FUZZY STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION WITH THE FIXED POINT ALTERNATIVE

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2015.
Aug,
22(3):
285-298

- Received : July 06, 2015
- Accepted : July 21, 2015
- Published : August 31, 2015

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In
[41]
, Th.M. Rassias proved that the norm defined over a real vector space
V
is induced by an inner product if and only if for a fixed positive integer
l
holds for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
V
. For the above equality, we can define the following functional equation
Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces.
is called a
quadratic functional equation
. In particular, every solution of the quadratic functional equation is said to be a
quadratic mapping
. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof
[50]
for mappings
f
:
X
→
Y
, where
X
is a normed space and
Y
is a Banach space. Cholewa
[8]
noticed that the theorem of Skof is still true if the relevant domain
X
is replaced by an Abelian group. Czerwik
[9]
proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see
[14
,
16
,
17
,
31
,
32
,
33
,
34
,
35
,
37
,
38
,
39
,
42
,
43
,
44
,
45
,
46
,
47
,
48
,
49]
).
In
[33]
, Park, Lee and Shin proved that an even mapping
f
:
V
→
W
satisfies the functional equation (0.1) if and only if the even mapping
f
:
V
→
W
is quadratic. Moreover, they proved the Hyers-Ulam stability of the quadratic functional equation (0.1) in real Banach spaces.
Katsaras
[18]
defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view
[11
,
20
,
52]
. In particular, Bag and Samanta
[2]
, following Cheng and Mordeson
[7]
, gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type
[19]
. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces
[3]
.
We use the definition of fuzzy normed spaces given in
[2
,
24
,
25]
to investigate a fuzzy version of the Hyers-Ulam stability for the functional equation (0.1) in the fuzzy normed vector space setting.
Definition 1.1
(
[2
,
24
,
25
,
26]
). Let
X
be a real vector space. A function
N
:
X
× ℝ → [0, 1] is called a
fuzzy norm
on
X
if for all
x
,
y
∈
X
and all
s
,
t
∈ ℝ,
The pair (
X
,
N
) is called a
fuzzy normed vector space
.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in
[24
,
27]
.
Definition 1.2
(
[2
,
24
,
25
,
26]
). Let (
X
,
N
) be a fuzzy normed vector space. A sequence {
x
_{n}
} in
X
is said to
be convergent
or
converge
if there exists an
x
∈
X
such that lim
_{n→∞}
N
(
x
_{n}
−
x
,
t
) = 1 for all
t
> 0. In this case,
x
is called the
limit
of the sequence {
x
_{n}
} and we denote it by
N
-lim
_{n→∞}
x
_{n}
=
x
.
Definition 1.3
(
[2
,
24
,
25]
). Let (
X
,
N
) be a fuzzy normed vector space. A sequence {
x
_{n}
} in
X
is called
Cauchy
if for each
ε
> 0 and each
t
> 0 there exists an
n
_{0}
∈ ℕ such that for all
n
≥
n
_{0}
and all
p
> 0, we have
N
(
x
_{n+p}
−
x
_{n}
,
t
) > 1 −
ε
.
It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be
complete
and the fuzzy normed vector space is called a
fuzzy Banach space
.
We say that a mapping
f
:
X
→
Y
between fuzzy normed vector spaces
X
and
Y
is continuous at a point
x
_{0}
∈
X
if for each sequence {
x
_{n}
} converging to
x
_{0}
in
X
, then the sequence {
f
(
x
_{n}
)} converges to
f
(
x
_{0}
). If
f
:
X
→
Y
is continuous at each
x
∈
X
, then
f
:
X
→
Y
is said to be
continuous
on
X
(see
[3]
).
Let
X
be a set. A function
d
:
X
×
X
→ [0, ∞] is called a
generalized metric
on
X
if
d
satisfies
Theorem 1.4
(
[4
,
10]
).
Let
(
X
,
d
)
be a complete generalized metric space and let
J
:
X
→
X
be a strictly contractive mapping with Lipschitz constant
L
< 1.
Then for each given element
x
∈
X
,
either
for all nonnegative integers n or there exists a positive integer
n
_{0}
such that
In 1996, G. Isac and Th.M. Rassias
[15]
were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see
[5
,
6
,
27
,
29
,
30
,
36]
).
Starting with the paper
[24]
, the stability of some functional equations in the framework of fuzzy normed spaces or random normed spaces has been investigated (see e.g.,
[21
,
22
,
23
,
24
,
25
,
26
,
27
,
28]
).
This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an even case.
Throughout this paper, assume that
X
is a vector space and that (
Y
,
N
) is a fuzzy Banach space. Let
l
be a fixed positive integer.
f
:
X
→
Y
, we define
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
.
Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation
C
f
(
x
_{1}
, ⋯ ,
x
_{2l}
) = 0 in fuzzy Banach spaces: an odd case.
Theorem 2.1
.
Let
φ
:
X
^{2l}
→ [0,∞)
and
be functions such that there exists an
L
< 1
with
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
.
Let f
:
X
→
Y
be an odd mapping satisfying
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
and all t
> 0.
Then
exists for each x
∈
X
and defines an additive mapping
A
:
X
→
Y
such that
for all x
∈
X
and all t
> 0.
Proof
.
Letting
x
_{1}
= ⋯ =
x
_{l}
=
x
and
x
_{l+1}
= ⋯ =
x
_{2l}
= 0 in (2.1), we get
for all
x
∈
X
.
Consider the set
and introduce the generalized metric on
S
:
where, as usual, inf
ϕ
= +∞. It is easy to show that (
S
,
d
) is complete. (See the proof of Lemma 2.1 of
[22]
.)
Now we consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
Let
g
,
h
∈
S
be given such that
d
(
g
,
h
) =
ε
. Then
for all
x
∈
X
and all
t
> 0. Hence
for all
x
∈
X
and all
t
> 0. So
d
(
g
,
h
) =
ε
implies that
d
(
Jg, Jh
) ≤
L
ε
. This means that
for all
g, h
∈
S
.
It follows from (2.3) that
By Theorem 1.4, there exists a mapping
A
:
X
→
Y
satisfying the following:
(1)
A
is a fixed point of
J
, i.e.,
for all
x
∈
X
. Since
f
:
X
→
Y
is odd,
A
:
X
→
Y
is an odd mapping. The mapping
A
is a unique fixed point of
J
in the set
This implies that
A
is a unique mapping satisfying (2.4) such that there exists a
µ
∈ (0, ∞) satisfying
for all
x
∈
X
;
(2)
d
(
J
^{n}
f, A
) → 0 as
n
→ ∞. This implies the equality
for all
x
∈
X
;
(3)
, which implies the inequality
This implies that the inequality (2.2) holds.
By (2.1),
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
, all
t
> 0 and all
n
∈ ℕ. So
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
, all
t
> 0 and all
n
∈ ℕ. Since
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
and all
t
> 0,
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
and all
t
> 0. Thus
CA
(
x
_{1}
, ⋯ ,
x
_{2l}
) = 0. Since
A
is odd, it follows from Lemma 2.1 of
[35]
that the mapping
A
:
X
→
Y
is additive, as desired. ☐
Corollary 2.2.
Let
θ
≥ 0
and let p be a real number with p
> 1.
Let X be a normed
vector space with norm
∥ · ∥.
Let f
:
X
→
Y
be an odd mapping satisfying
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
and all t
> 0.
Then
exists for each x
∈
X
and defines an additive mapping A
:
X
→
Y
such that
for all x
∈
X
and all t
> 0.
Proof
. The proof follows from Theorem 2.1 by taking
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
. Then we can choose
L
= 2
^{1−p}
and we get the desired result. ☐
Theorem 2.3.
Let
φ
:
X
^{2l}
→ [0,∞)
and
be functions
such that there exists an L
< 1
with
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
.
Let f
:
X
→
Y
be an odd mapping satisfying
(2.1).
Then
exists for each
x
∈
X
and defines an additive mapping
A
:
X
→
Y
such that
for all x
∈
X
and all t
> 0.
Proof
. Let (
S
,
d
) be the generalized metric space defined in the proof of Theorem 2.1.
Consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
It follows from (2.3) that
for all
x
∈
X
and all
t
> 0. Thus
for all
x
∈
X
and all
t
> 0. So
.
The rest of the proof is similar to the proof of Theorem 2.1. ☐
Corollary 2.4.
Let
θ
≥ 0
and let p be a real number with
0 <
p
< 1.
Let X be a normed vector space with norm
∥ · ∥.
Let f
:
X
→
Y
be an odd mapping satisfying
(2.5).
Then
exists for each x
∈
X
and defines an additive mapping
A
:
X
→
Y
such that
for all
x
∈
X
and all t
>0.
Proof
. The proof follows from Theorem 2.3 by taking
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
. Then we can choose
L
= 2
^{p−1}
and we get the desired result. ☐
Cf
(
x
_{1}
, ⋯ ,
x
_{2l}
) = 0 in fuzzy Banach spaces: an even case.
Theorem 3.1.
Let
φ
:
X
^{2l}
→ [0,∞)
and
be functions
such that there exists an L
< 1
with
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
.
Let f
:
X
→
Y
be an even mapping satisfying f
(0) = 0
and
(2.1).
Then
exists for each x
∈
X
and defines a quadratic mapping
Q
:
X
→
Y
such that
for all
x
∈
X
and all t
> 0.
Proof
. Letting
x
_{1}
= ⋯ =
x
_{l}
=
x
and
x
_{l+1}
= ⋯ =
x
_{2l}
= 0 in (2.1), we get
for all
x
∈
X
.
Let (
S
,
d
) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
Let
g, h
∈
S
be given such that
d(g, h)
=
ε
. Then
for all
x
∈
X
and all
t
> 0. Hence
for all
x
∈
X
and all
t
> 0. So
d
(
g, h
) =
ε
implies that
d
(
Jg, Jh
) ≤
L
ε
. This means that
for all
g, h
∈
S
.
It follows from (3.2) that
By Theorem 1.4, there exists a mapping
Q
:
X
→
Y
satisfying the following:
(1)
Q
is a fixed point of
J
, i.e.,
for all
x
∈
X
. Since
f
:
X
→
Y
is even,
Q
:
X
→
Y
is an even mapping. The mapping
Q
is a unique fixed point of
J
in the set
This implies that
Q
is a unique mapping satisfying (3.3) such that there exists a
µ
∈ (0, ∞) satisfying
for all
x
∈
X
;
(2)
d
(
J
^{n}
f
,
Q
) → 0 as
n
→ ∞. This implies the equality
for all
x
∈
X
;
(3)
, which implies the inequality
This implies that the inequality (3.1) holds.
The rest of the proof is similar to the proof of Theorem 2.1. ☐
Corollary 3.2.
Let
θ
≥ 0
and let p be a real number with p
> 2.
Let X be a normed vector space with norm
∥ · ∥.
Let f
:
X
→
Y be an even mapping satisfying f
(0) = 0
and
(2.5).
Then
exists for each x
∈
X
and defines a quadratic mapping
Q
:
X
→
Y
such that
for all
x
∈
X
and all t
> 0.
Proof
. The proof follows from Theorem 3.1 by taking
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
. Then we can choose
L
= 2
^{2−p}
and we get the desired result. ☐
Similarly, we can obtain the following. We will omit the proof.
Theorem 3.3.
Let
φ
:
X
^{2l}
→ [0,∞)
and
be functions
such that there exists an L
< 1
with
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
.
Let
f
:
X
→
Y
be an even mapping satisfying f
(0) = 0
and
(2.1).
Then
exists for each x
∈
X
and defines a quadratic mapping
Q
:
X
→
Y
such that
for all
x
∈
X
and all t
> 0.
Corollary 3.4.
Let
θ
≥ 0
and let p be a real number with
0 <
p
< 2.
Let X be a normed vector space with norm
∥ · ∥.
Let f
:
X
→
Y
be an even mapping satisfying
f
(0) = 0
and
(2.5).
Then
exists for each x
∈
X
and defines a quadratic mapping
Q
:
X
→
Y
such that
for all
x
∈
X
and all t
> 0.
Proof
. The proof follows from Theorem 3.3 by taking
for all
x
_{1}
, ⋯ ,
x
_{2l}
∈
X
. Then we can choose
L
= 2
^{p−2}
and we get the desired result. ☐

fuzzy Banach space
;
fixed point
;
functional equation related to inner product space
;
Hyers-Ulam stability
;
quadratic mapping
;
additive mapping.

1. INTRODUCTION AND PRELIMINARIES

The stability problem of functional equations originated from a question of Ulam
[51]
concerning the stability of group homomorphisms. Hyers
[13]
gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki
[1]
for additive mappings and by Th.M. Rassias
[40]
for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta
[12]
by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th.M. Rassias’ approach.
The functional equation
- f(x+y) +f(x−y) = 2f(x) + 2f(y)

- (N1)N(x,t) = 0 fort≤ 0;
- (N2)x= 0 if and only ifN(x,t) = 1 for allt> 0;
- (N3)ifc≠= 0;
- (N4)N(x+y,s+t) ≥ min{N(x,s),N(y,t)};
- (N5)N(x, ·) is a non-decreasing function of ℝ and limt→∞N(x,t) = 1;
- (N6) forx≠= 0,N(x, ·) is continuous on ℝ.

- (1)d(x,y) = 0 if and only ifx=y;
- (2)d(x,y) =d(y,x) for allx,y∈X;
- (3)d(x,z) ≤d(x,y) +d(y,z) for allx,y,z∈X.
- We recall a fundamental result in fixed point theory.

- d(Jnx,Jn+1x) = ∞

- (1), ∀n≥n0;
- (2)the sequence{Jnx}converges to a fixed point y* ofJ;
- (3)y*is the unique fixed point of J in the set Y= {y∈X|d(Jn0x,y) < ∞};
- (4)for all y∈Y.

2. HYERS-ULAM STABILITY OF THE FUNCTIONAL EQUATION (0.1): AN ODD CASE

For a given mapping
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3. HYERS-ULAM STABILITY OF THE FUNCTIONAL EQUATION (0.1): AN EVEN CASE

In this section, using the fixed point method, we prove the Hyers-Ulam stability of the functional equation
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Aoki T.
1950
On the stability of the linear transformation in Banach spaces
J. Math. Soc. Japan
2
64 -
66
** DOI : 10.2969/jmsj/00210064**

Bag T.
,
Samanta S.K.
2003
Finite dimensional fuzzy normed linear spaces
J. Fuzzy Math.
11
687 -
705

Bag T.
,
Samanta S.K.
2005
Fuzzy bounded linear operators
Fuzzy Sets and Systems
151
513 -
547
** DOI : 10.1016/j.fss.2004.05.004**

Cădariu L.
,
Radu V.
2003
Fixed points and the stability of Jensen’s functional equation
J. Inequal. Pure Appl. Math.
Art. ID 4
4
(1)

Cădariu L.
,
Radu V.
2004
On the stability of the Cauchy functional equation: a fixed point approach
Grazer Math. Ber.
346
43 -
52

Cădariu L.
,
Radu V.
2008
Fixed point methods for the generalized stability of functional equations in a single variable
Fixed Point Theory and Applications
Art. ID 749392
2008

Cheng S.C.
,
Mordeson J.M.
1994
Fuzzy linear operators and fuzzy normed linear spaces
Bull. Calcutta Math. Soc.
86
429 -
436

Cholewa P.W.
1984
Remarks on the stability of functional equations
Aequationes Math
27
76 -
86
** DOI : 10.1007/BF02192660**

Czerwik S.
1992
On the stability of the quadratic mapping in normed spaces
Abh. Math. Sem. Univ. Hamburg
62
59 -
64
** DOI : 10.1007/BF02941618**

Diaz J.
,
Margolis B.
1968
A fixed point theorem of the alternative for contractions on a generalized complete metric space
Bull. Amer. Math. Soc.
74
305 -
309
** DOI : 10.1090/S0002-9904-1968-11933-0**

Felbin C.
1992
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Citing 'FUZZY STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION WITH THE FIXED POINT ALTERNATIVE
'

@article{ SHGHCX_2015_v22n3_285}
,title={FUZZY STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION WITH THE FIXED POINT ALTERNATIVE}
,volume={3}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.3.285}, DOI={10.7468/jksmeb.2015.22.3.285}
, number= {3}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={SEO, JEONG PIL
and
LEE, SUNGJIN
and
SAADATI, REZA}
, year={2015}
, month={Aug}