International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 1 (2014), 10-19
http://www.etamaths.com

FUZZY STABILITY OF GENERALIZED SQUARE ROOT

FUNCTIONAL EQUATION IN SEVERAL VARIABLES: A FIXED

POINT APPROACH

K. RAVI1,∗ AND B.V. SENTHIL KUMAR2

Abstract. In this paper, we investigate the generalized Hyers-Ulam stability

of the generalized square root functional equation in several variables in fuzzy
Banach spaces, by applying the fixed point method.

1. INTRODUCTION

In the last forty years, fuzzy theory has gained paramount importance and
validty on the mathematical scenario by facilitating to focus an ardent attention
on multifarious avenues of development in the theory of fuzzy sets to explore the
fuzzy analogues of the classical set theory. In the effulgent light of the authentic
investigations executed in this branch, the fuzzy sets are being tapped to augment
a wide range of applications in science and Engineering with platonic dimensions.

Various mathematical visions, viewed in different perspectives, have triggered
scores of scholars to come out with different definitions of fuzzy norms on a vector
space. For Example, A.K. Katsaras [26] had accomplished a detailed survey to
define a fuzzy norm on a vector space to help to construct a fuzzy vector topological
structure. In 1991, R. Biswas [6] defined and studied fuzzy inner product spaces in
linear space. In 1992, C. Felbin [18] introduced an alternative definition of a fuzzy
norm on a linear topological structures of a fuzzy normed linear spaces. Similarly,
T. Bag and S.K. Samanta [4], gliding along the mathematical track of S.C. Cheng
and J.M. Mordeson [12], proved that the corresponding fuzzy metric of a fuzzy norm
would be the same as that of the metric executed by I. Kramosil and J. Michalek
[28]. They had initiated a decomposition theorem of a fuzzy norm into a family of
crisp norms by undertaking an analytical investigation of some of the properties of
fuzzy normed spaces.

An inquisitive question that was given a serious thought by S.M. Ulam [45]
concerning the stability of group homomorphisms gave rise to the stability problem
of functional equations. The laborious intellectual strivings of D.H. Hyers [23] did
not go in vain because he was the first to come out with a partial answer to solve
the question posed by Ulam on Banach spaces. In course of time, the theorem
formulated by Hyers was generalized by T. Aoki [2] for additive mappings and by
Th.M. Rassias [43] for linear mappings by taking into consideration an unbounded

2010 Mathematics Subject Classification. 46S40, 39B72, 39B52, 46S50, 26E50.
Key words and phrases. Fuzzy normed space, fixed point, generalized square root functional

equation, generalized Hyers-Ulam stability.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

10



FUZZY STABILITY OF GENERALIZED SQUARE ROOT FUNCTIONAL EQUATION 11

Cauchy difference.
The findings of Th.M. Rassias have exercised a delectable influence on the

development of what is addressed as the generalized Hyers-Ulam stability of Hyers-
Ulam-Rassias stability of functional equations. A generalized and modified form
of the theorem evolved by Th.M. Rassias was advocated by P. Gavruta [21] who
replaced the unbounded Cauchy difference by driving into study a general control
function within the viable approach designed by Th.M. Rassias. A further research
materialized by F. Skof [44] found solution to Hyers-Ulam-Rassias stability problem
for quadratic functional equation

(1) f(x + y) + f(x−y) = 2f(x) + 2f(y)
for a class of functions f : A → B, where A is a normed space and B is a Banach
space. The stability problems of several functional equations have been extensively
investigated by a number of scholars, possed with creative thinking and critial dis-
sent who have arrived at interesting results (see [3], [10], [11], [17], [22], [24], [27],
[42]).

In 1996, G. Isac and Th.M. Rassias [25] 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
[8], [9], [35], [36], [39]).

Functional equations find a lot of application in information theory, informa-
tion science, measure of information, coding theory, fuzzy system models, econom-
ics, social sciences and physics.

The paper presenters have made use of some basic concept concerning fuzzy
normed spaces and some fundamental results in fixed point theory.

Let X be a real linear space. A function N : X × R → [0, 1] is said to be a
fuzzy norm on X if for all x,y ∈ X and all u,v ∈ R :

(N1) N(x,u) = 0 for u ≤ 0
(N2) x = 0 if and only if N(x,u) = 1 for all u > 0

(N3) N(ux,v) = N
(
x, v|u|

)
if u 6= 0

(N4) N(x + y,u + v) ≥ min{N(x,u),N(y,v)}
(N5) N(x,.) is non-decreasing function on R and lim

t→∞
N(x,u) = 1

(N6) For x 6= 0, N(x,.) is (uppersemi) continuous on R.
The pair (X,N) is called a fuzzy normed linear space. One may regard N(x,u)
as the truth value of the statement the norm of x is less than or equal to the real
number u.

Definition 1.1. Let (X,N) be a fuzzy normed linear space. Let {xn} be a se-
quence in X. Then {xn} is said to be convergent if there exists x ∈ X such that
lim
n→∞

N(xn−x,u) = 1 for all u > 0. In this case, x is called the limit of the sequence
{xn} and we denote it by N- lim

n→∞
xn = x.

Definition 1.2. A sequence {xn} in X is called Cauchy if for each � > 0 and
each u > 0, there exists n0 ∈ N such that for all n ≥ n0 and all p > 0, we have
N(xn+p −xn,u) > 1 − �.

It is known that every convergent sequence in fuzzy normed space is Cauchy.
If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete



12 RAVI AND KUMAR

and the fuzzy normed space is called a fuzzy Banach space.
We say that a mapping f : X → Y between fuzzy normed linear spaces X and

Y is continuous at a point x0 ∈ X if for each sequence {xn} converging to x0 in X,
then the sequence {f(xn)} converges to f(x0). If f : X → Y is continuous at each
x ∈ X, then f : X → Y is said to be continuous on X.

Let X be a set. A function d : X ×X → [0,∞] is called a generalized metric
on X if d satisfies

(1) d(x,y) = 0 if and only if x = y;
(2) d(x,y) = d(y,x) for all x,y ∈ X;
(3) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X.

Theorem 1.3. Let (X,d) be a complete generalized metric space and let σ : X → X
be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given
element x ∈ X, either

d(σnx,σn+1x) = ∞
for all nonnegative integers n or there exists a positive integer n0 such that

(1) d(σnx,σn+1x) < ∞ for all n ≥ n0;
(2) the sequence {σnx} converges to a fixed point y∗ of σ;
(3) y∗ is the unique fixed point of σ in the set

Y = {y ∈ X/d(σn0x,y) < ∞};
(4) d(y,y∗) ≤ 1

1−Ld(y,σy) for all y ∈ Y .

K. Ravi and B.V. Senthil Kumar[41] introduced the generalized square root func-
tional equation (or GSRF equation) in several variables of the form

(2) s


 p∑
i=1

ρixi + 2

p−1∑
i=1

p∑
j=i+1

√
ρiρjxixj


 = p∑

i=1

√
ρis(xi)

for arbitrary but fixed real numbers (ρi,ρ2, . . . ,ρp) 6= (0, 0, . . . , 0), so that 0 <
ρ =

√
ρ1 +

√
ρ2 + · · · +

√
ρp =

∑p
i=1

√
ρi 6= 1 and s : X → R with X as space

of non-negative real numbers and investigated generalized Hyers-Ulam stability of
equation (2). It is easy to verify that the function f : X → R such that f(x) =

√
x

is a solution of the functional equation (2).
In this paper, we will show the generalized Hyers-Ulam stability of the equation

(2) on fuzzy normed spaces using fixed point method.
Throughout this paper, let us assume that X be space of non-negative real

numbers and Y be a fuzzy normed linear space.
For the sake of convenience, let us define

Dρs(x1,x2, . . . ,xp) = s


 p∑
i=1

ρixi + 2

p−1∑
i=1

p∑
j=i+1

√
ρiρjxixj


− p∑

i=1

√
ρis(xi)

for all x1,x2, . . . ,xp ∈ X and p ∈ N−{1}.

2. GENERALIZED HYERS-ULAM STABILITY OF THE

FUNCTIONAL EQUTION (2) IN FUZZY NORMED SPACES

Theorem 2.1. Let ϕ : Xp → R be a function such that there exists an L < 1 with

ϕ(ρ2x1,ρ
2x2, . . . ,ρ

2xp) ≤ ρLϕ(x1,x2, . . . ,xp)



FUZZY STABILITY OF GENERALIZED SQUARE ROOT FUNCTIONAL EQUATION 13

for all x1,x2, . . . ,xp ∈ X. Let f : X → R be a mapping satisfying

(1) N (Dρf(x1,x2, . . . ,xp), t) ≥
t

t + ϕ(x1,x2, . . . ,xp)

for all x1,x2, . . . ,xp ∈ X and all t > 0, where 0 < ρ =
∑p
i=1

√
ρi < 1. Then s(x) =

N- lim
n→∞

ρ−nf
(
ρ2nx

)
exists for each x ∈ X and defines a square root mapping s :

X → Y such that

(2) N(f(x) −s(x), t) ≥
(1 −L)t

(1 −L)t + Lϕ(x,x,. . . ,x)
for all x ∈ X and all t > 0.

Proof. Taking xi as x for 1 ≤ i ≤ p in (1), we get

(3) N
(
f(ρ2x) −ρf(x)

)
≥

t

t + ϕ(x,x,. . . ,x)

for all x ∈ X.
Consider the set

S = {g : X → Y/g is a function}

and introduce the generalized metric d on S as follows:

d(g,h) = inf{C ∈ R+ : N(g(x) −h(x),Ct) ≥
t

t + ϕ(x,x,. . . ,x)
,∀x ∈ X,∀t > 0},

where, as usual, inf φ = +∞. It is easy to show that (S,d) is complete. (See the
proof of Lemma 2.1 of [30]).

Define a mapping σ : S → S by

σh(x) =
1

ρ
h
(
ρ2x
)

(x ∈ X)

Let g,h ∈ S be given such that d(g,h) = �. Then

N (g(x) −h(x),�t) ≥
t

t + ϕ(x,x,. . . ,x)

for all x ∈ X and all t > 0. Hence

N(σg(x) −σh(x),L�t) = N
(

1

ρ
g(ρ2x) −

1

ρ
h(ρ2x),L�t

)
= N

(
g(ρ2x) −h(ρ2x),ρL�t

)
≥

ρLt

ρLt + ϕ(ρ2x,ρ2x,. . . ,ρ2x)

≥
ρLt

ρLt + ρLϕ(x,x,. . . ,x)

≥
t

t + ϕ(x,x,. . . ,x)



14 RAVI AND KUMAR

for all x ∈ X and all t > 0. So d(g,h) = � implies that d(σg,σh) ≤ L�. This means
that

d(σg,σh) ≤ Ld(g,h)

for all g,h ∈ S.
It follows from (3) that

N

(
1

ρ
f(ρ2x) −f(x),

Lt

ρ

)
≥

t

t + ϕ(x,x,. . . ,x)

for all x ∈ X and all t > 0. So d(σf,f) ≤ L
ρ

.

By Theorem 1.3, there exists a mapping s : X → Y satisfying the following:

(1) s is a fixed point of σ, ie.,

(4) s(ρ2x) = ρs(x)

for all x ∈ X. The mapping s is a unique fixed point of σ in the set

µ = {g ∈ S : d(f,g) < ∞}.

This implies that s is a unique mapping satisfying (4) such that there exists

a C ∈ (0,∞) satisfying

N(f(x) −s(x),Ct) ≥
t

t + ϕ(x,x,. . . ,x)

for all x ∈ X.
(2) d(σnf,s) → 0 as n →∞. This implies the equality

N- lim
n→∞

1

ρn
f(ρ2nx) = s(x)

for all x ∈ X.
(3) d(f,s) ≤ 1

1−Ld(σf,f), which implies the inequality

d(f,s) ≤
L

ρ−ρL
.

This implies that the inequality (2) holds.

By (1),

N

(
1

ρn
Dρf(ρ

2nx1,ρ
2nx2, . . . ,ρ

2nxp),
t

ρn

)
≥

t

t + ϕ(ρ2nx1,ρ2nx2, . . . ,ρ2nxp)

for all x1,x2, . . . ,xp ∈ X, all t > 0 and all n ∈ N. So

N

(
1

ρn
Dρf(ρ

2nx1,ρ
2nx2, . . . ,ρ

2nxp), t

)
≥

ρnt

ρnt + Lnρnϕ(x1,x2, . . . ,xp)

for all x1,x2, . . . ,xp ∈ X, all t > 0 and all n ∈ N. Since

lim
n→∞

ρnt

ρnt + Lnρnϕ(x1,x2, . . . ,xp)
= 1



FUZZY STABILITY OF GENERALIZED SQUARE ROOT FUNCTIONAL EQUATION 15

for all x1,x2, . . . ,xp ∈ X, all t > 0,

N (Dρs(x1,x2, . . . ,xp), t) = 1

for all x1,x2, . . . ,xp ∈ X, all t > 0. Thus the mapping s : Xp → Y is square root
as desired. �

Theorem 2.2. Let ϕ : Xp → Y be a function such that there exists an L < 1 with

ϕ

(
x1
ρ2
,
x2
ρ2
, . . . ,

xp
ρ2

)
≤
L

ρ
ϕ(x1,x2, . . . ,xp)

for all x1,x2, . . . ,xp ∈ X. Let f : X → Y be a mapping satisfying

(5) N (Dρf(x1,x2, . . . ,xp), t) ≥
t

t + ϕ(x1,x2, . . . ,xp)

for all x1,x2, . . . ,xp ∈ X and all t > 0, where 0 < ρ =
∑p
i=1

√
ρi > 1. Then s(x) =

N- lim
n→∞

ρnf
(
ρ−2nx

)
exists for each x ∈ X and defines a square root mapping s :

X → Y such that

(6) N(f(x) −s(x), t) ≥
(1 −L)t

(1 −L)t + Lϕ(x,x,. . . ,x)

for all x ∈ X and all t > 0.

Proof. Taking xi as
x
ρ2

for 1 ≤ i ≤ p in (5) and proceeding further using similar
arguments as in Theorem 2.1, the proof is complete. �

Corollary 2.3. Let c1 ≥ 0 and α be real numbers with α > 12 or α <
1
2

. Let

f : Xp → Y be a mapping satisfying

N (Dρf(x1,x2, . . . ,xp), t) ≥
t

t + c1 (
∑p
i=1 |xi|α)

for all x1,x2, . . . ,xp ∈ X and all t > 0. Then there exists a unique square mapping
s : X → Y such that

N(f(x) −s(x), t) ≥




(ρα−ρ
1
2 )t

(ρα−ρ
1
2 )t+ρ

1
2 pc1|x|α

for α > 1
2

and 0 < ρ =
∑p
i=1

√
ρi < 1

(ρ
1
2 −ρα)t

(ρ
1
2 −ρα)t+ραpc1|x|α

for α < 1
2

and 0 < ρ =
∑p
i=1

√
ρi > 1

for all x ∈ X and all t > 0.

Proof. By taking ϕ(x1,x2, . . . ,xp) = c1 (
∑p
i=1 |xi|

α) for all x1,x2, . . . ,xp ∈ X in
Theorem 2.1 and Theorem 2.2, and choosing respectively L = ρ

1
2
−α and L = ρα−

1
2 ,

we get the deisred result. �



16 RAVI AND KUMAR

Corollary 2.4. Let c2 ≥ 0 and α be real numbers with α > 12 or α <
1
2

. Let

f : Xp → Y be a mapping satisfying

(7) N (Dρf(x1,x2, . . . ,xp), t) ≥
t

t + c2

(∏p
i=1 |xi|

α
p

)
for all x1,x2, . . . ,xp ∈ X and all t > 0. Then there exists a unique square root
mapping s : X → Y such that

N(f(x) −s(x), t) ≥




(ρα−ρ
1
2 )t

(ρα−ρ
1
2 )t+ρ

1
2 c2|x|α

for α > 1
2

and 0 < ρ =
∑p
i=1

√
ρi < 1

(ρ
1
2 −ρα)t

(ρ
1
2 −ρα)t+ραc2|x|α

for α < 1
2

and 0 < ρ =
∑p
i=1

√
ρi > 1

for all x ∈ X and all t > 0.

Proof. By taking ϕ(x1,x2, . . . ,xp) = c2

(∏p
i=1 |xi|

α
p

)
for all x1,x2, . . . ,xp ∈ X in

Theorem 2.1 and Theorem 2.2, and choosing respectively L = ρ
1
2
−α and L = ρα−

1
2 ,

we get the desired result. �

Corollary 2.5. Let c3 ≥ 0 and α be real numbers with α > 12 or α <
1
2

. Let

f : Xp → Y be a mapping satisfying

N (Dρf(x1,x2, . . . ,xp), t) ≥
t

t + c3

[∑p
i=1

(∏p
j=1,j 6=i |xj|

α
p−1

)]
for all x1,x2, . . . ,xp ∈ X and all t > 0. Then there exists a unique square root
mapping s : X → Y such that

N(f(x) −s(x), t) ≥




(ρα−ρ
1
2 )t

(ρα−ρ
1
2 )t+ρ

1
2 pc3|x|α

for α > 1
2

and 0 < ρ =
∑p
i=1

√
ρi < 1

(ρ
1
2 −ρα)t

(ρ
1
2 −ρα)t+ραpc3|x|α

for α < 1
2

and 0 < ρ =
∑p
i=1

√
ρi > 1

for all x ∈ X and all t > 0.

Proof. By taking ϕ(x1,x2, . . . ,xp) = c3

[∑p
i=1

(∏p
j=1,j 6=i |xj|

α
p−1

)]
for all x1,x2, . . . ,xp ∈

X in Theorem 2.1 and Theorem 2.2, and choosing respectively L = ρ
1
2
−α and

L = ρα−
1
2 , we get the desired result. �

Corollary 2.6. Let c4 ≥ 0 and α be real numbers with α > 12 or α >
1
2

. Let

f : Xp → Y be a mapping satisfying

N (Dρf(x1,x2, . . . ,xp), t) ≥
t

t + c4

[∏p
i=1 |xi|

α
p + (

∑p
i=1 |xi|α)

]



FUZZY STABILITY OF GENERALIZED SQUARE ROOT FUNCTIONAL EQUATION 17

for all x1,x2, . . . ,xp ∈ X and all t > 0. Then there exists a unique square root
mapping s : X → Y such that

N(f(x)−s(x), t) ≥




(ρα−ρ
1
2 )t

(ρα−ρ
1
2 )t+ρ

1
2 (p+1)c4|x|α

for α > 1
2

and 0 < ρ =
∑p
i=1

√
ρi < 1

(ρ
1
2 −ρα)t

(ρ
1
2 −ρα)t+ρα(p+1)c4|x|α

for α < 1
2

and 0 < ρ =
∑p
i=1

√
ρi > 1

for all x ∈ X and all t > 0.

Proof. By taking ϕ(x1,x2, . . . ,xp) = c4

[∏p
i=1 |xi|

α
p + (

∑p
i=1 |xi|

α)
]

for all x1,x2, . . . ,xp ∈
X in Theorem 2.1 and Theorem 2.2, and choosing respectively L = ρ

1
2
−α and

L = ρα−
1
2 , we get the desired result. �

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1PG & Research Department of Mathematics, Sacred Heart College, Tirupattur -

635 601, TamilNadu, India

2Department of Mathematics, C. Abdul Hakeem College of Engg. and Tech., Melvisharam

- 632 509, TamilNadu, India

∗Corresponding author