Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 151547, 9 pages
doi:10.1155/2010/151547
Research Article
Stability of a Cauchy-Jensen Functional Equation in
Quasi-Banach Spaces
Jae-Hyeong Bae1 and Won-Gil Park2
1
2
College of Liberal Arts, Kyung Hee University, Yongin 446-701, South Korea
Division of Computational Sciences in Mathematics, National Institute for Mathematical Sciences,
385-16 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea
Correspondence should be addressed to Won-Gil Park, wgpark@nims.re.kr
Received 16 October 2009; Accepted 30 January 2010
Academic Editor: Yeol Je Cho
Copyright q 2010 J.-H. Bae and W.-G. Park. 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.
We obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation 2fx y, z w/2 fx, z fx, w fy, z fy, w.
1. Introduction
In 1940, Ulam proposed the general Ulam stability problem see 1.
Let G1 be a group and let G2 be a metric group with the metric d·, ·. Given ε > 0, does there
exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality dhxy, hxhy < δ for
all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with dhx, Hx < ε for all x ∈ G1 ?
In 1941, this problem was solved by Hyers 2 in the case of Banach space. Thereafter,
we call that type the Hyers-Ulam stability.
Throughout this paper, let X and Y be vector spaces. A mapping g : X → Y is called an
additive mapping respectively, an affine mapping if g satisfies the Cauchy functional equation
gxy gxgy respectively, the Jensen functional equation 2gxy/2 gxgy.
Aoki 3 and Rassias 4, 5 extended the Hyers-Ulam stability by considering variables for
Cauchy equation. Using the method introduced in 3, Jung 6 obtained a result for Jensen
equation. It also has been generalized to the function case by Găvruta 7 and Jung 8 for
Cauchy equation, and by Lee and Jun 9 for Jensen equation.
Definition 1.1. A mapping f : X × X → Y is called a Cauchy-Jensen mapping if f satisfies the
system of equations
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Journal of Inequalities and Applications
f x y, z fx, z f y, z ,
1.1
yz
f x, y fx, z.
2f x,
2
When X Y R, the function f : R × R → R given by fx, y : axy bx is a solution
of 1.1. In particular, letting x y, we get a function g : R → R given by gx : fx, x ax2 bx.
For a mapping f : X × X → Y , consider the functional equation
z w
2f x y,
fx, z fx, w f y, z f y, w .
2
1.2
Definition 1.2 see 10, 11. Let X be a real linear space. A quasi-norm is real-valued function
on X satisfying the following.
i x ≥ 0 for all x ∈ X and x 0 if and only if x 0.
ii λx |λ|x for all λ ∈ R and all x ∈ X.
iii There is a constant K ≥ 1 such that x y ≤ Kx y for all x, y ∈ X.
The pair X, · is called a quasi-normed space if · is a quasi-norm on X. The smallest
possible K is called the modulus of concavity of · . A quasi-Banach space is a complete quasinormed space. A quasi-norm · is called a p-norm 0 < p ≤ 1 if
x yp ≤ xp yp
1.3
for all x, y ∈ X. In this case, a quasi-Banach space is called a p-Banach space.
The authors 12 obtained the solutions of 1.1 and 1.2 as follows.
Theorem A. A mapping f : X × X → Y satisfies 1.1 if and only if there exist a biadditive mapping
B : X × X → Y and an additive mapping A : X → Y such that fx, y Bx, y Ax for all
x, y ∈ X.
Theorem B. A mapping f : X × X → Y satisfies 1.1 if and only if it satisfies 1.2.
In this paper, we investigate the generalized Hyers-Ulam stability of 1.1 and 1.2.
2. Stability of 1.1 and 1.2
Throughout this section, assume that X is a quasi-normed space with quasi-norm · X and
that Y is a p-Banach space with p-norm · Y . Let K be the modulus of concavity of · Y .
Let ϕ : X × X × X → 0, ∞ and ψ : X × X × X → 0, ∞ be two functions such that
lim
1 n
ϕ 2 x, 2n y, z 0,
n → ∞ 2n
1 ϕ x, y, 3n z 0,
n
n→∞3
lim
ψ 2n x, y, z 0,
2.1
1 ψ x, 3n y, 3n z 0
n
n→∞3
2.2
lim
1
n → ∞ 2n
lim
Journal of Inequalities and Applications
3
for all x, y, z ∈ X, and
∞
p
1 j
j
ϕ
2
x,
2
y,
z
< ∞,
M x, y, z :
pj
j0 2
2.3
∞
p
1 N x, y, z :
ψ x, 3j y, 3j z < ∞
pj
j0 3
2.4
for all x, y, z ∈ X.
Theorem 2.1. Suppose that a mapping f : X × X → Y satisfies the inequalities
fx y, z − fx, z − fy, z ≤ ϕ x, y, z ,
Y
2f x, y z − f x, y − fx, z ≤ ψ x, y, z
2
Y
2.5
2.6
for all x, y, z ∈ X. Then the limits
1 FC x, y : lim j f 2j x, y ,
j →∞2
1 FJ x, y : lim j f x, 3j y
j →∞3
2.7
exist for all x, y ∈ X and the mappings FC : X × X → Y and FJ : X × X → Y are Cauchy-Jensen
mappings satisfying
f x, y − FC x, y ≤ 1 M x, x, y 1/p ,
Y
2
fx, y − fx, 0 − FJ x, y ≤ K N x, y, y 1/p
Y
3
2.8
2.9
for all x, y ∈ X.
Proof. Letting y x and replacing z by y in 2.5 then,
f2x, y − 2fx, y ≤ ϕ x, x, y
Y
2.10
for all x, y ∈ X. Replacing x by 2n x in the above inequality and dividing by 2n1 , we get
1
1 n1
1 n
n
n
2n1 f 2 x, y − 2n f2 x, y ≤ 2n1 ϕ 2 x, 2 x, y
Y
2.11
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Journal of Inequalities and Applications
for all x, y ∈ X and all nonnegative integers n. Since Y is a p-Banach space, we have
p p
n 1
1
1 n1
1
m
j1
j
≤
f
2
x,
y
−
f2
x,
y
f2
x,
y
−
f2
x,
y
2n1
m
j
j1
2
2
2
Y
Y
jm
n
p
1 j
1
j
≤ p
ϕ
2
x,
2
y,
y
2 jm 2pj
2.12
for all x, y ∈ X and all nonnegative integers n and m with n ≥ m. Therefore we conclude
from 2.3 and 2.12 that the sequence {1/2n f2n x, y} is a Cauchy sequence in Y for all
x, y ∈ X. Since Y is complete, the sequence {1/2n f2n x, y} converges in Y for all x, y ∈ X.
So one can define the mapping FC : X × X → Y by
1 FC x, y : lim n f 2n x, y
n→∞2
2.13
for all x, y ∈ X. Letting m 0 and passing the limit n → ∞ in 2.12, we get 2.8. Now, we
show that FC is a Cauchy-Jensen mapping. It follows from 2.1, 2.11, and 2.13 that
FC 2x, y − 2FC x, y lim 1 f2n1 x, y − 1 f2n x, y
Y
n → ∞ 2n
2n−1
Y
1
1 n
n1
2 lim n1 f 2 x, y − n f 2 x, y n→∞ 2
2
Y
≤ lim
2.14
1 n
ϕ 2 x, 2n x, y 0
n → ∞ 2n
for all x, y ∈ X. So FC 2x, y 2FC x, y for all x, y ∈ X.
On the other hand it follows from 2.1, 2.5, 2.6, and 2.13 that
FC x y, z −FC x, z−FC y, z lim 1 f 2n x 2n y, z −f2n x, z−f 2n y, z Y
Y
n → ∞ 2n
≤ lim
1 n
ϕ 2 x, 2n y, z 0,
n → ∞ 2n
2FC x, y z −FC x, y −FC y, z lim 1 f 2n x, y z −f 2n x, y −f 2n y, z n → ∞ 2n 2
2
Y
Y
≤ lim
1
n → ∞ 2n
ψ 2n x, y, z 0
2.15
for all x, y, z ∈ X. Thus FC is a Cauchy-Jensen mapping. Next, setting z −y in 2.6 and
replacing y by −y and z by 3y in 2.6, one can obtain that
2fx, 0 − fx, y − fx, −y ≤ ψ x, y, −y ,
Y
2fx, y − fx, −y − fx, 3y ≤ ψ x, −y, 3y ,
Y
2.16
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5
respectively, for all x, y ∈ X. By two above inequalities,
3fx, y − 2fx, 0 − fx, 3y ≤ K ψ x, y, −y ψ x, −y, 3y
Y
2.17
for all x, y ∈ X. By the same method as above, one can find a Cauchy-Jensen mapping FJ
which satisfies 2.9. In fact, FJ x, y : limj → ∞ 1/3j fx, 3j y for all x, y ∈ X.
From now on, let χ : X × X × X × X → 0, ∞ be a function such that
lim
1 n
ϕ 2 x, 2n y, 3n z, 3n w 0,
n → ∞ 6n
∞
p
1 j
j
j
j
L x, y, z, w :
χ
2
x,
2
y,
3
z,
3
w
<∞
pj
j0 6
2.18
2.19
for all x, y, z, w ∈ X.
We will use the following lemma in order to prove Theorem 2.3.
Lemma 2.2 see 13. Let 0 < p ≤ 1 and let x1 , x2 , . . . , xn be nonnegative real numbers. Then
⎛
⎞p
n
n
⎝ xj ⎠ ≤
xj p .
j1
2.20
j1
Theorem 2.3. Suppose that a mapping f : X × X → Y satisfies fx, 0 f0, x 0 and the
inequality
z w
− fx, z − fx, w − fy, z − fy, w ≤ χ x, y, z, w
2f x y,
2
Y
2.21
for all x, y, z, w ∈ X. Then the limit Fx, y : limj → ∞ 1/6j f2j x, 3j y exists for all x, y ∈ X and
the mapping F : X × X → Y is the unique Cauchy-Jensen mapping satisfying
fx, y − Fx, y ≤ K χ x, y 1/p ,
Y
6
2.22
where
∞
p
1 p j
j
j
j
3
χ x, y :
χ
2
x,
2
x,
3
y,
−3
y
pj
j0 6
p
p K 3p χ 2j x, 2j x, −3j y, 3j y χ 2j x, 2j x, 3j y, 3j y
p K p p K 2p χ 2j x, 2j x, −3j y, 3j1 y p χ 2j x, 2j x, 3j1 y, 3j1 y
2
for all x, y ∈ X.
2.23
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Journal of Inequalities and Applications
Proof. Letting y x in 2.21, we get
z w
− 2fx, z − 2fx, w ≤ χx, x, z, w
2f 2x,
2
Y
2.24
for all x, z, w ∈ X. Putting z y and w −y in 2.24, we get
2fx, y 2fx, −y ≤ χ x, x, y, −y
Y
2.25
for all x, y ∈ X. Replacing z by −y and w by −y in 2.24, we get
f2x, −y − 2fx, −y ≤ 1 χ x, x, −y, −y
Y
2
2.26
for all x, y ∈ X. By 2.25 and 2.26, we have
2fx, y f2x, −y ≤ K χ x, x, y, −y 1 χ x, x, −y, −y
Y
2
2.27
for all x, y ∈ X. Setting z y and w −3y in 2.24, we get
f2x, −y − fx, y − fx, −3y ≤ 1 χ x, x, y, −3y
Y
2
2.28
for all x, y ∈ X. By 2.27 and the above inequality, we get
3fx, y fx, −3y ≤ K 2 χ x, x, y, −y 1 χ x, x, −y, −y K χ x, x, y, −3y
Y
2
2
2.29
for all x, y ∈ X. Replacing y by 3y in 2.26, we get
f2x, −3y − 2fx, −3y ≤ 1 χ x, x, −3y, −3y
Y
2
2.30
for all x, y ∈ X. By 2.29 and the above inequality, we have
6f x, y f 2x, −3y ≤ K 3 2χ x, x, y, −y χ x, x, −y, −y K 2 χ x, x, y, −3y
Y
K χ x, x, −3y, −3y
2
2.31
Journal of Inequalities and Applications
7
for all x, y ∈ X. Replacing y by −y in the above inequality, we get
6f x, −y f 2x, 3y ≤ K 3 2χ x, x, −y, y χ x, x, y, y K 2 χ x, x, −y, 3y
Y
K χ x, x, 3y, 3y
2
2.32
for all x, y ∈ X. By 2.25 and the above inequality, we get
6fx, y − f2x, 3y ≤ χ∗ x, y ,
Y
2.33
where
χ∗ x, y : 3Kχ x, x, y, −y K 4 2χ x, x, −y, y χ x, x, y, y K 3 χ x, x, −y, 3y
2.34
K2 χ x, x, 3y, 3y
2
for all x, y ∈ X. Replacing x by 2n x and y by 3n y in the above inequality and dividing 6n1 ,we
get
1 n
f 2 x, 3n y − 1 f 2n1 x, 3n1 y ≤ 1 χ∗ 2n x, 3n y
6n
n1
6n1
6
Y
2.35
for all x, y ∈ X and all nonnegative integers n. Since · Y is a p-norm, we have
p p
n 1
1
1 j1
1
n1
n1
m
m
j1
j
j
6n1 f2 x, 3 y − 6m f2 x, 3 y ≤
6j1 f 2 x, 3 y − 6j f2 x, 3 y
Y
Y
jm
≤
n
p
1 j
1
χ∗ 2 x, 3j y
p
pj
6 jm 6
2.36
for all x, y ∈ X and all nonnegative integers n and m with n ≥ m. Therefore we conclude
from 2.18 and 2.36 that the sequence {1/6n f2n x, 3n y} is a Cauchy sequence in Y for
all x, y ∈ X. Since Y is complete, the sequence {1/6n f2n x, 3n y} converges in Y for all
x, y ∈ X. So one can define the mapping F : X × X → Y by
1 F x, y : lim n f 2n x, 3n y
n→∞6
2.37
for all x, y ∈ X. Letting m 0, passing the limit n → ∞ in 2.36, and applying lemma,
we get 2.22. Now, we show that F is a Cauchy-Jensen mapping. By lemma, we infer that
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Journal of Inequalities and Applications
limn → ∞ 1/6n χ∗ 2n x, 3n y 0 for all x, y ∈ X. It follows from 2.18, 2.35, and the above
equality that
F2x, 3y − 6Fx, y lim 1 f2n1 x, 3n1 y − 1 f2n x, 3n y
Y
n−1
n → ∞ 6n
6
Y
1
1
n1
n1
n
n
6 lim f2
x,
3
y
−
f2
x,
3
y
n
n1
n→∞ 6
6
Y
2.38
1 n
χ∗ 2 x, 3n y 0
n
n→∞6
≤ lim
for all x, y ∈ X. So F2x, 3y 6Fx, y for all x, y ∈ X.
On the other hand it follows from 2.18, 2.21, and 2.37 that
zw
− Fx, z − Fx, w − Fy, z − Fy, w
2Fx y,
2
Y
1
3n z 3n w
n
n
f
2
− f2n x, 3n z − f2n x, 3n w
lim n x
2
y,
n→∞6 2
− f2 y, 3 z − f 2 y, 3 w n
n
n
2.39
n
Y
1 n
χ 2 x, 2n y, 3n z, 3n w 0
n
n→∞6
lim
for all x, y, z, w ∈ X. Hence the mapping F satisfies 1.2. To prove the uniqueness of F, let
G : X → Y be another Cauchy-Jensen mapping satisfying 2.22. It follows from 2.19 that
∞
p
1 n
1 j
n
n
n
L
2
x,
2
y,
3
z,
3
w
lim
χ 2 x, 2j y, 3j z, 3j w 0
pn
pj
n→∞6
n→∞
jn 6
lim
for all x, y, z, w ∈ X. Hence limn → ∞
2.22 and 2.37 that
1
χ2
n x, 3n y 0 for all x, y ∈ X. So it follows from
6pn
Fx, y − Gx, yp lim 1 f2n x, 3n y − G2n x, 3n yp
Y
Y
n → ∞ 6pn
1 Kp
≤ p lim pn χ 2n x, 3n y 0
6 n→∞6
for all x, y ∈ X. So F G.
2.40
2.41
Journal of Inequalities and Applications
9
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