Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 976284, 10 pages
doi:10.1155/2009/976284
Research Article
Stability of a Bi-Jensen Functional Equation II
Kil-Woung Jun,1 Il-Sook Jung,1 and Yang-Hi Lee2
1
2
Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea
Department of Mathematics Education, Gongju National University of Education,
Gongju 314-711, South Korea
Correspondence should be addressed to Kil-Woung Jun, kwjun@cnu.ac.kr and
Yang-Hi Lee, yanghi2@hanmail.net
Received 7 October 2008; Accepted 24 January 2009
Recommended by Alexander Domoshnitsky
We investigate the stability of the bi-Jensen functional equation II fxy/2, z−fx, z−fy, z 0, 2fx, y z/2 − fx, y − fx, z 0 in the spirit of Găvruta.
Copyright q 2009 Kil-Woung Jun et al. 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.
1. Introduction
In 1940, Ulam 1 raised a question concerning the stability of homomorphisms. 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
d hxy, hxhy < δ
1.1
for all x, y ∈ G1 , then there is a homomorphism H : G1 → G2 with
d hx, Hx < ε
1.2
for all x ∈ G1 . The case of approximately additive mappings was solved by Hyers 2 under
the assumption that G1 and G2 are Banach spaces. In 1978, Rassias 3 gave a generalization
of Hyers’ theorem by allowing the Cauchy difference to be controlled by a sum of powers
like
hx y − hx − hy ≤ ε xp yp .
1.3
2
Journal of Inequalities and Applications
Găvruta 4 provided a further generalization of Rassias’ theorem in which he replaced the
bound εxp yp by a general function.
Throughout this paper, let X and Y be a normed space and a Banach space,
respectively. A mapping g : X → Y is called a Cauchy mapping resp., a Jensen mapping if
g satisfies the functional equation gx y gx gy resp., 2gx y/2 gx gy.
For a given mapping f : X × X → Y , we define
xy
, z − fx, z − fy, z,
2
yz
− fx, y − fx, z
J2 fx, y, z : 2f x,
2
J1 fx, y, z : 2f
1.4
for all x, y, z ∈ X. A mapping f : X × X → Y is called a bi-Jensen mapping if f satisfies the
functional equations J1 f 0 and J2 f 0.
Bae and Park 5 obtained the generalized Hyers-Ulam stability of a bi-Jensen
mapping. Jun et al. 6 improved the results of Bae and Park in the spirit of Rassias.
In this paper, we investigate the stability of a bi-Jensen functional equation J1 f 0,
J2 f 0 in the spirit of Găvruta.
2. Stability of a Bi-Jensen Functional Equation
Jun et al. 7 established the basic properties of a bi-Jensen mapping in the following lemma.
Lemma 2.1. Let f : X × X → Y be a bi-Jensen mapping. Then, the following equalities hold:
n 1 n
1 n
1 2
1
n
f
2
x,
2
y
−
x,
0
f
0,
2
y
1
−
f0, 0
f
2
4n
2n 4n
2n
n
f 2 x, 2n y
2n − 1 n
1 2
n
n
n
x,
−2
y
f
−
2
x,
2
y
1
−
f0, 0,
f
2
2n
2n
22n1
n
y
1 n
x
1
n
n1
fx, y n f 2 x, 2 y 2 − 1 f n , 0 f 0, n
− 2 − 3 n f0, 0,
4
2
2
4
2
n
n
y
x y
x
fx, y 4n f n , n 2 − 4n f n , 0 f 0, n
2 − 1 f0, 0,
2 2
2
2
fx, y fx, y 1
n 1 n
1 2
1
f
2
x,
y
y
1
−
f0, 0
1
−
f
0,
2
2n
2n
2n
2n
for all x, y ∈ X and n ∈ N.
Now we have the stability of a bi-Jensen mapping.
2.1
Journal of Inequalities and Applications
3
Theorem 2.2. Let ϕ, ψ : X × X × X → 0, ∞ be two functions satisfying
∞ ϕ 2j x, 2j y, 2j z
j1
2j
j1
2j
∞ ψ 2j x, 2j y, 2j z
< ∞,
2.2
<∞
2.3
for all x, y, z ∈ X. Let f : X × X → Y be a mapping such that
J1 fx, y, z ≤ ϕx, y, z,
J2 fx, y, z ≤ ψx, y, z
2.4
for all x, y, z ∈ X. Then, there exists a unique bi-Jensen mapping F : X × X → Y such that
∞ ∞ Φ 2j x, 2j y
ψ 0, 0, 2j y ϕ 2j x, 0, 0
fx, y − Fx, y ≤
4j
2j
j1
j1
2.5
for all x, y ∈ X with F0, 0 f0, 0, where
Φx, y ϕx, 0, y ψx, 0, y
y
3ψ0, 0, y 3ϕx, 0, 0
x
ψ
, 0, y ϕ x, 0,
.
2
2
2
2
2.6
The mapping F : X × X → Y is given by
Fx, y : lim
j →∞
j
f 2 x, 2j y
4j
j
j f 2 x, 0 f 0, 2 y
2j
f0, 0
2.7
for all x, y ∈ X.
Proof. Let f , f , f be the maps defined by
f x, y fx, y − f0, y,
f x, y fx, y − fx, 0,
f x, y fx, y − fx, 0 − f0, y f0, 0
2.8
4
Journal of Inequalities and Applications
for all x, y ∈ X. By 2.4, we get
j
f 2 x, 0 f 2j1 x, 0 J1 f 2j1 x, 0, 0 ϕ2j1 x, 0, 0
−
,
≤
2j
2j1
2j1
2j1
j j1 j1 j1 f 0, 2 y
f 0, 2 y J2 f 0, 0, 2 y ψ 0, 0, 2 y
−
,
≤
2j
2j1
2j1
2j1
j
f 2 x, 2j y f 2j1 x, 2j1 y −
j1
4j
4
J1 f 2j1 x, 0, 2j1 y J2 f 2j1 x, 0, 2j1 y 2J2 f 2j x, 0, 2j1 y
2 · 4j1
j1
j1
j1 2J1 f 2 x, 0, 2j y − 3J2 f 0, 0, 2 y − 3J1 f 2 x, 0, 0 2 · 4j1
j1
Φ 2 x, 2j1 y
≤
4j1
2.9
for all x, y ∈ X and j ∈ N. For given integers l, m 0 ≤ l < m,
l
j1
f 2 x, 0 f 2m x, 0 m−1
ϕ 2 x, 0, 0
−
,
≤
jl
2m
2l
2j1
l m m−1 j1 f 0, 2 y
f 0, 2 y ψ 0, 0, 2 y
−
,
≤
jl
2m
2l
2j1
2.10
l
j1
j1
f 2 x, 2l y f 2m x, 2m y m−1
Φ 2 x, 2 y
−
≤
jl
4m
4l
4j1
for all x, y ∈ X. By 2.2 and 2.3, the sequences {1/2j f2j x, 0 − f0, 0},
{1/2j f0, 2j y − f0, 0}, and {1/4j f 2j x, 2j y} are Cauchy sequences for all x, y ∈ X.
Since Y is complete, the sequences {1/2j f2j x, 0 − f0, 0}, {1/2j f0, 2j y − f0, 0},
and {1/4j f 2j x, 2j y} converge for all x, y ∈ X. Define F1 , F2 , F3 : X × X → Y by
F1 x, y : lim
j →∞
F2 x, y : lim
j →∞
F3 x, y : lim
j →∞
j
f 2 x, 0
2j
j f 0, 2 y
2j
,
,
j
f 2 x, 2j y
4j
2.11
Journal of Inequalities and Applications
5
for all x, y ∈ X. Putting l 0 and taking m → ∞ in 2.10, one can obtain the inequalities
∞ ϕ 2j x, 0, 0
fx, 0 − f0, 0 − F1 x, y ≤
,
2j
j1
∞ ψ 0, 0, 2j y
f0, y − f0, 0 − F2 x, y ≤
,
2j
j1
∞ Φ 2j x, 2j y
fx, y − fx, 0 − f0, y f0, 0 − F3 x, y ≤
4j
j1
2.12
for all x, y ∈ X. By 2.4 and the definitions of F1 and F2 , we get
j
1
J1 f 2 x, 2j y, 0 0,
j
j →∞ 2
J1 F1 x, y, z lim
J2 F1 x, y, z 0,
J1 F2 x, y, z 0,
j
1
J2 f 0, 2 y, 2j z 0,
j
j →∞ 2
j
J1 f 2 x, 2j y, 2j z
J1 F3 x, y, z lim
0,
j →∞
4j
j
J2 f 2 x, 2j y, 2j z
J2 F3 x, y, z lim
0
j →∞
4j
J2 F2 x, y, z lim
2.13
for all x, y, z ∈ X. So F is a bi-Jensen mapping satisfying 2.5, where F is given by
Fx, y F1 x, y F2 x, y F3 x, y f0, 0.
2.14
Now, let F : X × X → Y be another bi-Jensen mapping satisfying 2.5 with F 0, 0 f0, 0.
By Lemma 2.1, we have
Fx, y − F x, y
F − F 2n x, 2n y
n
n 1
1 −
F − F 2 x, 0 F − F 0, 2 y 4n
2n 4n
F − f2n x, 2n y F − f0, 2n y F − f2n x, 0 ≤
4n
2n
2n
f − F 2n x, 2n y f − F 0, 2n y f − F 2n x, 0 4n
2n
2n
6
Journal of Inequalities and Applications
≤
∞ Φ 2j x, 2j y
jn1
2j−1
∞ ψ 0, 0, 2j y ϕ 2j x, 0, 0
2j−2
jn1
∞ Φ 2j x, 0 Φ 0, 2j y
2j−1
jn1
ϕ0, 0, 0 ψ0, 0, 0
2n−1
2n−1
2.15
for all x, y ∈ X and n ∈ N. As n → ∞, we may conclude that Fx, y F x, y for all x, y ∈ X.
Thus such a bi-Jensen mapping F : X × X → Y is unique.
Remark 2.3. Let ϕ, ψ : X × X × X → 0, ∞ be the functions defined by
ϕx, y, z ψx, y, z :
ε
3
2.16
for all x, y, z ∈ X. Let f, F, F : X × X → Y be the bi-Jensen mappings defined by
fx, y : 0,
Fx, y : ε,
F x, y : −ε
2.17
for all x, y ∈ X. Then, ϕ, ψ, and f satisfy 2.2, 2.3, 2.4 for all x, y, z ∈ X. In addition,
f, F satisfy 2.5 for all x, y ∈ X and f, F also satisfy 2.5 for all x, y ∈ X. But we get
F. Hence, the condition F0, 0 f0, 0 is necessary to show that the mapping F is
F /
unique.
We have another stability result applying for several cases.
Theorem 2.4. Let ϕ, ψ : X × X × X → 0, ∞ be two functions satisfying
∞
j0
j
4
ϕ
x y z
, ,
2j 2j 2j
ψ
x y z
, ,
2j 2j 2j
<∞
2.18
for all x, y, z ∈ X. Let f : X × X → Y be a mapping satisfying 2.4 for all x, y, z ∈ X. Then, there
exists a unique bi-Jensen mapping F : X × X → Y satisfying
∞
y
y
x
x
j
j
j
fx, y − Fx, y ≤
4Φ
2 ψ 0, 0, j 2 ϕ
,
, 0, 0
2j 2j
2
2j
j0
2.19
Journal of Inequalities and Applications
7
for all x, y ∈ X. The mapping F : X × X → Y is given by
j
Fx, y : lim 4 f
j →∞
x y
,
2j 2j
j
− 4 −2
j
f
j
2
y
x
2 − 1 f0, 0
, 0 f 0, j
2j
2
2.20
for all x, y ∈ X.
Proof. By 2.4 and the similar method in Theorem 2.2, we define the maps F1 , F2 , F3 : X×X →
Y by
x
, 0 − f0, 0 ,
F1 x, y : lim 2 f
j →∞
2j
y
j
F2 x, y : lim 2 f 0, j − f0, 0 ,
j →∞
2
y
y
x
x
F3 x, y : lim 4j f
−f
,
, 0 − f 0, j − f0, 0
j →∞
2j 2j
2j
2
j
2.21
for all x, y ∈ X. By 2.4 and the definitions of F1 , F2 , and F3 , we get
J1 F1 x, y, z lim 2 J1 f
j
j →∞
x y
, , 0 0,
2j 2j
J2 F1 x, y, z 0,
J1 F2 x, y, z 0,
y z
J2 F2 x, y, z lim 2 J2 f 0, j , j 0,
j →∞
2 2
y
y
x
z
x
J1 F3 x, y, z lim 4j J1 f
− J1 f
, ,
, ,0
0,
j →∞
2j 2j 2j
2j 2j
y z
x y z
j
J2 F3 x, y, z lim 4 J2 f
− J2 f 0, j , j
0
, ,
j →∞
2j 2j 2j
2 2
j
2.22
for all x, y, z ∈ X. By the similar method in Theorem 2.2, F is a bi-Jensen mapping satisfying
2.19, where F is given by
Fx, y F1 x, y F2 x, y F3 x, y f0, 0.
2.23
8
Journal of Inequalities and Applications
Now, let F : X × X → Y be another bi-Jensen mapping satisfying 2.19. Using Lemma 2.1,
ϕ0, 0, 0 ψ0, 0, 0 0, and F 0, 0 f0, 0 F0, 0, we have
Fx, y − F x, y
n
x y
x
y
n
n
4 F − F
0, n
, n 2 −4
,0 F − F
F−F
n
n
2 2
2
2
x y x
x y F
−
f
f
−
F
≤ 4n F − f
,
,
,
0
2n 2n 2n 2n 2n
x
y y F
−
f
0,
f
−
F
f − F
0,
,
0
2n
2n 2n ∞
y
y
x y
x
x
j
≤2 4 Φ
2ψ 0, 0, j 2ϕ
,
, 0, 0 Φ
, 0 Φ 0, j
2j 2j
2
2j
2j
2
jn
2.24
for all x, y ∈ X and n ∈ N. As n → ∞, we may conclude that Fx, y F x, y for all x, y ∈ X.
Thus such a bi-Jensen mapping F : X × X → Y is unique.
Theorem 2.5. Let ϕ, ψ : X × X × X → 0, ∞ be two functions satisfying
∞ ϕ 2j x, 2j y, 2j z
x y z
2ϕ
,
,
< ∞,
2j 2j 2j
4j
j0
j0
∞
∞ ψ 2j x, 2j y, 2j z
x y z
j
2ψ
, ,
<∞
2j 2j 2j
4j
j0
j0
∞
j
2.25
2.26
for all x, y, z ∈ X. Let f : X × X → Y be a mapping satisfying 2.4 for all x, y, z ∈ X. Then, there
exists a bi-Jensen mapping F : X × X → Y satisfying
∞ ∞ Φ 2j x, 2j y
y
x
j
fx, y − Fx, y ≤
2 ψ 0, 0, j ϕ
, 0, 0
4j
2
2j
j0
j1
2.27
for all x, y ∈ X, where the mapping F : X × X → Y is given by
j
j 1 j
f 2 x, 2j y − f 2 x, 0 − f 0, 2 y
j
4
j1
y
x
j
− 2 − 1 f0, 0
, 0 f 0, j
lim 2 f
j →∞
2j
2
Fx, y : lim
j →∞
for all x, y ∈ X.
2.28
Journal of Inequalities and Applications
9
Proof. We can obtain F1 , F2 as in Theorem 2.4 and F3 as in Theorem 2.2. Hence, F is a biJensen mapping satisfying 2.27, where F is given by
Fx, y F1 x, y F2 x, y F3 x, y f0, 0.
2.29
Theorem 2.6. Let ϕ, ψ : X × X × X → 0, ∞ be two functions satisfying 2.2 and 2.26 for all
x, y, z ∈ X. Let f : X × X → Y be a mapping satisfying 2.4 for all x, y, z ∈ X. Then, there exists a
bi-Jensen mapping F : X × X → Y satisfying
∞
∞ Φ 2j x, 2j y
∞ ϕ 2j x, 0, 0
y
j
fx, y − Fx, y ≤
2 ψ 0, 0, j 4j
2
2j
j0
j1
j1
2.30
for all x, y ∈ X, where the mapping F : X × X → Y is given by
j
j 1 j
f 2 x, 2j y − f 2 x, 0 − f 0, 2 y
j
j →∞ 4
j
y
1 j
j
lim j f 2 x, 0 2 f 0, j − 2 − 1 f0, 0
j →∞ 2
2
Fx, y : lim
2.31
for all x, y ∈ X.
Theorem 2.7. Let ϕ, ψ : X × X × X → 0, ∞ be two functions satisfying 2.3 and 2.25 for all
x, y, z ∈ X. Let f : X × X → Y be a mapping satisfying 2.4 for all x, y, z ∈ X. Then, there exists a
bi-Jensen mapping F : X × X → Y satisfying
∞
∞ Φ 2j x, 2j y
x
j
fx, y − Fx, y ≤
2ϕ
, 0, 0
4j
2j
j0
j1
2.32
for all x, y ∈ X, where the mapping F : X × X → Y is given by
j
j 1 j
f 2 x, 2j y − f 2 x, 0 − f 0, 2 y
j
4
x
1 j j
j
lim 2 f
, 0 j f 0, 2 y − 2 − 1 f0, 0
j →∞
2j
2
Fx, y : lim
j →∞
2.33
for all x, y ∈ X.
Applying Theorems 2.2–2.7, we easily get the following corollaries.
Corollary 2.8. Let 0 < p / 1, 2 and ε > 0. Let f : X × X → Y be a mapping such that
J1 fx, y, z ≤ ε xp yp zp ,
J2 fx, y, z ≤ ε xp yp zp
2.34
10
Journal of Inequalities and Applications
for all x, y, z ∈ X. Then, there exists a unique bi-Jensen mapping F : X × X → Y such that
fx, y − Fx, y ≤ ε
7 · 2p 2
2p
24 − 2p 2 − 2p xp yp
2.35
for all x, y ∈ X.
Proof. Applying Theorem 2.2 Theorems 2.4 and 2.5, resp. for the case 0 < p < 1 2 < p and
1 < p < 2, resp., we obtain the desired result.
Corollary 2.9. Let 0 < p, q < 2 (p, q /
1), and ε > 0. Let f : X × X → Y be a mapping such that
J1 fx, y, z ≤ ε xp yp zp ,
J2 fx, y, z ≤ ε xq yq zq
2.36
for all x, y, z ∈ X. Then, there exists a unique bi-Jensen mapping F : X × X → Y such that
fx, y − Fx, y
q
3 · 2p
2q 2
2p
2p 2
2q
p
q
p
p 3·2
q
q
x y
≤ε
x y x y 2−2p 2−2q 4−2p
4−2q
2 4−2q
24−2p
2.37
for all x, y ∈ X.
Proof. Applying Theorems 2.6 and 2.7, we obtain the desired result.
References
1 S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1968.
2 D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of
Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
3 Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American
Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
4 P. Găvruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive
mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
5 J.-H. Bae and W.-G. Park, “On the solution of a bi-Jensen functional equation and its stability,” Bulletin
of the Korean Mathematical Society, vol. 43, no. 3, pp. 499–507, 2006.
6 K.-W. Jun, I.-S. Jung, and Y.-H. Lee, “Stability of a bi-Jensen functional equation,” preprint.
7 K.-W. Jun, Y.-H. Lee, and J.-H. Oh, “On the Rassias stability of a bi-Jensen functional equation,” Journal
of Mathematical Inequalities, vol. 2, no. 3, pp. 363–375, 2008.