RESEARCH Open Access
Fuzzy stability of a mixed type functional
equation
Sun Sook Jin and Yang-Hi Lee
*
* Correspondence:
[email protected]
Department of Mathematics
Education, Gongju National
University of Education, Gongju
314-711, Republic of Korea
Abstract
In this paper, we investigate a fuzzy version of stability for the functional equation
f
(
x + y + z
)
− f
(
x + y
)
− f
(
y + z
)
− f
(
x + z
)
+ f
(

ing Cheng and Mordeson [20], gave an idea of a fuzzy norm in such a manner that the
corresp onding fuzzy metric is of Kramosil and Michalek type [21]. In 2008, Mirmosta-
faee and Moslehian [22] obtained a fuzzy version of stability for the Cauchy functional
equation:
f (
x + y
)
−
f (
x
)
−
f (
y
)
=0
.
(1:1)
In the same year, they [23] proved a fuzzy version of stability for the quadratic func-
tional equation:
f
(
x + y
)
+ f
(
x − y
)
− 2f
(

x + z
)
+ f
(
x
)
+ f
(
y
)
+ f
(
z
)
=0
.
(1:3)
which is called a mix ed type functional equation. We say a solution of (1.3) aquad-
ratic-additive mapping. In 2002, Jung [24] obtained a stability of the functional equa-
tion (1.3) by taking and composing an additive map A and a quadratic map Q to prove
the existence of a quadratic-additive mapping F, which is close to the given mapping f.
In his processing, A is approxi mate to the odd part
f (x)−f(−x)
2
of f and Q is close to the
even part
f
(x)+
f
(−x)

n
}issaidtobeconvergentifthere
exists x Î 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. A sequence {x
n
}inX
is called Cauchy if for each ε >0 and each t>0 there exists n
0
such that for all n ≥ n
0
and all p >0wehaveN(x
n+p
- x
n
, t)>1-ε . It is known that every convergent
sequence in a fuzzy normed space is Cauchy. If each Cauchy sequence is convergent,
then the fuzzy norm is said to be complete and the fuzzy normed space is called a
fuzzy Banach space.
Let (X, N)beafuzzynormedspaceand(Y, N’ ) a fuzzy Banach space. For a given
mapping f : X ® Y, we use the abbreviation

z
)
for all x, y, z Î X. For given q>0, the mapping f is called a fuzzy q-almost quadratic-
additive mapping,if
Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 2 of 12
N

(
Df
(
x, y, z
)
, s + t + u
)
≥ min{N
(
x, s
q
)
, N
(
y, t
q
)
, N
(
z, u
q

⎪
⎩
sup
t

<t
N

x,

2−2
p
3

q
t

q

if 1 < q,
sup
t

<t
N

x,

(4−2
p


q

if 0 < q <
1
2
(2:2)
where p =1/q.
Proof. It follows from (2.1) and (N4) that
N

(f (0), t) ≥ min

N

0,

t
3

q

, N

0,

t
3

q

2
(4
−n
(f (2
n
x)+f (−2
n
x)) + 2
−n
(f (2
n
x) − f (−2
n
x))
)
for all x Î X. Notice that J
0
f (x)=f (x) and
J
j
f (x) − J
j+1
f (x)=
Df (2
j
x,2
j
x, −2
j
x)

2
j+2
(2:3)
for all x Î X and j ≥ 0. Together with (N3), (N4) and (2.1), this equation implies that
if n + m>m≥ 0, then
N

⎛
⎝
J
m
f (x) − J
n+m
f (x),
n+m−1

j=m
3
2

2
p
2

j
t
p
⎞
⎠
= N

f (x) − J
j+1
f (x),
3 · 2
jp
2
j+1
t
p

≥ min
j=m, ,n+m−1

min

N


(2
j+1
+1)Df (2
j
x,2
j
x, −2
j
x)
2 · 4
j+1
,

p
2 · 4
j+1


≥ min
j=m, ,n+m−1
{N(2
j
x,2
j
t)}
= N
(
x, t
)
(2:4)
Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 3 of 12
for all x Î X and t>0. Let ε >0 be given. Since lim
t®∞
N (x, t)=1,thereist
0
>0
such that
N
(
x, t
0


j
=m
3 · 2
jp
2
j+1
˜
t
p
<
c
for each m ≥ n
0
and n>0. By (N5) and (2.4), we have
N

(J
m
f (x) − J
n+m
f (x), c) ≥ N

⎛
⎝
J
m
f (x) − J
n+m
f (x),

n
f (x
)
for all x Î X. Moreover, if we put m = 0 in (2.4), we have
N

(f (x) − J
n
f (x), t) ≥ N
⎛
⎜
⎝
x,
t
q


n−1
j=0
3·2
jp
2
j+1

q
⎞
⎟
⎠
(2:5)
for all x Î X. Next we will show that F is quadratic additive. Using (N4), we have

28

, N


(F − J
n
f )(z),
t
28

N


(J
n
f − F)(x + y),
t
28

, N


(J
n
f − F)(x + z),
t
28

,

≥ min

N


Df (2
n
x,2
n
y,2
n
z)
2 · 4
n
,
3t
16

, N


Df (−2
n
x, −2
n
y, −2
n
z)
2 · 4
n

z)
2 · 2
n
,
3t
16

Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 4 of 12
for all x, y, z Î X. By (N3) and (2.1), we obtain
N


Df (±2
n
x, ±2
n
y, ±2
n
z)
2 · 4
n
,
3t
16

= N



n
y,

4
n
t
8

q

, N

2
n
z,

4
n
t
8

q

≥ min

N

x,2
(2q−1)n−3q
t

,
3t
16

≥ min

N

x,2
(q−1)n−3q
t
q

, N

y,2
(q−1)n−3q
t
q

, N

z,2
(q−1)n−3q
t
q

for all x, y, z Î X and n Î N. Since q>1, together with (N5), we can deduce that the
last term of (2.6) also tends to 1 as n ® ∞. It follows from (2.6) that
N

, t − t

)
≥ 1 − ε
.
By (2.5), we have
N

(F ( x ) − f (x), t) ≥ min{N

(F ( x ) − J
n
f (x), t − t

), N

(J
n
f (x) − f(x), t

)
}
≥ min
⎧
⎪
⎨
⎪
⎩
1 − ε, N
⎛

(2 − 2
p
)t

3

q

.
Because 0 <ε < 1 is arbitrary, we get the inequality (2.2) in this case.
Finally, to prove the uniqueness of F,letF’ : X ® Y be another quadratic-additive
mapping satisfying (2.2). Then by (2.3), we get
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
F( x ) − J
n
F( x )=
n−1

j=0
(J
j
F( x ) − J


(x), t)
= N

(J
n
F(x) − J
n
F

(x), t)
≥ min

N


J
n
F(x) − J
n
f (x),
t
2

, N


J
n
f (x) − J

n
,
t
8

,
N


(F − f )(−2
n
x)
2 · 4
n
,
t
8

, N


(f − F

)(−2
n
x)
2 · 4
n
,
t

N


(F − f )(−2
n
x)
2 · 2
n
,
t
8

, N


(f − F

)(−2
n
x)
2 · 2
n
,
t
8


≥ sup
t


(
x
)
for all x Î X by (N2).
Case 2. Let
1
2
< q <
1
and let J
n
f : X ® Y be a mapping defined by
J
n
f (x)=
1
2

4
−n
(f (2
n
x)+f (−2
n
x)) + 2
n

f

x

j
x,2
j
x, −2
j
x)
2 · 4
j+1
− 2
j−1

Df

x
2
j+1
,
x
2
j+1
,
−x
2
j+1

− Df

−x
2
j+1

j
+
3
2
p

2
2
p

j

t
p

≥ min
j=m, ,n+m−1

min

N


Df (2
j
x,2
j
x, −2
j
x)


,
N


−2
j−1
Df

x
2
j+1
,
x
2
j+1
,
−x
2
j+1

,
3 · 2
j−1
t
p
2
(j+1)
p


≥ min
j=m, ,n+m−1

N(2
j
x,2
j
t), N

x
2
j+1
,
t
2
j+1

= N
(
x, t
)
Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 6 of 12
for all x Î X and t>0. In the similar argument following (2.4) of the previous case,
we can define the limit F(x):=N’ -lim
n®∞
J
n
f (x) of the Cauchy sequence {J

p
(
2
2
p
)
j

q
⎞
⎟
⎠
(2:8)
for each x Î X and t>0. To prove that F is a quadratic-additive function, we have
enough to show that the last term of (2.6) in Case 1 tends to 1 as n ® ∞. By (N3) and
(2.1), we get
N


DJ
n
f (x, y, z),
3t
4

≥ min

N



N


2
n−1
Df

x
2
n
,
y
2
n
,
z
2
n

,
3t
16

, N


2
n−1
Df


t
q

, N

z,2
(2q−1)n−3q
t
q

,
N

x,2
(1−q)n−3q
t
q

, N

y,2
(1−q)n−3q
t
q

, N

z,2
(1−q)n−3q
t



J
n
F(x) − J
n
f (x),
t
2

, N


J
n
f (x) − J
n
F

(x),
t
2

≥ min

N


(F − f )(2
n

t
8

, N


(f − F

)(−2
n
x)
2 · 4
n
,
t
8

,
N


2
n−1

(F − f )

x
2
n



−x
2
n

,
t
8

, N


2
n−1

(f − F

)

−x
2
n

,
t
8


≥ min



(4 − 2
p
)(2
p
− 2)
6

q
t

q

for all x Î X and n Î N. Since lim
n®∞
2
(2q -1)n-2q
= lim
n®∞
2
(1 - q)n -2q
= ∞ in this
case, both terms on the right-hand side of the above inequality tend to 1 as n ® ∞ by
(N5). This implies that N’(F(x)-F’(x), t) = 1 and so F(x)=F’(x) for all x Î X by (N2).
Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 7 of 12
Case 3. Finally, we take
0 < q <
1

n


for all x Î X. Then we have J
0
f (x)=f (x) and
J
j
f (x) − J
j+1
f (x)=−
4
j
2

Df

−x
2
j+1
,
−x
2
j+1
,
x
2
j+1

+ Df

− Df

−x
2
j+1
,
−x
2
j+1
,
x
2
j+1

which implies that if n + m>m≥ 0, then
N

⎛
⎝
J
m
f (x) − J
n+m
f (x),
n+m−1

j=m
3
2
p

j+1
,
−x
2
j+1
)
2
,
3(4
j
+2
j
) t
p
2 · 2
(j+1)
p

,
N


−
(4
j
− 2
j
)Df (
−x
2

j+1
,
t
2
j+1

= N(x, t)
for all x Î X and t>0. Similar to the previ ous cases, it leads us to define the map-
ping F : X ® Y by F(x):=N’ -lim
n®∞
J
n
f (x). Putting m =0intheaboveinequality,
we have
N

(f (x) − J
n
f (x), t) ≥ N
⎛
⎜
⎝
x,
t
q


n−1
j=0
3

n
2
Df

x
2
n
,
y
2
n
,
z
2
n

,
3t
16

, N


4
n
2
Df

−x
2

z
2
n

,
3t
16

, N


2
n−1
Df

−x
2
n
,
−y
2
n
,
−z
2
n

,
3t
16

t
q

, N

y,2
(1−q)n−3q
t
q

, N

z,2
(1−q)n−3q
t
q

for each x, y, z Î X and t>0. Since
0 < q <
1
2
, all terms on the right-hand side tend
to 1 as n ® ∞, which implies that the last term of (2.6) tends to 1 as n ® ∞.There-
fore, we can say that DF ≡ 0. Moreover, using the similar argument after (2.6) in Case
1, we get the inequality (2.2) from (2.9) in this case. To prove the uniqueness of F,let
F’ : X ® Y be another quadratic-additive function satisfying (2.2). Then by (2.7), we get
Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 8 of 12
N

≥ min

N


4
n
2

(F − f )

x
2
n

,
t
8

,
4
n
2


f − F

)

x

n
2

(f − F

)

−
x
2
n

,
t
8

,
N


2
n−1

(F − f )

x
2
n

,

−x
2
n

,
t
8

, N


2
n−1

(f − F

)

−x
2
n

,
t
8


≥ sup
t


x, y, z
)
, t
)
≥ N

(
Df
(
x, y, z
)
,3s
)
≥ min{N
(
x, s
q
)
, N
(
y, s
q
)
, N
(
z, s
q
)}
for all x, y, z Î X. Since q<0, we have
lim

1
and so
N

(
Df
(
x, y, z
)
, t
)
=
1
for all x, y, z Î X and t>0. By (N2), it allows us to get Df(x, y, z) = 0 for all x, y, z Î
X. In other words, f is itself a quadratic-additive mapping if f is a fuzzy q-almost quad-
ratic-additive mapping for the case q<0.
Corollary 2.4. Let f be an even mapping satisfying all of the conditions of Theorem
2.2. Then there is a unique quadratic mapping F : X ®Y such that
N

(F ( x ) − f (x), t) ≥ sup
t

<t
N

x,

|4 − 2
p

(f (2
−n
x)+f (−2
−n
x))) if 0 < q <
1
2
for all x Î X. Notice that J
0
f (x)=f (x) and
J
j
f (x) − J
j+1
f (x)=
⎧
⎪
⎨
⎪
⎩
Df (2
j
x,2
j
x,−2
j
x)
2·4
j+1
+

j+1

+Df

x
2
j+1
,
x
2
j+1
,
−x
2
j+1

if 0 < q <
1
2
Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
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Page 9 of 12
for all x Î X and j Î N ∪ {0}. From these, using the similar method in Theorem 2.2 ,
we obtain the quadratic-additive function F satisfying (2.10). Notice that F(x):=N’ -
lim
n®∞
J
n
f (x)forallx Î X, F is even, and DF (x, y, z)=0forallx, y, z Î X.Hence,
we get

<t
N

x,

|2 − 2
p
|t

3

q

(2:11)
for all x Î X and t >0, where p =1/q.
Proof. Let J
n
f be defined as in Theorem 2.2. Since f is an odd mapping, we obtain
J
n
f (x)=

f (2
n
x)+f (−2
n
x)
2
n+1
if q > 1,

x,2
j
x,−2
j
x)
2
j+2
−
Df (−2
j
x,−2
j
x,2
j
x)
2
j+2
if q ¿1,
−2
j−1

Df

x
2
j+1
,
x
2
j+1


x − y
2
,
x + y
2
,
−x + y
2

=
0
for all x, y Î X. This means that F is an additive mapping.
We can use Theorem 2.2 to get a classical result in the framework of normed spaces.
Let (X, || · ||) be a normed linear space. Then we can define a fuzzy norm N
X
on X by
following
N
X
(x, t)=

0, t ≤x 
1, t >  x

where x Î X and t Î ℝ, see [14]. Suppose that f : X ® Y is a mapping into a Banach
space (Y, ||| · |||) such that
|
||Df
(

+  z
p
≥|Df
(
x, y, z
)
|≥s + t +
u
and so either ||x||
p
≥ s or ||y||
p
≥ t or ||z ||
p
≥ u in this case. Hence, for
q =
1
p
,we
have
min{N
X
(
x, s
q
)
, N
X
(
y, t

q
)
, N
X
(
z, u
q
)}
hol ds. It means that f is a fuzzy q-almost quadratic-additive mapping, and by Theo-
rem 2.2, we get the following stability result.
Corollary 2.6. Let (X, || · ||) be a normed linear space and let (Y, |||·|||) be a Banach
space. If f : X ® Y satisfies
|
||Df
(
x, y, z
)
||| ≤  x
p
+  y
p
+  z
p
for all x, y, z Î X, where p >0 and p ≠ 1, 2, then there is a unique quadratic-additive
mapping F : X ® Y such that
|||F(x ) − f(x)||| ≤
⎧
⎨
⎩
3

References
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Jin and Lee Journal of Inequalities and Applications 2011, 2011:70
http://www.journalofinequalitiesandapplications.com/content/2011/1/70
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