RESEARCH Open Access
Approximate Cauchy functional inequality in
quasi-Banach spaces
Hark-Mahn Kim and Eunyoung Son
*
* Correspondence:

Department of Mathematics,
Chungnam National University, 79
Daehangno, Yuseong-gu, Daejeon
305-764, Korea
Abstract
In this article, we prove the generalized Hyers-Ulam stability of the following Cauchy
functional inequality:
|
|f (x)+f (y)+nf (z)|| ≤ nf




x + y
n
+ x




in the class of mappings from n-divisible abelian groups to p-Banach spaces for any
fixed positive integer n ≥ 2.
1 Introduction
The stability problem of functional equations originated from a question of Ulam [1]

1
® E
2
satisfies the following condition: there is a constan t  ≥ 0 such
that
|f
(
x + y
)
− f
(
x
)
−
(
y
)
|| ≤
ε
for all x,y Î E
1
. Then, the limit
h(x) = lim
n→∞
f (2
n
x)
2
n
exists fo r all x Î E

any medium, provided the original work is properly cited.
there exists  > 0 and 0 ≤ p < 1 such that
||f
(
x + y
)
− f
(
x
)
− f
(
y
)
|| ≤ ε
(
||x||
p
+ ||y||
p
)
, ∀x, y ∈ E
1
.
Then, there exists a unique R-linear mapping T : E
1
® E
2
such that
|

x + y
n
+ z

, n ≥ 2
for all x,y, z Î G, which has been introduced in [13].
Proposition 1.1. For a mapping f : G ® X, the following statements are equivalent.
(a) f is additive,
(b)
f (x)+f (y)+nf (z)=nf

x + y
n
+ z

,
(c)
|
|f (x)+f (y)+nf (z)|| ≤



nf

x + y
n
+ z




||
p
for all x,y Î X. In this case, a quasi-Banach space is called a p-Banach space.
Given a p-norm, the formula d(x,y):=||x - y||
p
gives us a translation invariant
metric on X. By the Aoki-Rolewicz theorem [20], each qua si-norm is equiva lent to
some p-norm (see also [19]). Since it is much easier to work with p-norms, henceforth,
we restrict our attention mainly to p-norms. We observe that if x
1
, x
2
, , x
n
are non-
negative real numbers, then

n

i=1
x
i

p
≤
n

i=1
x
i

∞

i=0
ϕ(n
i
x, n
i
y, n
i
z)
p
n
ip
<
∞
for all x,y,z Î G. Then, there exists a unique additive mapping h : G ® Y, defined as
h(x) = lim
k→∞
f (n
k
x) − f (−n
k
x)
2
n
k
, such that
|
|f (x) − h(x)|| ≤
M

|f
(
nx
)
+ nf
(
−x
)
|| ≤ ϕ
(
nx,0,−x
)
(4)
Kim and Son Journal of Inequalities and Applications 2011, 2011:102
/>Page 3 of 11
for all x Î G. Replacing x by -x in (4), one has
|
|f
(
−nx
)
+ nf
(
x
)
|| ≤ ϕ
(
−nx,0,x
)
(5)

for all x Î G. It follows from (6) that




g(n
l
x
n
l
−
g(n
m
x)
n
m




p
≤
m−1

k=l




1

1
n
g(n
k+1
x)




p
≤
m−1

k
=1
M
p
2
p
n
(k+1)
p
[ϕ(n
k+1
x,0,−n
k
x)
p
+ ϕ(−n
k+1

x)
n
m
= lim
m→∞
f (n
m
x) − f (−n
m
x)
2
n
m
, x ∈ G
.
Moreover, letting l = 0 and taking m ® ∞ in (7), we get




f (x) − f(−x)
2
− h(x)




≤||g(x) − h(x)|| ≤
M
2n

n
kp
||g(n
k
x)+g(n
k
y)+g(−n
k
(x + y))||
p
≤ lim
k→∞
1
2
p
n
kp
(||f (n
k
x)+f (n
k
y)+nf (−n
k−1
(x + y))||
p
+|| −f (−n
k
x) −f (−n
k
y) −nf (n

+ ϕ(−n
k
x, −n
k
y, n
k−1
(x + y))
p
+ϕ(−n
k
(x + y), 0, n
k−1
(x + y))
p
+ ϕ(n
k
(x + y), 0, −n
k−1
(x + y))
p
)
=
0
for all x,y ÎG. This implies that the mapping h is additive.
Next, let h’ : G ® Y be another additive mapping satisfying
|
|f (x) − h

(x)|| ≤
M

h

(n
k
x)




p
≤
1
n
kp
(||h(n
k
x) − f (n
k
x)||
p
+ ||f(n
k
x) − h

(n
k
x)||
p
)
≤


i
=
k
2M
2p
2
p
n
(i+1)p
[ϕ(n
i+1
x,0,−n
i
x)
p
+ ϕ(−n
i+1
x,0,n
i
x)
p
]+
2M
p
ϕ(n
k
x, −n
k
x,0)

then we get the following Corollaries 2.2 and 2.3.
Corollary 2.2. Let q + r + s <1,q, r, s >0,θ > 0. If a mapping f : X ® Y with f(0) =
0 satisfies the following functional inequality:
||f (x)+f (y)+nf (z)|| ≤



nf

x + y
n
+ x




+ θ (||x||
q
||y||
r
||z||
s
for all x, y, z Î X, then f is additive.
Corollary 2.3. Let 0 <q,r,s <1, θ
1
,θ
2
> 0. If a mapping f : X ® Y with f(0) = 0 satisfies
the following functional inequality:
||f (x)+f (y)+nf (z)|| ≤

2
n
k
, such that
Kim and Son Journal of Inequalities and Applications 2011, 2011:102
/>Page 5 of 11
|
|f (x) − h(x)|| ≤
M
2
p
√
2
2

n
pq
θ
p
1
||x||
pq
n
p
− n
pq
+
θ
p
1

for all x Î X.
Noting the inequality
||f
(
nx
)
− nf
(
x
)
|| ≤ M[ϕ
(
nx,0,−x
)
+ nϕ
(
x, −x,0
)]
according to the inequalities (3) and (4), then we can similarly prove another stability
theorem under the same condition as in Theorem 2.1:
Remark 2.4. Let  : G
3
® R+ and f : G ® Y satisfy the assumptions of Theorem 2.1.
Then, there exists a unique additive mapping h :G® Y,definedby
h(x) = lim
k→∞
f (n
k
x)
n

q
+ ||y||
r
+ ||z||
s
)+θ
2
for all x,y,z in a normed space X,where0<q,r,s <1,θ
1
,θ
2
>0,thenthereexistsa
unique additive mapping h : X ® Y such that
||f (x) − h(x)|| ≤ M

(n
pq
+ n
p
)θ
p
1
||x||
pq
n
p
− n
pq
+
n

1
p
for all x Î X.
We may obtain more simple and sharp approximation than that of Theorem 2.1 for
the stability result under the oddness condition.
Remark 2.5. Let  : G
3
® R
+
and f : G ® Y satisfy the assumptions of Theorem 2.1.
Moreover, if the mapping f is odd, then there exists a unique additive mapping h : G
® Y, defined by
h(x) = lim
k→∞
f (n
k
x)
n
k
, such that
|
|f (x) − h(x)|| ≤
1
n
(nx,0,−x)
1
p
for all x Î G.
Now, we consider another stability result of functional inequality (c) in the
followings.

n
ip
ϕ

x
n
i
,
y
n
i
,
z
n
i

p
<
∞
for all x,y,z Î G. T hen, there exists a unique additive mapping h : G ® Y, defined
h(x)lim
k→∞
n
k
2

f (
x
n
k


|| ≤
M
2

ϕ

x,0,−
x
n

+ ϕ

−x,0,
x
n

,
where
g
(x)=
f (x) − f(−x)
2
, x Î G. It follows from the last inequality that



g(x) −n
m
g

i+1

p
+ ϕ

−
x
n
i
,0,
x
n
i+1

p

(11)
for all x Î G.
The remaining proof is similar to the corresponding proof of Theorem 2.1. This
completes the proof.
Suppose that X is a normed space in the following co rollaries. If we put (x,y,z):=
θ(||x||
q
||y||
r
||z||
s
) and (x,y,z):=θ(||x||
q
+||y||

functional inequality:
|
|f (x)+f (y)+nf (z)|| ≤



nf

x + y
n
+ z




+ θ
1
(||x||
q
+ ||y||
r
+ ||z||
s
)
for a ll x,y,z Î X, then there exists a unique additive mapping h : X ® Y, defined as
h(x)lim
k→∞
n
k
2

− n
p
+
||x||
ps
n
ps
− n
p

1
p
+
Mθ
1
2
(||x||
q
+ ||x||
r
)
for all x Î X.
We can similarly prove another stability theorem under somewhat different condi-
tions as follows:
Remark 2.9. Let  : G
3
® R
+
and f : G ® Y satisfy the assumptions of Theorem 2.6.
Then, there exists a unique a dditive mapping h : G ® Y,definedbyh(x)=





+ θ
1
(||x||
q
+ ||y||
r
+ ||z||
s
)
for all x,y, z in a normed space X, where q,r,s >1,θ
1
> 0, then there exists a unique
additive mapping h : X ® Y such that
|
|f (x) − h(x)|| ≤ Mθ
1

(n
pq
+ n
p
)||x||
pq
n
pq
− n

k→∞
n
k
f (
x
n
k
)
, such that
|
|f (x) − h(x)|| ≤
1
n
(nx,0,−x)
1
p
for all x Î G.
3 A lternative generalized Hyers-Ulam stability of (c)
From now on, we investigate the generalized Hyers-Ulam stability of the functional
inequality (c).
Theorem 3.1. Suppose that a mapping f : G ® Y with f(0) = 0 satisfies the functional
inequality
|
|f (x)+f (y)+nf (z)|| ≤



nf

x + y

x) − f (−n
k
x)
2
n
k
, such that
|
|f (x) − h(x)|| ≤
M
2
2n
p
√
1 − L
p
[ϕ(nx,0,−x)+ϕ(−nx,0,x)] +
M
2
ϕ(x, −x,0
)
for all x Î G.
Proof. It follows from (7) and (12) that




g(n
1
x)

k
x)]
p
≤
m−1

k
=1
M
p
L
kp
2
p
n
p
[ϕ(nx,0,−x)+ϕ(−nx,0,x)]
p
for all nonnegative integers m and l with m >l ≥ 0andx Î G,where
g
(x)=
f (x) − f(−x)
2
. Since the sequence

g(n
m
x
n
m





≤
M
2n
p
√
1 − L
p
[ϕ(nx,0,−x)+ϕ(−nx,0,x)
]
(13)
for all x Î G. It follows from (3) and (13) that
|
|f (x) − h(x)|| ≤
M
2
2n
p
√
1 − L
p
[ϕ(nx,0,−x)+ϕ(−nx,0,x)] +
M
2
ϕ(x, −x,0
)
for all x Î G.

/>Page 9 of 11
||f (x)+f (y)+nf (z)|| ≤



nf

x + y
n
+ z




+ θ
1
(||x||
r
+ ||y||
r
+ ||z||
r
)+θ
2
for all x, y, z in a normed space X,where0<r <1,θ
1
, θ
2
>0,thenthereexistsa
unique additive mapping h : X ® Y such that

n
+ z




+ ϕ(x, y, z
)
for all x,y,z Î G and there exists a constant L with 0 <L < 1 for which the perturbing
function  : G
3
® R
+
satisfies
ϕ

x
n
,
y
n
,
z
n

≤
L
n
ϕ(x, y, z
)

M
2
ϕ(x, −x,0
)
for all x Î G.
Proof. We observe that f(0) = 0 because (0,0,0) = 0, which follows from the condi-
tion
ϕ(0,0,0) ≤
L
n
ϕ(0,0,0
)
. It follows from the inequality (11) and (14) that



g(x) −n
m
g

x
n
m




p
≤
M

p
≤
M
p
2
p
n
p
m−1

i
=
0
L
(i+1)p
[ϕ(nx,0,−x)+ϕ(−nx,0,x)]
p
for all x Î G, where
g
(x)=
f (x) − f(−x)
2
, x Î G.
The remaining proof is similar to the corresponding proof of Theorem 2.1. This
completes the proof.
Remark 3.4. Let  : G
3
® R
+
and f : G ® Y satisfy the assumptions of Theorem 3.3.


x + y
n
+ z




+ θ
1
(||x||
r
+ ||y||
r
+ ||z||
r
)
Kim and Son Journal of Inequalities and Applications 2011, 2011:102
/>Page 10 of 11
for all x, y, z in a normed space X,wherer >1,θ
1
> 0, then there exists a unique
additive mapping h : X®Y such that
||f (x) − h(x)|| ≤
M
p
√
n
pr
− n

6. Găvruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl.
184, 431–436 (1994). doi:10.1006/jmaa.1994.1211
7. Jung, S: On the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl. 204, 221–226
(1996). doi:10.1006/jmaa.1996.0433
8. Rassias, TM: The stability of mappings and related topics, in ‘Report on the 27th ISFE’. Aequ Math. 39, 292–293 (1990)
9. Gajda, Z: On the stability of additive mappings. Int J Math Math Sci. 14, 431–434 (1991). doi:10.1155/
S016117129100056X
10. Rassias, TM, Semrl, P: On the behaviour of mappings which do not satisfy Hyers-Ulam-Rassias stability. Proc Am Math
Soc. 114, 989–993 (1992). doi:10.1090/S0002-9939-1992-1059634-1
11. Rassias, JM: On approximation of approximately linear mappings by linear mappings. J Funct Anal. 46, 126–130 (1982).
doi:10.1016/0022-1236(82)90048-9
12. Rassias, JM: On approximation of approximately linear mappings by linear mappings. Bull Sci Math. 108, 445–446 (1984)
13. Gao, ZX, Cao, HX, Zheng, WT, Xu, L: Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional
equations. J Math Inequal. 3(1), 63–77 (2009)
14. Moghimi, M, Najati, A, Park, C: A functional inequality in restricted domains of Banach modules. Adv Difference
Equations 14 (2009). 2009, Article ID 973709
15. Moslehian, MS, Najati, A: An application of a fixed point theorem to a functional inequality. Fixed Point Theory. 10(1),
141–149 (2009)
16. Najati, A, Zamani, EG: Stability of a mixed additive and cubic functional equation in quasi-Banach spaces. J Math Anal
Appl. 342, 1318–1331 (2008). doi:10.1016/j.jmaa.2007.12.039
17. Najati, A, Moradlou, F: Stability of a quadratic functional equation in quasi Banach spaces. Bull Korean Math Soc. 45,
587–600 (2008)
18. Najati, A, Lee, J, Park, C: On a Cauchy-Jensen functional inequality. Bull Malaysian Math Sci Soc. 33(2):253–263 (2010)
19. Benyamini, Y, Lindenstrauss, J: Geometric Nonlinear Functional Analysis. American Mathematical Society, Providence1
(2000) Colloq. Publ. vol. 48
20. Rolewicz, S: Metric Linear Spaces. PWN-Polish Scientific Publishers, Reidel, Warszawa, Dordrecht (1984)
21. Najati, A, Moghimi, MB: Stability of a functional equation deriving from quadratic and additive function in quasi-Banach
spaces. J Math Anal Appl. 337, 399–415 (2008). doi:10.1016/j.jmaa.2007.03.104
doi:10.1186/1029-242X-2011-102
Cite this article as: Kim and Son: Approximate Cauchy functional inequality in quasi-Banach spaces. Journal of