In this paper, we prove the generalized HyersUlam stability of the following cubic funtional equation
by the direct method in 2normed spaces.
AMS Mathematics Subject Classification : 39B52, 39B72, 39B82.
1. Introduction and preliminaries
G
ä
hler
[4
,
5]
has introduced the concept of 2normed spaces and G
ä
hler and White [16] introduced the concept of 2Banach spaces. Lewandowska published a series of papers on 2normed sets and generalized 2normed spaces
[10
,
11]
. Recently, Park
[12]
investigated approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2Banach spaces.
We list some definitions related to 2normed spaces.
Definition 1.1.
Let
X
be a linear space over ℝ with dim
X
> 1 and let ∥· , ·∥ :
X
×
X
→ ℝ be a function satisfying the following properties :
(1) ∥
x, y
∥ = 0 if and only if
x
and
y
are linearly dependent,
(2) ∥
x, y
∥ = ∥
y, x
∥,
(3) ∥
ax, y
∥ = 
a
∥
x, y
∥, and
(4) ∥
x, y
+
z
∥ ≤ ∥
x, y
∥ + ∥
x, z
∥
for all
x, y, z
∈
X
and
a
∈ ℝ. Then the function ∥· , ·∥ is called
a 2

norm on X
and (
X
, ∥· , ·∥) is called a
2

normed space
.
Let (
X
, ∥· , ·∥) be a 2normed space. Suppose that
x
∈
X
and ∥
x, y
∥ = 0 for all
y
∈
X
. Suppose that
x
≠ 0. Since dim
X
> 1, choose
y
∈
X
such that {
x, y
} is linearly independent and so by (1) in Definition1.1, we have ∥
x, y
∥ ≠ 0, which is a contradiction. Hence we have the following lemma.
Lemma 1.2.
Let
(
X
, ∥· , ·∥)
be a 2

normed space. If x
∈
X and
∥
x, y
∥ = 0
for all y
∈
X, then x
= 0.
Definition 1.3.
A sequence {
x_{n}
} in a 2normed space (
X
, ∥· ; ·∥) is called
a 2

Cauchy sequence
if
for all
x
∈
X
.
Definition 1.4.
A sequence {
x_{n}
} in a 2normed space (
X
, ∥· , ·∥) is called
2

convergent
if
for all
y
∈
X
and for some
x
∈
X
. In case, {
x_{n}
} said to be
converge to x
and denoted by
x_{n}
→
x as n
→ ∞ or lim
_{n→∞}
x_{n}
=
x
.
A 2normed space (
X
, ∥· , ·∥) is called a
2

Banach space
if every 2Cauchy sequence in
X
is 2convergent. Now, we state the following results as lemma
[12]
.
Lemma 1.5.
Let
(
X
, ∥· , ·∥)
be a 2

normed space. Then we have the following
:
(1) ∥
x, z
∥ − ∥
y, z
∥ ≤ ∥
x
−
y, z
∥
for all x, y, z
∈
X
,
(2)
if
∥
x, z
∥ = 0
for all z
∈
X, then x
= 0,
and
(3)
for any 2

convergent sequence
{
x_{n}
}
in X
,
for all z
∈
X
.
In 1940, S.M.Ulam [15] proposed the following stability problem :
“Let
G
_{1}
be a group and
G
_{2}
a metric group with the metric
d
. Given a constant
δ
> 0, does there exists a constant
c
> 0 such that if a mapping
f
:
G
_{1}
→
G
_{2}
satisfies
d
(
f
(
xy
);
f
(
x
)
f
(
y
)) < 0 for all
x, y
∈
G
_{1}
, then there exists a unique homomorphism
h
:
G
_{1}
→
G
_{2}
with
d
(
f
(
x
);
h
(
x
)) <
δ
for all
x
∈
G
_{1}
?”
In the next year, D. H. Hyers
[7]
gave a partial solution of Ulam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by T. Aoki
[1]
for additive mappings and by TH. M. Rassias [14] for linear mappings, to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of funtional equations have been extensively investigated by a number of mathematicians.
Rassias
[13]
introduced the cubic functional equation
and Jun and Kim
[8]
introduced the following cubic funtional equation
In this paper, we inverstigate the following cubic funtional equation
which is a linear combination of (1) and (2) and proved the generalized HyersUlam stability of (3) in 2normed spaces.
2. Stability of (3) from a normed space to a 2Banach space
Thoughout this section, (
X
, ∥·∥) or simply
X
is a real normed space and (
Y
, ∥·,·∥) or simply
Y
is a 2Banach space. We start the following theorem.
Theorem 2.1.
A mapping f
:
X
→
Y
satisfies
(3)
if and only if f is cubic
.
Proof
. Suppose that
f
satisfies (3). Letting
x
=
y
= 0 in (3), we have
f
(0) = 0 and letting
y
= 0 in (3), we have
for all
x
∈
X
. Letting
x
= 0 in (3), by (4), we have
f
(
y
) = −
f
(−
y
) for all
y
∈
X
and so
f
is odd. Letting
y
= −
y
in (3), we have
for all
x, y
∈
X
and by (3) and (5), we have
for all
x, y
∈
X
. Hence by (3) and (6), we have
for all
x, y
∈
X
. Interching
x
and
y
in (7), since
f
is odd,
f
satisfies (2) and hence
f
is cubic.
For any function
f
:
X
→
Y
, we define the difference operator
D_{f}
:
X
×
X
→
Y
by
Now we prove the generalized HyersUlam stability of (3).
Theorem 2.2.
Let ε
≥ 0,
p and q be positive real numbers with p
+
q
< 3
and r
> 0.
Suppose that f
:
X
→
Y is a function such that
for all x, y
∈
X and z
∈
Y. Then there exists a unique cubic function C
:
X
→
Y satisfying
(3)
and
for all x
∈
X and z
∈
Y
.
Proof
. Letting
x
=
y
= 0 in (8), we have ∥2
f
(0);
z
∥ = 0 for all
z
∈
Y
and by the definition of 2norm, we have
f
(0) = 0. Putting
y
= 0 in (8), we have
for all
x
∈
X
and
z
∈
Y
and so
for all
x
∈
X
and
z
∈
Y
. Replacing
x
by 2
x
in (11), we get
for all
x
∈
X
and
z
∈
Y
. By (11) and (12), we get
for all
x
∈
X
and
z
∈
Y
. By induction on
n
, we can show that
for all
x
∈
X
and
z
∈
Y
. For
m, n
∈ ℕ with
n
<
m
and
x
∈
X
, by (13), we have
Since
is a 2 Cauchy sequence in
Y
for all
x
∈
X
. Since
Y
is a 2Banach space, the sequence
is a 2convergent in
Y
for all
x
∈
X
and so we can define a mapping
C
:
X
→
Y
as
for all
x
∈
X
. By (14), we have
for all
x
∈
X
and
z
∈
Y
and by Lemma 1.5 , we have
for all
x
∈
X
and
z
∈
Y
. Next we will show that C satisfies (3). By (8), we have
for all
z
∈
Y
, because
p
< 3,
q
< 3,
p
+
q
< 3 and so
D_{C}
(
x, y
) = 0 for all
x, y
∈
X
. By Theoem 2.1,
C
is cubic.
To show that
C
is unique, suppose there exists another cubic function
C′
:
X
→
Y
which satisfies (3) and (9). Since
C
and
C′
are cubic,
for all
x
∈
X
. It follows that
So ∥
C′
(
x
)−
C
(
x
),
z
∥= 0 for all
z
∈
Y
and hence
C′
(
x
) =
C
(
x
) for all
x
∈
X
.
Related with Theorem 2.2, we can also the following theorem.
Theorem 2.3.
Let ε
≥ 0,
p and q be positive real numbers with p, q
> 3
and r
> 0.
Suppose that f
:
X
→
Y is a function satisfying
(8).
Then there exists a unique cubic function C
:
X
→
Y satisfying
(3)
and
for all x
∈
X and z
∈
Y
.
Proof
. Letting
x
=
y
= 0 in (8), we have ∥2
f
(0),
z
∥ = 0 for all
z
∈
Y
and so we have
f
(0) = 0. Putting
y
= 0 and replacing
x
by
in (8), we get
for all
x
∈
X
and
z
∈
Y
and so
for all
x
∈
X
and
z
∈
Y
. Replacing
x
by
in (16), we get
for all
x
∈
X
and
z
∈
Y
. By (16) and (17), we get
for all
x
∈
X
and
z
∈
Y
. By induction on
n
, we can show that
for all
x
∈
X
and
z
∈
Y
. For
m, n
∈ ℕ with
n
<
m
and
x
∈
X
, by (18), we have
and since
p
> 3,
is a 2 Cauchy sequence in
Y
for all
x
∈
X
. Since
Y
is a 2Banach space, the sequence
is a 2convergent in
Y
for all
x
∈
X
. Define
C
:
X
→
Y
as
for all
x
∈
X
. By (18), we have
for all
x
∈
X
and
z
∈
Y
and by Lemma 1.5, we have
for all
x
∈
X
and
z
∈
Y
. Next we will show that
C
satisfies (3).
for all
z
∈
Y
, because
p, q
> 3 and so
D_{C}
(
x, y
) = 0 for all
x, y
∈
X
. By Theoem 2.1,
C
is cubic.
To show that
C
is unique, suppose there exists another cubic function
C′
:
X
→
Y
which satisfies (3) and (15). Since
C
and
C′
are cubic,
for all
x
∈
X
. It follows that
So ∥
C′
(
x
)−
C
(
x
),
z
∥= 0 for all
z
∈
Y
and hence
C′
(
x
) =
C
(
x
) for all
x
∈
X
.
3. Stability of (3) from a 2 normed space to a 2Banach space
In this section, we study similar problems of (3). Let (
X
, ∥· , ·∥) be a 2normed space and (
Y
, ∥· , ·∥) a 2 Banach space.
Theorem 3.1.
Let ε
≥ 0
and p and q be positive real numbers with p
+
q
< 3.
Suppose that f
:
X
→
Y is a function such that
for all x, y
∈
X and z
∈
Y. Then there exists a unique cubic function C
:
X
→
X satisfying
(3)
and
for all x
∈
X and z
∈
Y
.
Proof
. Letting
x
=
y
= 0 in (19). We have ∥2
f
(0),
z
∥ = 0 for all
z
∈
Y
, so we have
f
(0) = 0. Putting
y
= 0 in (19), we have
for all
x
∈
X
and
z
∈
Y
. Therefore
for all
x
∈
X
and
z
∈
Y
. Replacing
x
by 2
x
in (21), we get
for all
x
∈
X
and
z
∈
Y
. By induction on
n
, we can show that
for all
x
∈
X
and
z
∈
Y
. For
m, n
∈ ℕ with
n
<
m
and
x
∈
X
, by (22), we get
for all
x
∈
X
and
z
∈
Y
. Since
p
< 3,
is a 2 Cauchy sequence in
Y
for all
x
∈
X
. Since
Y
is a 2Banach space, the sequence
is a 2convergent in
Y
for all
x
∈
X
. Define
C
:
X
→
Y
as
for all
x
∈
X
and by Lemma 1.5 and (22), we have (20). Next we show that
C
satisfies (3). By (19), we have
for all
z
∈
Y
and so
D_{C}
(
x, y
) = 0 for all
x, y
∈
X
. By Theoem 2.1,
C
is cubic.
To show that
C
is unique, suppose that there exists another cubic function
C′
:
X
→
Y
which satisfies (3) and (20). Since
C
and
C′
are cubic,
for all
x
∈
X
. Since
p
< 3,
So ∥
C′
(
x
)−
C
(
x
),
z
∥= 0 for all
z
∈
Y
and hence
C′
(
x
) =
C
(
x
) for all
x
∈
X
.
Similar to Theorem 3.1, we have the following theorem.
Theorem 3.2.
Let
(
X
, ∥· , ·∥)
be a 2

Banach space. Let ε
≥ 0,
p and q be positive real numbers with p, q
> 3.
Suppose that f
:
X
→
X is a function satisfying
(19).
Then there exists a unique cubic function C
:
X
→
X satisfying
(3)
and
for all x, z
∈
X
.
BIO
Chang Il Kim received M.Sc. from Sogang Uninversity and Ph.D at Sogang Uninersity. Since 1993 he has been at Dankook University. His research interests include general topology and functional analysis.
Department of Mathematics Education, Dankook University, 152, Jukjeon, Suji, Yongin, Gyeonggi, 448701, Korea.
email:kci206@hanmail.net
Kap Hun Jung received M.Sc. and Ph.D. from Dankook University. He is now teaching at Seoul National University of Science and Technology as a lecturer. His research interests include functional analysis.
School of Liberal Arts, Seoul National University of Science and Technology, Seoul 139743, Korea.
email:jkh58@hanmail.net
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