2 , . . . , un 2.17
for all x, u2 , . . . , un ∈ X. By 2.6, 2.7, 2.13, 2.15, and 2.16, we get
kf2k 1xkf−2k − 1x6fkx − 2f3kx − kf4x2kf3x − 6kfx, u2 , . . . , un Y
≤ ϕx, 2k 1x, u2 , . . . , un ϕx, 2k − 1x, u2 , . . . , un ϕ3x, x, u2 , . . . , un ϕx, x, u2 , . . . , un 1
ϕ0, k 1x, u2 , . . . , un |k − 1|
1
|k|
ϕ0, k − 1x, u2 , . . . , un ϕ0, 2kx, u2 , . . . , un |k − 1|
|k − 1|
2.18
for all x, u2 , . . . , un ∈ X. By 2.6, 2.10, and 2.11, we have
f2k 1x f2k − 1x − kfk 2x − kf−k − 2x − 4fkx 4kfx, u2 , . . . , un Y
≤ ϕx, k 1x, u2 , . . . , un ϕx, k − 1x, u2 , . . . , un k ϕ0, kx, u2 , . . . , un k − 1
1
ϕ0, x, u2 , . . . , un |k − 1|
2.19
Abstract and Applied Analysis
7
for all x, u2 , . . . , un ∈ X. It follows from 2.12 and 2.19 that
kfk 2x kf−k − 2x − 2f2kx 4fkx − kf3x 2kf2x − 5kfx, u2 , . . . , un Y
≤ ϕx, k 1x, u2 , . . . , un ϕx, k − 1x, u2 , . . . , un ϕ2x, x, u2 , . . . , un k 1
ϕ0, kx, u2 , . . . , un ϕ0, x, u2 , . . . , un k − 1
|k − 1|
2.20
for all x, u2 , . . . , un ∈ X. By 2.14 and 2.20, we have
f3kx − 4f2kx 5fkx − kf3x 4kf2x − 5kfx, u2 , . . . , un Y
≤ ϕx, k 1x, u2 , . . . , un ϕx, k − 1x, u2 , . . . , un ϕ2x, x, u2 , . . . , un k 1
ϕ0, kx, u2 , . . . , un ϕ0, x, u2 , . . . , un ϕ2x, kx, u2 , . . . , un k − 1
|k − 1|
2.21
for all x, u2 , . . . , un ∈ X. By 2.6, 2.15, 2.16, and 2.17, we have
kf−k 1x − kf−k − 1x − k2 f2k 1x k2 f−2k − 1x
k2 f2kx − k2 − 1 f−2kx f4kx − 2fkx 2kfx, u2 , . . . , un Y
≤ |k|ϕx, 2k 1x, u2 , . . . , un |k|ϕx, 2k − 1x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un k ϕ0, 3k − 1x, u2 , . . . , un k − 1
2.22
for all x, u2 , . . . , un ∈ X. It follows from 2.6, 2.8, 2.9, and 2.22 that
f4kx − 2f2kx − k3 f4x 2k3 f2x, u2 , . . . , un Y
≤ |k|ϕx, 2k 1x, u2 , . . . , un |k|ϕx, 2k − 1x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un k ϕ0, 3k − 1x, u2 , . . . , un ϕx, kx, u2 , . . . , un k2 ϕ2x, 2x, u2 , . . . , un k − 1
2.23
2
k ϕ0, k 1x, u2 , . . . , un k ϕ0, 2k − 1x, u2 , . . . , un k−1
|k − 1|
k2 − 1
ϕ0, 2kx, u2 , . . . , un |k − 1|
8
Abstract and Applied Analysis
for all x, u2 , . . . , un ∈ X. Hence,
f2kx − 2fkx − k 3 f2x 2k3 fx, u2 , . . . , un Y
x 2k 1x
x 2k − 1x
x 3kx
,
,
u
,
,
u
,
,
u
≤ |k|ϕ
2 , . . . , un |k|ϕ
2 , . . . , un ϕ
2 , . . . , un
2
2
2
2
2 2
k x kx
ϕ 0, 3k − 1x , u2 , . . . , un
ϕ
,
, u2 , . . . , un k2 ϕx, x, u2 , . . . , un 2 2
k − 1
2
2
k ϕ 0, k 1x , u2 , . . . , un k ϕ0, k − 1x, u2 , . . . , un k − 1
2
|k − 1|
k2 − 1
ϕ0, kx, u2 , . . . , un |k − 1|
2.24
for all x, u2 , . . . , un ∈ X. By 2.9, we have
f4kx − kf2k 1x − kf−2k − 1x − 2f2kx 2kf2x, u2 , . . . , un Y
≤ ϕ2x, 2kx, u2 , . . . , un 2.25
for all x, u2 , . . . , un ∈ X. From 2.23 and 2.25, we have
kf2k 1x kf−2k − 1x − k3 f4x 2k3 − 2k f2x
Y
≤ |k|ϕx, 2k 1x, u2 , . . . , un |k|ϕx, 2k − 1x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un ϕx, kx, u2 , . . . , un k2 ϕ2x, 2x, u2 , . . . , un ϕ2x, 2kx, u2 , . . . , un k ϕ0, 3k − 1x, u2 , . . . , un k ϕ0, k 1x, u2 , . . . , un k−1
k − 1
2.26
k2
k2 − 1
ϕ0, 2k − 1x, u2 , . . . , un ϕ0, 2kx, u2 , . . . , un |k − 1|
|k − 1|
for all x, u2 , . . . , un ∈ X. Also, from 2.18 and 2.26, we get
2f3kx − 6fkx k − k 3 f4x − 2kf3x 2k3 − 2k f2x 6kfx, u2 , . . . , un Y
≤ |k| 1 ϕx, 2k 1x, u2 , . . . , un ϕx, 2k − 1x, u2 , . . . , un ϕ3x, x, u2 , . . . , un ϕx, x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un ϕx, kx, u2 , . . . , un k2 ϕ2x, 2x, u2 , . . . , un ϕ2x, 2kx, u2 , . . . , un |k| 1
ϕ0, k 1x, u2 , . . . , un |k − 1|
Abstract and Applied Analysis
9
1
k2 |k| − 1
ϕ0, k − 1x, u2 , . . . , un ϕ0, 2kx, u2 , . . . , un |k − 1|
|k − 1|
2
k ϕ0, 3k − 1x, u2 , . . . , un k ϕ0, 2k − 1x, u2 , . . . , un k−1
|k − 1|
2.27
for all x, u2 , . . . , un ∈ X.
On the other hand, it follows from 2.21 and 2.27 that
8f2kx − 16fkx k − k 3 f4x 2k3 − 10k f2x 16kfx, u2 , . . . , un Y
≤ |k| 1 ϕx, 2k 1x, u2 , . . . , un ϕx, 2k − 1x, u2 , . . . , un ϕ3x, x, u2 , . . . , un ϕx, x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un ϕx, kx, u2 , . . . , un k2 ϕ2x, 2x, u2 , . . . , un ϕ2x, 2kx, u2 , . . . , un 2ϕx, k 1x, u2 , . . . , un 2ϕx, k − 1x, u2 , . . . , un 2ϕ2x, x, u2 , . . . , un 2ϕ2x, kx, u2 , . . . , un 2
2|k|
|k| 1
ϕ0, x, u2 , . . . , un ϕ0, kx, u2 , . . . , un ϕ0, k 1x, u2 , . . . , un |k − 1|
|k − 1|
|k − 1|
1
k2 |k| − 1
ϕ0, k − 1x, u2 , . . . , un ϕ0, 2kx, u2 , . . . , un |k − 1|
|k − 1|
2
k ϕ0, 3k − 1x, u2 , . . . , un k ϕ0, 2k − 1x, u2 , . . . , un k − 1
|k − 1|
2.28
for all x, u2 , . . . , un ∈ X. Therefore by 2.24 and 2.28, we get
f4x − 10f2x 16fx, u2 , . . . , un Y
≤
1
|k3 − k|
× |k| 1 ϕx, 2k 1x, u2 , . . . , un ϕx, 2k − 1x, u2 , . . . , un ϕ3x, x, u2 , . . . , un 8k2 1 ϕx, x, u2 , . . . , un ϕx, 3kx, u2 , . . . , un ϕx, kx, u2 , . . . , un k2 ϕ2x, 2x, u2 , . . . , un ϕ2x, 2kx, u2 , . . . , un 2ϕx, k 1x, u2 , . . . , un 2ϕx, k − 1x, u2 , . . . , un 2ϕ2x, x, u2 , . . . , un x 2k − 1x
x kx
,
, u2 , . . . , un 8|k|ϕ
,
, u2 , . . . , un
2ϕ2x, kx, u2 , . . . , un 8ϕ
2 2
2
2
x 2k 1x
x 3kx
8|k|ϕ
,
, u2 , . . . , un 8ϕ
,
, u2 , . . . , un
2
2
2 2
10
Abstract and Applied Analysis
8k2 1
|k| 1
ϕ0, k 1x, u2 , . . . , un ϕ0, k − 1x, u2 , . . . , un |k − 1|
|k − 1|
k 2
ϕ0, 3k − 1x, u2 , . . . , un ϕ0, x, u2 , . . . , un k − 1
|k − 1|
k2
k2 |k| − 1
ϕ0, 2k − 1x, u2 , . . . , un ϕ0, 2kx, u2 , . . . , un |k − 1|
|k − 1|
8|k|
3k − 1x
, u2 , . . . , un
ϕ 0,
2
|k − 1|
8|k|
8k2 2|k| − 8
k 1x
, u2 , . . . , un ϕ 0,
ϕ0, kx, u2 , . . . , un 2
|k − 1|
|k − 1|
: ϕx,
u2 , . . . , un 2.29
for all x, u2 , . . . , un ∈ X.
Now, let g : X → Y be the mapping defined by gx : f2x − 8fx for all
x, u2 , . . . , un ∈ X. Then, 2.29 means that
f4x − 10f2x 16fx, u2 , . . . , un ≤ ϕx,
u2 , . . . , un Y
2.30
for all x, u2 , . . . , un ∈ X. Also, we get
g2x − 2gx, u2 , . . . , un ≤ ϕx,
u2 , . . . , un Y
2.31
for all x ∈ X. Replacing x by 2j x in 2.31 and dividing both sides of 2.31 by 2j1 , we get
1
g2j x − 1 g2j1 x, u2 , . . . , un ≤ 1 ϕ 2j x, u2 , . . . , un
2j
2j1
2j1
Y
2.32
for all x, u2 , . . . , un ∈ X and all integers j ≥ 0. For all integers l, m with 0 ≤ l < m, we have
m−1
1 l 1
1 j m
g 2x −
≤
g 2 x − 1 g 2j1 x , u2 , . . . , un g2
x,
u
,
.
.
.
,
u
2
n
2l
m
j
j1
2
2
Y
Y
jl 2
≤
m−1
1
jl
2j1
ϕ 2j x, u2 , . . . , un
2.33
for all x, u2 , . . . , un ∈ X. So, we get
1
1
l
m
lim g2 x − m g2 x, u2 , . . . , un 0
l,m → ∞ 2l
2
Y
2.34
Abstract and Applied Analysis
11
for all x, u2 , . . . , un ∈ X. This shows that the sequence {1/2j g2j x} is a Cauchy sequence in
Y . Since Y is an n-Banach space, the sequence {1/2j g2j x} converges. So, we can define a
mapping A : X → Y by
1 j g 2x
j → ∞ 2j
Ax : lim
2.35
for all x ∈ X. Putting l 0, then passing the limit m → ∞ in 2.33, and using Lemma 1.64,
we get
∞
1 j
gx − Ax, u2 , . . . , un ≤
ϕ
2
x,
u
,
.
.
.
,
u
2
n
Y
j1
j0 2
2.36
for all x, u2 , . . . , un ∈ X.
Now we show that A is additive. By Lemma 1.6, 2.2, 2.32, and 2.35, we have
1 j1 1 j g
2
x
−
g
2
x
,
u
,
.
.
.
,
u
A2x − 2Ax, u2 , . . . , un Y lim 2
n
j
j−1
j →∞ 2
2
Y
1 j1 1 j 2 lim g 2 x − j g 2 x , u2 , . . . , un j → ∞ 2j1
2
Y
1
≤ lim j ϕ 2j x, u2 , . . . , un 0
j →∞2
2.37
for all x, u2 , . . . , un ∈ X. By Lemma 1.63, A2x 2Ax for all x ∈ X. Also, by
Lemma 1.64, 2.2, 2.3, and 2.35, we get
DAx, y, u2 , . . . , un Y
1
Dg 2j x, 2j y , u2 , . . . , un j
Y
j →∞2
1
lim j Df 2j1 x, 2j1 y − 8Df 2j x, 2j y , u2 , . . . , un Y
j →∞2
1 ≤ lim j Df 2j1 x, 2j1 y , u2 , . . . , un 8Df 2j x, 2j y , u2 , . . . , un Y
Y
j →∞2
1 ≤ lim j ϕ 2j1 x, 2j1 y, u2 , . . . , un 8ϕ 2j x, 2j y, u2 , . . . , un 0
j →∞2
lim
2.38
for all x, y, u2 , . . . , un ∈ X. By Lemma 1.63, DAx, y 0 for all x, y ∈ X. Hence, the
mapping A satisfies 1.1. By 11, Lemma 2.3, the mapping x → A2x − 8Ax is additive.
Therefore, A2x 2Ax implies that the mapping A is additive.
12
Abstract and Applied Analysis
To prove the uniqueness of A, let B : X → Y be another additive mapping satisfying
2.4. Fix x ∈ X. Clearly, A2l x 2l Ax and B2l x 2l Bx for all l ∈ N. It follows from
2.4 that
A 2l x
B 2l x
−
,
u
,
.
.
.
,
u
Ax − Bx, u2 , . . . , un Y 2
n
l
2l
2
Y
≤
1 l1
l
l
2
x
−
8f
2
x
−
A
2
x
,
u
,
.
.
.
,
u
f
2
n
Y
2l
B 2l x − f 2l1 x 8f 2l x , u2 , . . . , un Y
≤
∞
1 jl
1
ϕ
2
x,
u
,
.
.
.
,
u
2
n
2l j0 2j
≤
∞
∞
1 jl
1 j
ϕ 2 x, u2 , . . . , un ϕ 2 x, u2 , . . . , un
j
jl
j0 2
jl 2
2.39
for all x, u2 , . . . , un ∈ X, and l ∈ N. By 2.2, we see that the right-hand side of the above
inequality tends to 0 as l → ∞. Therefore, Ax − Bx, u2 , . . . , un Y 0 for all u2 , . . . , un ∈ X.
By Lemma 1.6, we can conclude that Ax Bx for all x ∈ X. So, A B. This proves the
uniqueness of A.
Theorem 2.2. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
x y
2 ϕ j , j , u2 , . . . , un < ∞,
2 2
j1
∞
j
Dfx, y, u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.40
for all x, y, u2 , . . . , un ∈ X. Then, there is a unique additive mapping A : X → Y such that
∞
x
j−1
f2x − 8fx − Ax, u2 , . . . , un ≤
2
ϕ
,
u
,
.
.
.
,
u
2
n
Y
2j
j1
2.41
for all x, u2 , . . . , un ∈ X, where ϕx,
u2 , . . . , un is defined as in Theorem 2.1.
Proof. The proof is similar to the proof of Theorem 2.1.
Corollary 2.3. Let X be a normed space and Y an n-Banach space. Let θ ∈ 0, ∞, p, r2 , . . . , rn ∈
0, ∞ such that p /
1, and let f : X → Y be a mapping with f0 0 such that
Dfx, y, u2 , . . . , un ≤ θ xp yp u2 r2 · · · un rn
X
X
X
X
Y
2.42
Abstract and Applied Analysis
13
for all x, y, u2 , . . . , un ∈ X. Then, there exists a unique additive mapping A : X → Y such that
p
θxX u2 rX2 · · · un rXn
f2x − 8fx − Ax, u2 , . . . , un ≤
Y
|2 − 2p k3 − k|
2.43
for all x, u2 , . . . , un ∈ X, where
1 |k| 23−p |k| 2k 1p 2k − 1p 2|k| 13 3p 3|k|p 16k2 3p |k|p 2p1 k2
|k| 1|k 1|p
2p 5 |k|p 2|k 1|p 2|k − 1|p 23−p 2 |k| |k|p 3p |k|p |k − 1|
2p |k|p k2 |k| − 1
23−p |k|
|k 1|p 1 8k2 2p k2 |k − 1|p−1 |k − 1|
|k − 1|
3−p
|k| 2 1
8k2 2|k| − 8 p
2
|3k − 1|p |k| .
|k − 1|
|k − 1|
|k − 1|
p
2.44
p
Proof. Define ϕx, y θxX yX u2 rX2 · · · un rXn for all x, y, u2 , . . . , un ∈ X, and apply
Theorems 2.1 and 2.2.
The following example shows that the assumption p /
1 cannot be omitted in
Corollary 2.3.
Example 2.4. Let X C be a linear space over R. Define ·,
√ · : X × X → R by x1 , x2 |a1 b2 − a2 b1 |, where xj aj bj i ∈ C, aj , bj ∈ R, j 1, 2 i −1 is the imaginary unit. Then,
X, ·, · is a 2-normed linear space.
Let φ : C → C defined by
φx ⎧
⎨x, for |x| < 1,
⎩1,
for |x| ≥ 1.
2.45
Consider the function f : C → C defined by
fx ∞
α−m φαm x
2.46
m0
for all x ∈ C, where α > |k|. Then, f satisfies the functional inequality
2
Df x, y , u ≤ 4α |k| 1 |x| y |u|
α−1
2.47
for all x, y, u ∈ C, but there do not exist an additive mapping A : C → C and a constant d > 0
such that fx − Ax, u ≤ d |x||u| for all x, u ∈ C.
14
Abstract and Applied Analysis
It is clear that |fx| ≤ α/α − 1 for all x ∈ C. If |x| |y| 0 or |x| |y| ≥ 1/α for all
x, y ∈ C, then the inequality 2.47 holds. Now suppose that 0 < |x| |y| < 1/α. Then, there
exists an integer n ≥ 1 such that
1
αn1
1
≤ |x| y < n .
α
2.48
Hence, αm |kx ± y| < 1, αm |x ± y| < 1, αm |x| < 1 for all m 0, 1, . . . , n − 1. From the definition of
f and 2.48, we obtain that
Df x, y , u
∞
∞
∞
α−m φ αm kx y α−m φ αm kx − y
− k α−m φ αm x y
mn
mn
mn
∞
∞
∞
m
−m
−m
m
−m
m
−k α φ α x − y
− 2 α φα kx 2k α φα x, u
mn
mn
mn
≤
2.49
4α2 |k| 1 |x| y |u|.
α−1
Therefore, f satisfies 2.47. Now, we claim that the functional equation 1.1 is not stable
for p 1 in Corollary 2.3. Suppose on the contrary that there exist an additive mapping
A : C → C and a constant d > 0 such that fx − Ax, u ≤ d |x||u| for all x, u ∈ C. Then,
there exists a constant c ∈ C such that Ax cx for all rational numbers x. So, we obtain that
fx, u ≤ d |c| |x||u|
2.50
for all rational numbers x and all u ∈ C. Let s ∈ N with s 1 > d |c|. If x is a rational number
in 0, α−s and u bi b ∈ R, then αm x ∈ 0, 1 for all m 0, 1, . . . , s, and we get
∞
s
m
φαm x φα x
fx, u ≥
,
u
|b| s 1x|b| > d |c|x|b| d |c||x||u|,
m0 αm
m0 αm
2.51
which contradicts 2.50.
Theorem 2.5. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
∞
1 j
ϕ 2 x, 2j y, u2 , . . . , un < ∞,
j
j0 8
2.52
Dfx, y, u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.53
Abstract and Applied Analysis
15
for all x, y, u2 , . . . , un ∈ X. Then, there is a unique cubic mapping C : X → Y such that
∞
1 j
f2x − 2fx − Cx, u2 , . . . , un ≤
ϕ
2
x,
u
,
.
.
.
,
u
2
n
Y
j1
j0 8
2.54
for all x, u2 , . . . , un ∈ X, where ϕx,
u2 , . . . , un is defined as in Theorem 2.1.
Proof. As in the proof of Theorem 2.1, we have
f4x − 10f2x 16fx, u2 , . . . , un ≤ ϕx,
u2 , . . . , un Y
2.55
for all x ∈ X, where ϕx,
u2 , . . . , un is defined as in Theorem 2.1.
Now, let h : X → Y be the mapping defined by hx : f2x − 2fx. By 2.55, we
have
u2 , . . . , un h2x − 8hx, u2 , . . . , un Y ≤ ϕx,
2.56
for all x ∈ X. Replacing x by 2j x in 2.56 and dividing both sides of 2.56 by 8j1 , we get
1
h2j x − 1 h2j1 x, u2 , . . . , un ≤ 1 ϕ 2j x, u2 , . . . , un
8j
8j1
8j1
Y
2.57
for all x, u2 , . . . , un ∈ X and all integers j ≥ 0. For all integers l, m with 0 ≤ l < m, we have
m−1
1
1
h2l x − 1 h2m x, u2 , . . . , un ≤
h2j x − 1 h2j1 x, u2 , . . . , un 8l
j
8m
8j1
Y
Y
jl 8
≤
m−1
jl
j
ϕ
2
x,
u
,
.
.
.
,
u
2
n
8j1
2.58
1
for all x, u2 , . . . , un ∈ X. So, we get
1
1
l
m
lim l h2 x − m h2 x, u2 , . . . , un 0
l,m → ∞ 8
8
Y
2.59
for all x, u2 , . . . , un ∈ X. This shows that the sequence {1/8j h2j x} is a Cauchy sequence in
Y . Since Y is an n-Banach space, the sequence {1/8j h2j x} converges. So, we can define a
mapping C : X → Y by
1 j h 2x
j → ∞ 8j
Cx : lim
2.60
16
Abstract and Applied Analysis
for all x ∈ X. Putting l 0, then passing the limit m → ∞ in 2.58, and using Lemma 1.64,
we get
hx − Cx, u2 , . . . , un Y ≤
∞
1 j
ϕ 2 x, u2 , . . . , un
j1
j0 8
2.61
for all x, u2 , . . . , un ∈ X.
Now we show that C is cubic. By Lemma 1.6, 2.52, 2.58, and 2.60, we have
1
1
j1
j
h2
x
−
h2
x,
u
,
.
.
.
,
u
C2x − 8Cx, u2 , . . . , un Y lim 2
n
j
j →∞ 8
8j−1
Y
1
1
8 lim h2j1 x − j h2j x, u2 , . . . , un j1
j →∞ 8
8
Y
2.62
1 j
0
ϕ
2
x,
u
,
.
.
.
,
u
2
n
j → ∞ 8j
≤ lim
for all x, u2 , . . . , un ∈ X. By Lemma 1.63, C2x 8Cx for all x ∈ X. Also, by Lemma 1.64,
2.52, 2.53, and 2.60, we get
DCx, y, u2 , . . . , un Y
1
j
j
x,
2
y,
u
,
.
.
.
,
u
Dh2
2
n
Y
j → ∞ 8j
lim
1
j1
j1
j
j
x,
2
y
−
2Df2
x,
2
y,
u
,
.
.
.
,
u
Df2
2
n
Y
j → ∞ 8j
lim
2.63
1 Df2j1 x, 2j1 y, u2 , . . . , un 2Df2j x, 2j y, u2 , . . . , un j
Y
Y
j →∞8
≤ lim
1 j1
j1
j
j
ϕ
2
2ϕ
2
0
x,
2
y,
u
,
.
.
.
,
u
x,
2
y,
u
,
.
.
.
,
u
2
n
2
n
j → ∞ 8j
≤ lim
for all x, y, u2 , . . . , un ∈ X. By Lemma 1.63, DCx, y 0 for all x, y ∈ X. Hence the mapping
C satisfies 1.1. By 11, Lemma 2.3, the mapping x → C2x − 2Cx is cubic. Therefore,
C2x 8Cx implies that the mapping C is cubic.
Abstract and Applied Analysis
17
To prove the uniqueness of C, let S : X → Y be another cubic mapping satisfying
2.54. Fix x ∈ X. Clearly, C2l x 8l Ax and S2l x 8l Sx for all l ∈ N. It follows from
2.54 that
C 2l x
S 2l x
−
,
u
,
.
.
.
,
u
Cx − Sx, u2 , . . . , un Y 2
n
l
8l
8
Y
≤
1 l1
l
l
x
−
2f2
x
−
C2
x,
u
,
.
.
.
,
u
f2
2
n
Y
8l
S2l x − f2l1 x 2f2l x, u2 , . . . , un Y
≤
∞
1 jl
1
ϕ
2
x,
u
,
.
.
.
,
u
2
n
8l j0 8j
≤
∞
∞
1 jl
1 j
ϕ 2 x, u2 , . . . , un ϕ 2 x, u2 , . . . , un
j
jl
j0 8
jl 8
2.64
for all x, u2 , . . . , un ∈ X, and l ∈ N. By 2.52, we see that the right-hand side of the above
inequality tends to 0 as l → ∞. Therefore, Cx − Sx, u2 , . . . , un Y 0 for all u2 , . . . , un ∈ X.
By Lemma 1.6, we can conclude that Cx Sx for all x ∈ X. So C S. This proves the
uniqueness of C.
Theorem 2.6. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
x y
8 ϕ j , j , u2 , . . . , un < ∞,
2 2
j1
∞
j
Dfx, y, u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.65
for all x, y, u2 , . . . , un ∈ X. Then, there is a unique cubic mapping C : X → Y such that
∞
x
j−1
f2x − 2fx − Cx, u2 , . . . , un ≤
8
ϕ
,
u
,
.
.
.
,
u
2
n
Y
2j
j1
2.66
for all x, u2 , . . . , un ∈ X, where ϕx,
u2 , . . . , un is defined as in Theorem 2.1.
Proof. The proof is similar to the proof of Theorem 2.5.
Corollary 2.7. Let X be a normed space and Y an n-Banach space. Let θ ∈ 0, ∞, p, r2 , . . . , rn ∈
0, ∞ such that p /
3, and let f : X → Y be a mapping with f0 0 such that
Dfx, y, u2 , . . . , un ≤ θ xp yp u2 r2 · · · un rn
X
X
X
X
Y
2.67
18
Abstract and Applied Analysis
for all x, y, u2 , . . . , un ∈ X. Then, there exists a unique cubic mapping C : X → Y such that
p
θxX u2 rX2 · · · un rXn
f2x − 2fx − Cx, u2 , . . . , un ≤
Y
|8 − 2p k3 − k|
2.68
for all x, u2 , . . . , un ∈ X, where is defined as in Corollary 2.3.
p
p
Proof. Define ϕx, y θxX yX u2 rX2 · · · un rXn for all x, y, u2 , . . . , un ∈ X, and apply
Theorems 2.5 and 2.6.
The following example shows that the the generalized Hyers-Ulam stability problem
for the case of p 3 was excluded in Corollary 2.7.
Example 2.8. Let X C be a linear space over R, and let ·, · : X × X → R be defined as in
Example 2.4. Then, X, ·, · is a 2-normed linear space.
Let φ : C → C be defined by
φx ⎧
⎨x3 ,
for |x| < 1,
⎩1,
for |x| ≥ 1.
2.69
Consider the function f : C → C defined by
fx ∞
α−3m φαm x
2.70
m0
for all x ∈ C, where α > |k|. Then, f satisfies the functional inequality
6
Df x, y , u ≤ 4α |k| 1 |x|3 y3 |u|
α3 − 1
2.71
for all x, y, u ∈ C, but there do not exist a cubic mapping C : C → C and a constant d > 0
such that fx − Cx, u ≤ d |x|3 |u| for all x, u ∈ C.
It is clear that |fx| ≤ α3 /α3 − 1 for all x ∈ C. If |x|3 |y|3 0 or |x|3 |y|3 ≥ 1/α3 for
all x, y ∈ C, then inequality 2.71 holds. Now suppose that 0 < |x|3 |y|3 < 1/α3 . Then, there
exists an integer n ≥ 1 such that
1
α3n1
3
1
≤ |x|3 y < 3n .
α
2.72
Abstract and Applied Analysis
19
Hence, αm |kx ± y| < 1, αm |x ± y| < 1, αm |x| < 1 for all m 0, 1, . . . , n − 1. From the definition of
f and 2.72, we obtain that
∞
∞
∞
Df x, y , u α−3m φ αm kx − y − k α−3m φ αm x y
α−3m φ αm kx y mn
mn
mn
∞
∞
∞
m
−3m
−3m
m
−3m
m
−k α φ α x − y − 2 α φα kx 2k α φα x, u
mn
mn
mn
≤
4α6 |k| 1 3 3 |x| y |u|.
α3 − 1
2.73
Therefore, f satisfies 2.71. Now, we claim that the functional equation 1.1 is not stable for
p 3 in Corollary 2.7. Suppose on the contrary that there exist a cubic mapping C : C → C
and a constant d > 0 such that fx − Cx, u ≤ d |x|3 |u| for all x, u ∈ C. Then, there exists a
constant β ∈ C such that Cx βx3 for all rational numbers x. So, we obtain that
fx, u ≤ d β |x|3 |u|
2.74
for all rational numbers x and all u ∈ C. Let s ∈ N with s 1 > d |β|. If x is a rational number
in 0, α−s and u bi b ∈ R, then αm x ∈ 0, 1 for all m 0, 1, . . . , s, and we get
∞
s
m
φα
x
φαm x
fx, u , u ≥
|b|
3m
m0 α
m0 α3m
s 1x3 |b| > d β x3 |b| d β |x|3 |u|,
2.75
which contradicts 2.74.
Theorem 2.9. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
∞
1 j
j
< ∞,
ϕ
2
x,
2
y,
u
,
.
.
.
,
u
2
n
j
j0 2
2.76
Df x, y , u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.77
for all x, y, u2 , . . . , un ∈ X. Then, there exist a unique additive mapping A : X → Y and a unique
cubic mapping C : X → Y such that
∞ 1
1
fx − Ax − Cx, u2 , . . . , un ≤ 1
ϕ 2j x, u2 , . . . , un
Y
j1
j1
6 j0 2
8
for all x, u2 , . . . , un ∈ X, where ϕx,
u2 , . . . , un is defined as in Theorem 2.1.
2.78
20
Abstract and Applied Analysis
Proof. By Theorems 2.1 and 2.5, there exist an additive mapping A : X → Y and a cubic
mapping C : X → Y such that
∞
1 j
f2x − 8fx − A x, u2 , . . . , un ≤
,
ϕ
2
x,
u
,
.
.
.
,
u
2
n
Y
j1
j0 2
∞
1 j
f2x − 2fx − C x, u2 , . . . , un ≤
ϕ
2
x,
u
,
.
.
.
,
u
2
n
Y
j1
j0 8
2.79
for all x, u2 , . . . , un ∈ X. Hence,
∞ 1
1
j
fx 1 A x − 1 C x, u2 , . . . , un ≤ 1
x,
u
,
.
.
.
,
u
ϕ
2
2
n
6
6
6 j0 2j1 8j1
Y
2.80
for all x ∈ X. So, we obtain 2.78 by letting Ax −1/6A x and Cx 1/6C x for
all x ∈ X.
To prove the uniqueness of A and C, let A , C : X → Y be another additive and cubic
mapping satisfying 2.78. Fix x ∈ X. Let A1 A − A and C1 C − C . So,
A1 x C1 x, u2 , . . . , un Y
≤ fx − Ax − Cx, u2 , . . . , un Y fx − A x − C x, u2 , . . . , un Y
∞ 1
1
1
j
ϕ
2
≤
x,
u
,
.
.
.
,
u
2
n
3 j0 2j1 8j1
2.81
for all x, u2 , . . . , un ∈ X. Then 2.76 implies that
lim
1
n → ∞ 8n
A1 2n x C1 2n x, u2 , . . . , un Y 0
2.82
for all x, u2 , . . . , un ∈ X. Thus, C1 0. So, it follows from 2.81 that
A1 x, u2 , . . . , un Y ≤
∞ 1
1
1
ϕ 2j x, u2 , . . . , un
j1
j1
3 j0 2
8
for all u2 , . . . , un ∈ X. Therefore, A1 0.
Similarly to Theorem 2.9, one can prove the following result.
2.83
Abstract and Applied Analysis
21
Theorem 2.10. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
x y
8j ϕ j , j , u2 , . . . , un < ∞,
2 2
j0
∞
Dfx, y, u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.84
for all x, y, u2 , . . . , un ∈ X. Then, there exist a unique additive mapping A : X → Y and a unique
cubic mapping C : X → Y such that
∞ x
j−1
j−1
fx − Ax − Cx, u2 , . . . , un ≤ 1
2
ϕ
8
,
u
,
.
.
.
,
u
2
n
Y
6 j1
2j
2.85
for all x, u2 , . . . , un ∈ X, where ϕx,
u2 , . . . , un is defined as in Theorem 2.1.
Proof. The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems
2.2 and 2.6.
Theorem 2.11. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with
f0 0 for which there is a function ϕ : X n1 → 0, ∞ such that
∞
1 j
x y
j
< ∞,
2 ϕ j , j , u2 , . . . , un < ∞,
ϕ
2
x,
2
y,
u
,
.
.
.
,
u
2
n
j
2 2
j0 8
j1
∞
j
Df x, y , u2 , . . . , un ≤ ϕ x, y, u2 , . . . , un
Y
2.86
for all x, y, u2 , . . . , un ∈ X. Then, there exist a unique additive mapping A : X → Y and a unique
cubic mapping C : X → Y such that
fx − Ax − Cx, u2 , . . . , un Y
⎤
⎡
∞
∞
1
1 ⎣
x
≤
2j−1 ϕ j , u2 , . . . , un ϕ 2j x, u2 , . . . , un ⎦
j1
6 j1
2
j0 8
2.87
u2 , . . . , un is defined as in Theorem 2.1.
for all x, u2 , . . . , un ∈ X, where ϕx,
Proof. The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems
2.2 and 2.5.
Corollary 2.12. Let X be a normed space and Y an n-Banach space. Let θ ∈ 0, ∞, r2 , . . . , rn ∈
0, ∞, p ∈ 0, 1 ∪ 1, 3 ∪ 3, ∞, and let f : X → Y be a mapping with f0 0 such that
Df x, y , u2 , . . . , un ≤ θ xp yp u2 r2 · · · un rn
X
X
X
X
Y
2.88
22
Abstract and Applied Analysis
for all x, y, u2 , . . . , un ∈ X. Then, there exist a unique additive mapping A : X → Y and a unique
cubic mapping C : X → Y such that
fx − Ax − Cx, u2 , . . . , un ≤
Y
1
1
1
p
θxX u2 rX2 · · · un rXn
p
p
3
6|k − k| |2 − 2 | |8 − 2 |
2.89
for all x, u2 , . . . , un ∈ X, where is defined as in Corollary 2.3.
p
p
Proof. Define ϕx, y θxX yX u2 rX2 · · · un rXn for all x, y, u2 , . . . , un ∈ X, and apply
Theorems 2.9–2.11.
Remark 2.13. The generalized Hyers-Ulam stability problem for the cases of p 1 and p 3
was excluded in Corollary 2.12 see Examples 2.4 and 2.8.
Acknowledgments
The authors would like to thank the Editor Professor Krzysztof Ciepliński and anonymous
referees for their valuable comments and suggestions. The first author was supported by the
National Natural Science Foundation of China NNSFC grant No. 11171022.
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