In this paper, using the fixed point method, we investigate the generalized HyersUlam stability of the system of quintic functional equation
in nonArchimedean 2Banach spaces.
AMS Mathematics Subject Classification : 39B82, 46S10.
1. Introduction and preliminaries
In 1940, Ulam
[22]
posed the following problem concerning the stability of functional equations:
Let G
_{1}
be a group and let G
_{2}
be a metric group with the metric d
(·, ·).
Given ϵ
> 0,
does there exist a δ
> 0
such that if a mapping h
:
G
_{1}
→
G
_{2}
satisfies the inequality d
(
h
(
xy
),
h
(
x
)
h
(
y
)) <
δ for all x, y
∈
G
_{1}
,
then there exists a homomorphism H
:
G
_{1}
→
G
_{2}
with d
(
h
(
x
),
H
(
x
)) <
ϵ for all x
∈
G
_{1}
?
Hyers
[8]
solved the Ulam's problem for the case of approximately additive functions in Banach spaces. Since then, the stability of several functional equations has been extensively investigated by several mathematicians
[3
,
5
,
9
,
10
,
11
,
14
,
17]
. The HyersUlam stability for the quadratic functional equation
was proved by Skof
[21]
for a function
f
:
E
_{1}
→
E
_{2}
, where
E
_{1}
is a normed space and
E
_{2}
is a Banach space and later by Jung
[13]
on unbounded domains.
Rassias
[20]
investigated the stability for the following cubic functional equation
and Jun and Kim
[12]
investigated the stability for the following cubic funtional equation
A
valuation
is a function  ·  from a field K into [0,∞) such that for any
r
,
s
∈ K, the following conditions hold: (i) 
r
 = 0 if and only if
r
= 0, (ii) 
rs
 = 
r
║
s
, and (iii) 
r
+
s
 ≤ 
r
 + 
s
. A field K is called
a valued field
if K carries a valuation. The usual absolute values of ℝ and ℂ are examples of valuations. If the triangle inequality is replaced by 
r
+
s
 ≤
max
{
r
, 
s
} for all
r
,
s
∈ K, then the valuation  ·  is called a
nonArchimedean valuation
and the field with a nonArchimedean valuation is called a
nonArchimedean field
. If  ·  is a nonArchimedean valuation on K, then clearly, 1 =  − 1 and 
n
 ≤ 1 for all
n
∈ ℕ.
Definition 1.1.
Let
X
be a vector space over a nonArchimedean field K. A function ║ · ║ :
X
→ ℝ is called
a nonArchimedean norm
if it satisfies the following conditions:
(a) ║
x
║ = 0 if and only if
x
= 0,
(b) ║
rx
║ = 
r
║
x
║, and
(c) ║
x
+
y
║ ≤
max
{║
x
║, ║
y
║} for all
x
,
y
∈
X
and all
r
∈ K.
If ║ · ║ is a nonArchimedean norm, then (
X
, ║ · ║) is called
a nonArchimedean normed space
.
Let (
X
, ║ · ║) be a nonArchimedean normed space and {
x_{n}
} a sequence in
X
. Then {
x_{n}
} is said to be
convergent
in (
X
, ║ · ║) if there exists an
x
∈
X
such that lim
_{n→∞}
║
x_{n}
−
x
║ = 0. In case,
x
is called
the limit of the sequence
{
x_{n}
}, and one denotes it by lim
_{n→∞}
x_{n}
=
x
. A sequence {
x_{n}
} is said to be
Cauchy
in (
X
, ║ · ║) if lim
_{n→∞}
║
x_{n+p}
−
x_{n}
║ = 0 for all
p
∈ ℕ. By (c) in Definition 1.1,
a sequence {
x_{n}
} is Cauchy in (
X
, ║ · ║) if and only if {
x
_{n+1}
−
x_{n}
} converges to zero in (
X
, ║ · ║). By
a complete nonArchimedean space
we mean one in which every Cauchy sequence is convergent.
G
ä
hler
[6
,
7]
has introduced the concept of 2normed spaces and White
[23]
introduced the concept of 2Banach spaces. In 1999 to 2003, Lewandowska published a series of papers on 2normed sets and generalized 2normed spaces
[15
,
16]
.
Definition 1.2.
Let
X
be a linear space over a nonArchimedean field K with dim
X
> 1 and ║·, ·║ :
X
×
X
→ ℝ a function satisfying the following properties
(
N
A1) ║
x
,
y
║ = 0 if and only if
x
and
y
are linearly dependent,
(
N
A2) ║
x
,
y
║ = ║
y
,
x
║,
(
N
A3) ║
x
,
ay
║ = 
a
║
x
,
y
║ , and
(
N
A4) ║
x
,
y
+
z
║ ≤
max
{║
x
,
y
║, ║
x
,
z
║}
for all
x
,
y
,
z
∈
X
and all
a
∈ K. Then ║· , ·║ is called
a nonArchimedean 2norm
and (
X
, ║· , ·║) is called a
nonArchimedean 2normed spaces
.
Definition 1.3.
A sequence {
x_{n}
} in a nonArchimedean 2normed space (
X
, ║· , ·║) is called
a Cauchy sequence
if
for all
x
∈
X
.
Definition 1.4.
A sequence {
x_{n}
} in a nonArchimedean 2normed space (
X
, ║· , ·║) is called
convergent
if
for all
y
∈
X
and for some
x
∈
X
. In case,
x
is called
the limit of the sequence
{
x_{n}
}, and we denoted by
x_{n}
→
x as n
→ ∞ or lim
_{n→∞}
x_{n}
=
x
.
Let {
x_{n}
} be a sequence in a nonArchimedean 2normed space (
X
, ║· , ·║). It follows from (NA4) that
for all
y
∈
X
and so a sequence {
x_{n}
} is a Cauchy sequence in (
X
, ║· , ·║) if and only if {
x
_{m+1}
−
x_{m}
} converges to zero in (
X
, ║· , ·║).
A nonArchimedean 2normed space (
X
, ║· , ·║) is called a
nonArchimedean 2Banach space
if every Cauchy sequence in (
X
, ║· , ·║) is convergent. Now, we state the following results as lemma
[18]
.
Lemma 1.5.
Let
(
X
, ║· , ·║)
be a nonArchimedean 2normed space. Then we have the following
:
(1) ║
x
,
z
║ − ║
y
,
z
║ ≤ ║
x
−
y
,
z
║
for all x
,
y
,
z
∈
X
,
(2) ║
x
,
z
║ = 0 for all
z
∈
X
if and only if x
= 0,
and
(3)
for any convergent sequence
{
x_{n}
}
in
(
X
, ║· , ·║),
for all z
∈
X
.
In 2003, Radu
[19]
proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also
[1
,
2]
).
We recall the following theorem by Margolis and Diaz.
Theorem 1.6
(
[4]
).
Let
(
X
,
d
)
be a complete generalized metric space and let J
:
X
→
X be a strictly contractive mapping with some Lipschitz constant L with
0 <
L
< 1.
Then for each given element x
∈
X, either d
(
J^{n}x
,
J
^{n+1}
x
) = ∞
for all nonnegative integers n or there exists a positive integer n
_{0}
such that
(1)
d
(
J^{n}x
,
J
^{n+1}
x
) < ∞
for all n
≥
n
_{0}
(2)
the sequence
{
J^{n}x
}
converges to a fixed point y
^{∗}
of J
(3)
y
^{∗}
is the unique fixed point of J in the set Y
= {
y
∈
X

d
(
J
^{n0}
x
,
y
) < ∞}
(4)
d
(
y
,
y
^{∗}
) ≤
for all y
∈
Y
.
In this paper, we investigate the following cubic functional equation
and using fixed point method, we inverstigate the generalized HyersUlam stability for the system of the quintic functional equation
and prove the generalized HyersUlam stability for (3) in nonArchimedean 2Banach spaces. In this paper, we will assume that (
X
, ║·║) is a a nonArchimedean norm space and (
Y
, ║· , ·║) is a nonArchimedean 2Banach space.
2. Stability of quintic mappings
In this section, using the fixed point method, we investigate the generalized HyersUlam stability for the system of quintic functional equation (3) in nonArchimedean 2Banach spaces. We start the following lemma.
Lemma 2.1.
Let f
:
X
→
Y be a mapping with
(2).
Then f is a cubic mapping
.
Proof
. Suppose that
f
satisfies (2). Letting
x
=
y
= 0 in (2), we have
f
(0) = 0 and letting
y
= 0 in (2), we have
for all
x
∈
X
. Letting
x
= 0 in (2), by (4), we have
f
(
y
) = −
f
(−
y
) for all
y
∈
X
and so
f
is odd. Letting
y
= −
y
in (2), we have
for all
x
,
y
∈
X
and by (2) and (5), we have
for all
x
,
y
∈
X
. Interching
x
and
y
in (6), since
f
is odd,
f
satisfies (1) and hence
f
is cubic. □
The function
f
: ℝ × ℝ → ℝ given by
f
(
x
,
y
) =
cx
^{2}
y
^{3}
is a solution of (3). In partcular, letting
y
=
x
in (3), we get a quintic function
g
: ℝ → ℝ in one variable given by
g
(
x
) =
f
(
x
,
x
) =
cx
^{5}
.
Proposition 2.2.
If a mapping f
:
X
^{2}
→
Y satisfies
(3),
then f
(
λx
,
μy
) =
λ
^{2}
μ
^{3}
f
(
x
,
y
)
for all x
,
y
∈
X and all rational numbers λ
,
μ
.
Theorem 2.3.
Let ϕ
_{1}
,
ϕ
_{2}
:
X
^{3}
×
Y
→ [0,∞)
be functions such that
for all x
,
y
,
z
∈
X
,
w
∈
Y and some L with
0 <
L
< 1.
Suppose that f
:
X
^{2}
→
Y is a mapping such that f
(
x
, 0) =
f
(0,
x
) = 0
for all x
∈
X
,
and
for all w
∈
Y and all x
,
y
,
x
_{1}
,
x
_{2}
,
y
_{1}
,
y
_{2}
∈
X
.
Then there exists a unique quintic mapping T
:
X
^{2}
→
Y
satisfying
(3)
and
for all w
∈
Y and all x
,
y
∈
X
,
where
Proof
. Putting
y
_{2}
= 0 and
y
_{1}
=
y
in (9), we get
for all
w
∈
Y
and all
x
,
y
∈
X
. Putting
x
_{1}
=
x
_{2}
=
x
in (8), we get
for all
w
∈
Y
and all
x
,
y
∈
X
. Thus by (11) and (12), we have
for all
w
∈
Y
and all
x
,
y
∈
X
. It follows from (13) that
for all
w
∈
Y
and all
x
,
y
∈
X
.
Consider the set
S
= {
h

h
:
X
×
X
→
Y with h
(
x
, 0) =
h
(0,
x
) = 0, ∀
x
∈
X
} and the generalized metric
d
on
S
defined by
Then (
S
,
d
) is a complete metric space
[2]
. Define a mapping
J
:
S
→
S
by
Jg
(
x
,
y
) = 2
^{−5}
g
(2
x
, 2
y
) for all
x
,
y
∈
X
and all
g
∈
S
. Let
g
,
h
∈
S
and
d
(
g
,
h
) ≤
ε
for some nonnegative real number
ε
. Then by (7), we have
and so
d
(
Jg
,
Jh
) ≤
εL
. This mean that
d
(
Jg
,
Jh
) ≤
Ld
(
g
,
h
) for all
g
,
h
∈
S
and so
J
is a strictly contractive mapping. By (14), we get
d
(
Jf
,
f
) ≤ 1 < ∞. By Theorem 1.6, there exists a mapping
T
:
X
^{2}
→
Y
which is a fixed point of
J
such that
d
(
J^{n}f
,
T
) → 0 as
n
→ ∞, which implies the equality
T
(
x
,
y
) = lim
_{n→∞}
2
^{−5n}
f
(2
^{n}x
, 2
^{n}y
). Since
d
(
Jf
,
f
) ≤ 1 < ∞, by (4) in Theorem 1.6, we have (10). By (8) and (9), we get
and
for all
w
∈
Y
and all
x
,
y
,
x
_{1}
,
x
_{2}
,
y
_{1}
,
y
_{2}
∈
X
. Hence
T
satisfies (3).
To prove the uniquness of
T
, assume that
T
_{1}
:
X
^{2}
→
Y
is another solution of (3) satisfying (10). Then
T
_{1}
is a fixed point of
J
and by (10),
By (3) in Theorem 1.6, we have
T
=
T
_{1}
. □
Theorem 2.4.
Let
ϕ
_{1}
,
ϕ
_{2}
:
X
^{3}
×
Y
→ [0,∞)
be functions such that
for all x
,
y
,
z
∈
X
,
w
∈
Y and some L with
0 <
L
< 1.
Suppose that f
:
X
^{2}
→
Y is a mapping satisfying f
(
x
, 0) =
f
(0,
x
) = 0
for all x
∈
X
, (8)
and
(9).
Then there exists a unique quintic mapping T
:
X
^{2}
→
Y satisfying
(3)
and
for all w
∈
Y and all x
,
y
∈
X
,
where
Proof
. Putting
y
_{2}
= 0 and
in (9), we get
for all
w
∈
Y
and all
x
,
y
∈
X
. Putting
in (8), we get
for all
w
∈
Y
and all
x
,
y
∈
X
. Thus by (17) and (18), we have
for all
x
,
y
∈
X
and all
w
∈
Y
. That is, we have
for all
x
,
y
∈
X
and all
w
∈
Y
.
Consider the set
S
= {
h

h
:
X
×
X
→
Y with h
(
x
, 0) =
h
(0,
x
) = 0, ∀
x
∈
X
} and the generalized metric
d
on
S
defined by
Then (
S
,
d
) is a complete metric space(
[2]
). Define a mapping
J
:
S
→
S
by
for all
x
,
y
∈
X
and all
g
∈
S
. Let
g
,
h
∈
S
and
d
(
g
,
h
) ≤
ε
for some nonnegative real number
ε
. Then by (15), we have
and so
d
(
Jg
,
Jh
) ≤
εL
. This mean that
d
(
Jg
,
Jh
) ≤
L d
(
g
,
h
) for all
g
,
h
∈
S
and so
J
is a strictly contractive mapping. By (19), we get
d
(
Jf
,
f
) ≤
L
< ∞. By Theorem 1.6, there exists a mapping
T
:
X
^{2}
→
Y
which is a fixed point of
J
such that
d
(
J^{n}f
,
T
) → 0 as
n
→ ∞, which imples the equality
T
(
x
,
y
) = lim
_{n→∞}
. Since
d
(
Jf
,
f
) ≤
L
, by (4) in Theorem 1.6, we have (16) and by (8) and (9), we get
and
for all
w
∈
Y
and all
x
,
y
,
x
_{1}
,
x
_{2}
,
y
_{1}
,
y
_{2}
∈
X
. Hence
T
satisfies (3).
To prove the uniquness of
T
, assume that
T
_{1}
:
X
^{2}
→
Y
is another solution of (3) satisfying (16). Then
T
_{1}
is a fixed point of
J
and by (16),
By (3) in Theorem 1.6, we have
T
=
T
_{1}
. □
As example of
ϕ
_{1}
(
x
_{1}
,
x
_{2}
,
y
,
w
) and
ϕ
_{2}
(
x
,
y
_{1}
,
y
_{2}
,
w
) in Theorem 2.3 and Theorem 2.4, we can take
ϕ
_{1}
(
x
_{1}
,
x
_{2}
,
y
,
w
) =
θ
(║
x
_{1}
║
^{p}
+ ║
x
_{2}
║
^{p}
+ ║
y
║
^{p}
)║
w
║ and
ϕ
_{2}
(
x
,
y
_{1}
,
y
_{2}
,
w
) = 2
^{4}
θ
(║
x
║
^{p}
+║
y
_{1}
║
^{p}
+║
y
_{2}
║
^{p}
)║
w
║ for all
x
,
y
,
x
_{1}
,
x
_{2}
,
y
_{1}
,
y
_{2}
∈
X
, all
w
∈
Y
and some positive real number
θ
. Then we have the following corollary.
Corollary 2.5.
Let θ
,
p be positive real numbers with p
≠ 5.
Suppose that f
:
X
^{2}
→
Y is a mapping satisfying f
(
x
, 0) =
f
(0,
x
) = 0,
and
for all w
∈
Y and all x
,
y
,
x
_{1}
,
x
_{2}
,
y
_{1}
,
y
_{2}
∈
X
.
Then there exists a unique quintic mapping T
:
X
^{2}
→
Y satisfying
for all w
∈
Y and x
,
y
∈
X
.
Proof
. Let
ϕ
_{1}
(
x
_{1}
,
x
_{2}
,
y
,
w
) =
θ
(║
x
_{1}
║
^{p}
+║
x
_{2}
║
^{p}
+║
y
║
^{p}
)║
w
║ and
ϕ
_{2}
(
x
,
y
_{1}
,
y
_{2}
,
w
) = 2
^{4}
θ
(║
x
║
^{p}
+ ║
y
_{1}
║
^{p}
+ ║
y
_{2}
║
^{p}
)║
w
║. Note that
So if
p
> 5, by Theorem 2.3, we have (20). Note that
So if
p
< 5, by Theorem 2.4, we have (20). □
As another example of
ϕ
_{1}
(
x
,
y
,
z
,
w
) and
ϕ
_{2}
(
x
,
y
,
z
,
w
) in Theorem 2.3 and Theorem 2.4, we can take
ϕ
_{1}
(
x
,
y
,
z
,
w
) =
ϕ
_{2}
(
x
,
y
,
z
,
w
) =
θ
║
x
║
^{p}
║
y
║
^{q}
║
z
║
^{r}
║
w
║ for all
x
,
y
,
z
∈
X
, all
w
∈
Y
and some positive real number
p
,
q
,
r
,
θ
. Then we have the following corollary:
Corollary 2.6.
Let p
,
q
,
r and θ be positive real numbers with p
+
q
+
r
≠ 5.
Suppose that f
:
X
^{2}
→
Y is a mapping satisfying f
(
x
, 0) = 0,
and
for all w
∈
Y and all x
,
y
,
x
_{1}
,
x
_{2}
,
y
_{1}
,
y
_{2}
∈
X
.
Then there exists a unique quintic mapping T
:
X
^{2}
→
Y satisfying
for all w
∈
Y and all x
,
y
∈
X
.
Proof
. Let
ϕ
_{1}
(
x
,
y
,
z
,
w
) =
ϕ
_{2}
(
x
,
y
,
z
,
w
) =
θ
║
x
║
^{p}
║
y
║
^{q}
║
z
║
^{r}
║
w
║. Then we have
Hence if
p
+
q
+
r
> 5, by Theorem 2.3, we have (21). Note that
Thus if
p
+
q
+
r
< 5, by Theorem 2.4, we have (21). □
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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