RESEARCH Open Access
A fixed point approach to the Hyers-Ulam
stability of a functional equation in various
normed spaces
Hassan Azadi Kenary
1
, Sun Young Jang
2
and Choonkil Park
3*
* Correspondence: baak@hanyang.
ac.kr
3
Department of Mathematics,
Research Institute for Natural
Sciences, Hanyang University, Seoul
133-791, Korea
Full list of author information is
available at the end of the article
Abstract
Using direct method, Kenary (Acta Universitatis Apulensis, to appear) proved the
Hyers-Ulam stability of the following functional equation
f (mx + ny)=
(m + n)f(x + y)
2
+
(m − n)f(x − y)
2
in non-Archimedean normed spaces and in random normed spaces, where m, n are
different integers greater than 1. In this article, using fixed point method, we prove
the Hyers-Ulam stability of the above functional equation in various normed spaces.

provided the original wo rk is p roperly cited.
The stability problems of several functional equ ations have been extensively investi-
gated by a number of authors, and there are many interesting results concerning this
problem (see [8-12]).
Using fixed point method, we prove the Hyers-Ulam stability of the following func-
tional equation
f (mx + ny)=
(m + n)f(x + y)
2
+
(m − n)f(x − y)
2
(1)
in various spaces, which was introduced and investigated in [13].
2. Preliminaries
In this section, we give some definitions and lemmas for the main results in this
article.
A valuation is a function | · | from a field
K
into [0, ∞)suchthat,forall
r
,
s ∈
K
,the
following conditions hold:
(a) |r| = 0 if and only if r =0;
(b) |rs|=|r||s|;
(c) |r + s| ≤ |r|+|s|.
A field

Definition 2.1.LetX be a vector space over a field
K
with a non-Archimedean
valuation | · |. A function || · || : X ® [0, ∞ )iscalledan on-Archimed ean norm if the
following conditions hold:
(a)||x|| = 0 if and only if x = 0 for all x Î X;
(b)||rx|| = |r|||x|| for all
r
∈ K
and x Î X;
(c) the strong triangle inequality holds:
|
|x +
y
|| ≤ max{||x||, ||
y
||
}
for all x, y Î X. Then (X, || · ||) is called a non-Archimedean normed space.
Definition 2.2. Let {x
n
} be a sequence in a non-Archimedean normed space X.
(a) The sequence {x
n
}iscalledaCauchy sequence if, for any ε >0, there is a positive
integer N such that ||x
n
- x
m
|| ≤ ε for all n, m ≥ N.

for all m, n ≥ 1 with n>m.
In the sequel (in random stability section), we adopt the usual terminology, notions,
and conventions of the theory of random normed spaces as in [15].
Throughout this article (in random stability section), let Γ
+
denote the set of all
probability distribution f unctions F : ℝ ∪ [-∞,+∞] ® [0,1] such that F is left-continu-
ous and nondecreasing on ℝ and F(0) = 0, F(+∞) = 1. It is clear that the set D
+
={F Î
Γ
+
: l
-
F(-∞)=1},where
l
−
f
(
x
)
= lim
t→x
−
f
(
t
)
, is a subset of Γ
+

Three typical examples of continuous t-norms are as follows: T(x, y)=xy, T(x, y)=
max{a + b - 1, 0}, and T(x, y) = min(a, b).
Definition 2.4.[16]Arando m normed space (briefly, RN-space) is a triple (X, μ, T),
where X is
a vector space, T is a continuous t-norm, and μ : X ® D
+
is a m apping
such that the following conditions hold:
(a) μ
x
(t)=H
0
(t) for all x Î X and t>0 if and only if x =0;
(b)
μ
αx
(t )=μ
x
(
t
|
α
|
)
for all a Î ℝ with a ≠ 0, x Î X and t ≥ 0;
(c) μ
x+y
(t + s) ≥ T ( μ
x
(t), μ

for
all t>0.
(3) The RN-space (X, μ, T)issaidtobecomplete if every Cauchy sequence i n X is
convergent.
Theorem 2.1. [15]If (X, μ, T) is an RN-space and {x
n
} is a sequence such that x
n
®
x, then
lim
n→∞
μ
x
n
(t )=μ
x
(t )
.
Definition 2.6.[17]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 Î X and all s, t Î ℝ,
(N1) N(x, t)=0for t ≤ 0;
(N2) x =0if and only if N(x, t)=1for all t >0;
Kenary et al. Fixed Point Theory and Applications 2011, 2011:67
/>Page 3 of 14
(N3)
N( cx, t)=N(x,
t
|
c

denote it by N - lim
t®∞
x
n
= x.
Definition 2.8. [17]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
Î 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.
Example 2.2. Let N : ℝ × ℝ ® [0, 1] be a fuzzy norm on ℝ defined by
N( x , t)=

t
t+|x|
t > 0
0 t ≤ 0
.
Then (ℝ, N) is a fuzzy Banach space.

)
=
∞
for all non-negative integers n or there exists a positive integer n
0
such that
(a) d(J
n
x, J
n+1
x)<∞ for all n
0
≥ n
0
;
(b) the sequence {J
n
x} converges to a fixed point y* of J;
(c) y* is the unique fixed point of J in the set
Y = {y ∈ X : d
(
J
n
0
x, y
)
< ∞
}
;
Kenary et al. Fixed Point Theory and Applications 2011, 2011:67

(
m
n
x
)
= m
n
f
(
x
)
for all x Î X and all n Î N.□
Theorem 3.1. Let ζ : X
2
® [0, ∞) be a function such that there exists an L <1 with
|m|ζ
(
x, y
)
≤ Lζ
(
mx, my
)
for all x, y Î X. If f : X ® Y is a mapping satisfying f(0) = 0 and the inequality




f (mx + ny) −
(m + n)f(x + y)


mf

x
m

− f (x)



≤ ζ

x
m
,0

≤
L
|
m
|
ζ (x,0
)
(4)
for all x Î X. Consider the set
S := {g : X → Y; g
(
0
)
=0

for all x Î X and so
 Jg(x) − Jh(x) =



mg

x
m

− mh

x
m




≤|m|εζ

x
m
,0

≤|m|ε
L
|
m
|
ζ (x,0

(5)
for all x Î X. The mapping A is a unique fixed point of J in the set

= {h ∈ S : d
(
g, h
)
< ∞}
.
This implies that A is a unique mapping satisfying (5) such that there exists μ Î (0, ∞)
satisfying
 f
(
x
)
− A
(
x
)
≤ μζ
(
x,0
)
for all x Î X.
(2) d(J
n
f, A) ® 0asn ® ∞. This implies the equality
lim
n→∞
m

L
.
This implies that the inequality (3) holds. By (2), we have





m
n
f

mx + ny
m
n

−
m
n
(m + n)f(
x+y
m
n
)
2
−
m
n
(m − n)f(
x−y

ζ (x, y)
for all x, y Î X and n ≥ 1 and so




A(mx + ny) −
(m + n)A(x + y)
2
−
(m − n)A(x − y)
2




=
0
for all x, y Î X.
On the other hand
mA

x
m

− A(x) = lim
n→∞
m
n+1
f

(m − n)f(x − y)
2




≤ θ (||x||
p
+ ||y||
p
)
(6)
for all x, y Î X. Then, for all x Î X,
A(x) = lim
n→∞
m
n
f

x
m
n

exists and A : X ® Y is a unique additive mapping such that
 f (x) − A(x) ≤
|m|θ||x||
p
|
m
|

≤ Lζ (x, y
)
for all x, y Î X. Let f : X ® Y be a mapping satisfying f(0) = 0 and (2). Then there is
a unique additive mapping A : X ® Y such that
 f (x) − A(x) ≤
ζ (x,0)
|m|−|m|L
.
Proof. The proof is similar to the proof of Theorem 3.1. □
Corollary 3.2. Let θ ≥ 0 and p be a real number with p >1. Let f : X ® Y be a map-
ping satisfying f(0) = 0 and (6). Then, for all x Î X
A(x) = lim
n→∞
f (m
n
x)
m
n
exists and A : X ® Y is a unique additive mapping such that
 f (x) − A(x) ≤
θ||x||
p
|
m
|
−
|
m
|
p

Theorem 4.1. Let X be a linear space,(Y, μ, T) a complete RN-space and F a mapping
from X
2
to D
+
(F(x, y) is denoted by F
x,y
) such that there exists
0 <α<
1
m
such that

mx,my

t
α

≤ 
x,y
(t
)
(8)
for all x, y Î X and t >0. Let f : X ® Y be a mapping satisfying f(0) = 0 and
μ
f (mx+ny)−
(m + n)f(x + y)
2
−
(m − n)f(x − y)

m
, we have
μ
mf

x
m

−f ( x)
(t ) ≥ 
x
m
,0
(t
)
(11)
for all x Î X and t>0. Consider the set
S
∗
:= {g : X → Y; g
(
0
)
=0
}
and the generalized metric d*inS* defined by
d
∗
(f , g)= inf
u∈

(εt) ≥ 
x,0
(t
)
for all x Î X and t>0 and so
μ
Jg(x)−Jh(x)
(mαεt)=μ
mg

x
m

−mh

x
m

(mαεt)=μ
g

x
m

−h

x
m

(αεt

≤ α
.
By Theorem 2.2, there exists a mapping A : X ® Y satisfying the following:
(1) A is a fixed point of J, that is,
A

x
m

=
1
m
A(x
)
(12)
for all x Î X. The mapping A is a unique fixed point of J in the set

= {h ∈ S
∗
: d
∗
(
g, h
)
< ∞}
.
This implies that A is a unique mapping satisfying (12) such that there exists u Î (0,
∞) satisfying
μ
f

∗
(f , A) ≤
d
∗
(f ,Jf)
1−
m
α
with f Î Ω, which implies the inequality
d
∗
(f , A) ≤
α
1 −
m
α
and so
μ
f (x)−A(x)

αt
1 − mα

≥ 
x,0
(t
)
for all x Î X and t>0. This implies that the inequality (10) holds.
On the other hand
μ

m
n
,
y
m
n

t
m
n

for all x, y Î X, t>0 and n ≥ 1 and so, from (8), it follows that

x
m
n
,
y
m
n

t
m
n

≥ 
x,y

t
m

1
for all x, y Î X and t>0.
On the other hand
A(mx) − mA(x) = lim
n→∞
m
n
f

x
m
n−1

− m lim
n→∞
m
n
f

x
m
n

= m

lim
n→∞
m
n−1
f

t + θ

||x||
p
+ ||y||
p

(13)
for all x, y Î X and t >0. Then, for all x Î X,
A(x) = lim
n→∞
m
n
f

x
m
n

exists and A : X ® Y is a unique additive mapping such that
μ
f (x)−A(x)
(t ) ≥
m
p
(1 − m
1−p
)t
m
p

x,y
) such that for some 0<a <m

x
m
,
y
m
(t ) ≤ 
x,y
(αt
)
for all x, y Î X and t >0. Let f : X ® Y be a mapping satisfying f(0) = 0 and
μ
f (mx+ny)−
(m + n)f(x + y)
2
−
(m − n)f(x − y)
2
(t ) ≥ 
x,y
(t
)
for all x, y Î X and t >0. Then, for all x Î X,
A(x) := lim
n→∞
f (m
n
x)

t
t + θ

||x||
p
+ ||y||
p

for all x, y Î X and t >0. Then, for all x Î X,
A(x) = lim
n→∞
f (m
n
x)
m
n
exists and A : X ® Y is a unique additive mapping such that
μ
f (x)−A(x)
(t ) ≥
(m − m
p
)t
(
m − m
p
)
t + θ ||x||
p
for all x Î X and t >0.

Theorem 5.1. Let  : X
2
® [0, ∞) be a function such that there exists an L <1 with
ϕ

x
m
,
y
m

≤
L
m
ϕ(x, y
)
for all x, y Î X. Let f : X ® Y be a mapping satisfying f(0) = 0 and
N

f (mx + ny) −
(m + n)f(x + y)
2
−
(m − n)f(x − y)
2
, t

≥
t
t + ϕ

x
m
, we have
N

mf

x
m

− f (x), t

≥
t
t + ϕ

x
2
,0

for all x Î X and t>0. Consider the set
s
∗∗
:= {g : X → Y, g
(
0
)
=0
}
and the generalized metric d** in S** defined by

f (mx + ny) −
(m + n)f(x + y)
2
−
(m − n)f(x − y)
2
, t

≥
t
t + θ

||x||
p
+ ||y||
p

(17)
for all x, y Î X and all t >0. Then
A(x):=N - lim
n→∞
m
n
f

x
m
n

exists for each x Î X and defines an additive mapping A : X ® Y such that

1-p
and we get the desired result. □
Theorem 5.2. Let  : X
2
® [0, ∞) be a function such that there exists an L <1 with
ϕ
(
mx, my
)
≤ mLϕ
(
x, y
)
for all x, y Î X. Let f : X ® Y be a mapping satisfying f(0) = 0 and
N

f (mx + ny) −
(m + n)f(x + y)
2
−
(m − n)f(x − y)
2
, t

≥
t
t + φ
(
x, y
)

2
−
(m − n)f(x − y)
2
, t

≥
t
t + θ

||x||
p
+ ||y||
p

for all x, y Î X and all t >0. Then the limit
A(x):=N - lim
n→∞
f (m
n
x)
m
n
exists for each x Î X and defines a unique additive mapping A : X ® Y such that
N( f (x) − A(x), t) ≥
(m − m
p
)t
(
m − m

We linked here five different disciplines, namely, the random normed spa ces, non-
Archimedean normed spaces, fu zzy normed spaces, f unctional equations, and f ixed
point theory. We established th e Hyers-Ulam stability of the functional equation (1) in
various normed spaces by using fixed point method.
Acknowledgements
The second author was supported by Basic Science Research Program through the National Research Foundation of
Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0013211).
Author details
1
Department of Mathematics, College of Sciences, Yasouj Universi ty, Yasouj 75914-353, Iran
2
Department of
Mathematics, University of Ulsan, Ulsan 680-749, Korea
3
Department of Mathematics, Research Institute for Natural
Sciences, Hanyang University, Seoul 133-791, Korea
Kenary et al. Fixed Point Theory and Applications 2011, 2011:67
/>Page 13 of 14
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in
the sequence alignment, and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 2 June 2011 Accepted: 25 October 2011 Published: 25 October 2011
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doi:10.1186/1687-1812-2011-67
Cite this article as: Kenary et al.: A fixed point approach to the Hyers-Ulam stability of a functional equation in
various normed spaces. Fixed Point Theory and Applications 2011 2011:67.
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