Fixed Points and Fuzzy Stability of Functional Equations ... · It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is

Available online at www.ispacs.com/jnaa

Volume 2012, Year 2012 Article ID jnaa-00109, 15 Pages

doi:10.5899/2012/jnaa-00109

Research Article

Fixed Points and Fuzzy Stability of Functional

Equations Related to Inner Product

Sun-Young Jang1, Choonkil Park2∗, Hassan Azadi Kenary3

(1) Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea

(2) Department of Mathematics, Hanyang University, Seoul 133-791, Korea

(3) Department of Mathematics, College of Science, Yasouj University, Yasouj 75914-353, Iran

Copyright 2012 c⃝ Sun-Young Jang, Choonkil Park, Hassan Azadi Kenary. This is an open access article

distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

AbstractIn [40], Th.M. Rassias introduced the following equality

m∑i,j=1

∥xi − xj∥2 = 2mm∑i=1

∥xi∥2,m∑i=1

xi = 0

for a fixed integer m ≥ 3.Let V,W be real vector spaces. It is shown that if a mapping f : V →W satisfies

m∑i,j=1

f(xi − xj) = 2mm∑i=1

f(xi) (0.1)

for all x1, . . . , xm ∈ V with∑m

i=1 xi = 0, then the mapping f : V → W is realized as thesum of an additive mapping and a quadratic mapping. From the above equality we candefine the functional equation

f(x− y) + f(2x+ y) + f(x+ 2y) = 3f(x) + 3f(y) + 3f(x+ y),

which is called a quadratic functional equation. Every solution of the quadratic functionalequation is said to be a quadratic mapping.

Using fixed point theorem we prove the Hyers-Ulam stability of the functional equation(0.1) in fuzzy Banach spaces.Keywords : Hyers-Ulam stability; fuzzy Banach space; fixed point; additive mapping; quadratic

mapping.

∗Corresponding author. Email address: [email protected]; Tel: +82-2-2220-0892

1

2 Journal of Nonlinear Analysis and Application

1 Introduction and preliminaries

The theory of fuzzy space has much progressed as developing the theory of randomness.Some mathematicians have defined fuzzy norms on a vector space from various points ofview [2, 16, 26, 30, 53]. Following Cheng and Mordeson [8], Bag and Samanta [2] gave anidea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosiland Michalek type [25] and investigated some properties of fuzzy normed spaces [3].

We use the definition of fuzzy normed spaces given in [2, 30, 31] to investigate a fuzzyversion of the Hyers-Ulam stability for the functional equation (0.1) in the fuzzy normedvector space setting.

Definition 1.1. ([2, 30, 31, 32]) 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,

(N1) N(x, t) = 0 for t ≤ 0;(N2) x = 0 if and only if N(x, t) = 1 for all t > 0;(N3) N(cx, t) = N(x, t

|c|) if c = 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) for x = 0, N(x, ·) is continuous on R.

The pair (X,N) is called a fuzzy normed vector space.

Definition 1.2. ([2, 30, 31, 32]) Let (X,N) be a fuzzy normed vector space. A se-quence {xn} in X is said to be convergent or converge if there exists an x ∈ X suchthat limn→∞N(xn − x, t) = 1 for all t > 0. In this case, x is called the limit of thesequence {xn} and we denote it by N -limn→∞ xn = x.

Definition 1.3. ([2, 30, 31]) Let (X,N) be a fuzzy normed vector space. A sequence {xn}in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such thatfor all n ≥ n0 and all p > 0, we have N(xn+p − xn, t) > 1− ε.

It is well-known that every convergent sequence in a fuzzy normed vector space isCauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be completeand 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 iscontinuous at a point x0 ∈ X if for each sequence {xn} converging to x0 in X, then thesequence {f(xn)} converges to f(x0). If f : X → Y is continuous at each x ∈ X, thenf : X → Y is said to be continuous on X (see [3]).

The stability problem of functional equations originated from a question of Ulam [52]concerning the stability of group homomorphisms. Hyers [19] gave a first affirmative par-tial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalizedby Aoki [1] for additive mappings and by Th.M. Rassias [39] for linear mappings by con-sidering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theoremwas obtained by Gavruta [18] by replacing the unbounded Cauchy difference by a generalcontrol function in the spirit of Th.M. Rassias’ approach.

The functional equation f(x+y)+f(x−y) = 2f(x)+2f(y) is called a quadratic func-tional equation. A generalized Hyers-Ulam stability problem for the quadratic functionalequation was proved by Skof [51] for mappings f : X → Y , where X is a normed spaceand Y is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the

Journal of Nonlinear Analysis and Application 3

relevant domain X is replaced by an Abelian group. Czerwik [11] proved the Hyers-Ulamstability of the quadratic functional equation. The stability problems of several functionalequations have been extensively investigated by a number of authors and there are manyinteresting results concerning this problem (see [15, 20, 23], [35]–[37], [41]–[50]).

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

We recall a fundamental result in fixed point theory.

Theorem 1.1. ([4, 14]) 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 givenelement x ∈ X, either

d(Jnx, Jn+1x) = ∞

for all nonnegative integers n or there exists a positive integer n0 such that

(1) d(Jnx, Jn+1x) <∞, ∀n ≥ n0;

(2) the sequence {Jnx} converges to a fixed point y∗ of J ;

(3) y∗ is the unique fixed point of J in the set Y = {y ∈ X | d(Jn0x, y) <∞};(4) d(y, y∗) ≤ 1

1−Ld(y, Jy) for all y ∈ Y .

In 1996, G. Isac and Th.M. Rassias [21] were the first to provide applications of stabilitytheory 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 havebeen extensively investigated by a number of authors (see [5]–[7], [29, 33, 34, 38, 49]).

This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability ofthe functional equation (0.1) in fuzzy Banach spaces for an even case. In Section 3, weprove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces foran odd case.

Throughout this paper, assume that X is a vector space and that (Y,N) is a fuzzyBanach space.

2 Hyers-Ulam stability of the functional equation (0.1): aneven mapping case

The square norm in an inner product space satisfies

m∑i,j=1

∥xi − xj∥2 = 2mm∑i=1

∥xi∥2,m∑i=1

xi = 0

for all x1, . . . , xm ∈ R with∑m

i=1 xi = 0 (see [40]). We can induce the following:

Df(x1, . . . , xm) =

m∑i,j=1

f(xi − xj)− 2m

m∑i=1

f(xi)


i=1 xi = 0.


Lemma 2.1. [9], Let V and W be real vector spaces. If a mapping f : V →W satisfies

m∑i,j=1

f(xi − xj) = 2m

m∑i=1

f(xi)


i=1 xi = 0, then the mapping f : V → W is realized as thesum of an additive mapping and a quadratic mapping.

Using fixed point method, we prove the Hyers-Ulam stability of the functional equationDf(x1, . . . , xm) = 0 in fuzzy Banach spaces: an even case.

Theorem 2.1. Let φ : Xm → [0,∞) be a function such that there exists an L < 1 with

φ(x1, . . . , xm) ≤ L

4φ (2x1, . . . , 2xm)

for all x1, . . . , xm ∈ X. Let f : X → Y be an even mapping satisfying

N (Df(x1, . . . , xm), t) ≥ t

t+ φ (x1, . . . , xm)(2.2)

for all x1, . . . , xm ∈ X and all t > 0. Then Q(x) := N − limn→∞ 4nf(

x2n

)exists for each

x ∈ X and defines a quadratic mapping Q : X → Y such that

N (f(x)−Q(x), t) ≥ (8− 8L)t

(8− 8L)t+ Lφ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)(2.3)

for all x ∈ X and all t > 0.

Proof. Letting x1 = x, x2 = −x and x3 = · · · = xm = 0 in (2.2), we get

N (2f (2x)− 8f(x), t) ≥ t

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)(2.4)

for all x ∈ X.Consider the set

S := {g : X → Y }

and introduce the generalized metric on S:

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

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

), ∀x ∈ X, ∀t > 0},

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see the proof of[28, Lemma 2.1]).

Now we consider the linear mapping J : S → S such that

Jg(x) := 4g(x2

)


for all x ∈ X.Let g, h ∈ S be given such that d(g, h) = ε. Then

N (g(x)− h(x), εt) ≥ t

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)

for all x ∈ X and all t > 0. Hence

N (Jg(x)− Jh(x), Lεt) = N(4g

(x2

)− 4h

(x2

), Lεt

)= N

(g(x2

)− h

(x2

),L

4εt

)≥

Lt4

Lt4 + φ(x2 ,−

x2 , 0, . . . , 0︸︷︷︸

m−2 times

)

≥Lt4

Lt4 + L

4φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)

=t

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

d(Jg, Jh) ≤ Ld(g, h)

for all g, h ∈ S.It follows from (2.4) that

N

(f(x)− 4f

(x2

),L

8t

)≥ t

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)

for all x ∈ X and all t > 0. So d(f, Jf) ≤ L8 .

By Theorem 1.1, there exists a mapping Q : X → Y satisfying the following:(1) Q is a fixed point of J , i.e.,

Q(x2

)=

1

4Q(x) (2.5)

for all x ∈ X. Since f : X → Y is even, Q : X → Y is an even mapping. The mapping Qis a unique fixed point of J in the set

M = {g ∈ S : d(f, g) <∞}.

This implies that Q is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0,∞)satisfying

N(f(x)−Q(x), µt) ≥ t

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)


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

N - limn→∞

4nf( x2n

)= Q(x)

for all x ∈ X;(3) d(f,Q) ≤ 1

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

d(f,Q) ≤ L

8− 8L.

This implies that the inequality (2.3) holds true.By (2.2),

N(4kDf

(x12k, . . . ,

xm2k

), 4kt

)≥ t

t+ φ(x1

2k, . . . , xm

2k

)for all x1, . . . , xm ∈ X, all t > 0 and all k ∈ N. So

N(4kDf

(x12k, . . . ,

xm2k

), t)≥

t4k

t4k

+ Lk

4kφ (x1, . . . , xm)

for all x1, . . . , xm ∈ X, all t > 0 and all n ∈ N. Since limk→∞t

4k

t

4k+Lk

4kφ(x1,...,xm)

= 1 for all

x1, . . . xm ∈ X and all t > 0,

N (DQ(x1, . . . , xm), t) = 1

for all x1, . . . , xm ∈ X and all t > 0. Thus Q(x1, . . . , xm) = 0. Since Q is even, it followsfrom [9, Lemma 2.1] that the mapping Q : X → Y is quadratic, as desired.

Corollary 2.1. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normedvector space with norm ∥ · ∥. Let f : X → Y be an even mapping satisfying

N (Df(x1, . . . , xm), t) ≥ t

t+∑m

i=1 θ(∥xi∥p)(2.6)

for all x1, . . . , xm ∈ X and all t > 0. Then Q(x) := N − limk→∞ 4kf(

x2k

)exists for each

x ∈ X and defines a quadratic mapping Q : X → Y such that

N (f(x)−Q(x), t) ≥ 2(2p − 4)t

2(2p − 4)t+ θ∥x∥p


Proof. The proof follows from Theorem 2.1 by taking

φ(x1, . . . , xm) :=

m∑i=1

θ(∥xi∥p)

for all x1, . . . , xm ∈ X. Then we can choose L = 22−p and we get the desired result.



φ(x1, . . . , xm) ≤ 4Lφ(x12, . . . ,

xm2

)for all x1, . . . , xm ∈ X. Let f : X → Y be an even mapping satisfying (2.2). ThenQ(x) := N − limk→∞

14kf(2kx

)exists for each x ∈ X and defines a quadratic mapping

Q : X → Y such that

N (f(x)−Q(x), t) ≥ (8− 8L)t

(8− 8L)t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)(2.7)


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

Jg(x) :=1

4g (2x)

for all x ∈ X.It follows from (2.4) that

N

(f(x)− 1

4f(2x),

1

8t

)≥ t

t+ φ(x,−x, 0, . . . , 0︸︷︷︸m−2 times

)

for all x ∈ X and all t > 0. So d(f, Jf) ≤ 18 .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.2. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normedvector space with norm ∥ · ∥. Let f : X → Y be an even mapping satisfying (2.6). ThenQ(x) := N − limk→∞

14kf(2kx

)exists for each x ∈ X and defines a quadratic mapping

Q : X → Y such that

N (f(x)−Q(x), t) ≥ 2(4− 2p)t

2(4− 2p)t+ θ∥x∥p



φ (x1, . . . , xm) := θm∑i=1

∥xi∥p

for all x1 . . . , xm ∈ X. Then we can choose L = 2p−2 and we get the desired result.


3 Hyers-Ulam stability of the functional equation (0.1): anodd mapping case

Using fixed point method, we prove the Hyers-Ulam stability of the functional equationDf(x1, . . . , xm) = 0 in fuzzy Banach spaces: an odd case.


φ(x1, . . . , xm) ≤ L

2φ (2x1, . . . , 2xm)

for all x1, . . . , xm ∈ X. Let f : X → Y be an odd mapping satisfying (2.2). ThenA(x) := N − limn→∞ 2nf

(x2n

)exists for each x ∈ X and defines an additive mapping

A : X → Y such that

N (f(x)−A(x), t) ≥ (4m− 4mL)t

(4m− 4mL)t+ Lφ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)(3.8)


Proof. Letting x1 = x, x2 = x, x3 = −2x and x4 = · · · = xm = 0 in (2.2), we get

N (2mf (2x)− 4mf(x), t) ≥ t

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)(3.9)

for all x ∈ X.Consider the set

S := {g : X → Y }

and introduce the generalized metric on S:

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

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

), ∀x ∈ X, ∀t > 0},

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see the proof of[28, Lemma 2.1]).

Now we consider the linear mapping J : S → S such that

Jg(x) := 2g(x2

)for all x ∈ X.Let g, h ∈ S be given such that d(g, h) = ε. Then

N (g(x)− h(x), εt) ≥ t

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)


for all x ∈ X and all t > 0. Hence

N (Jg(x)− Jh(x), Lεt) = N(2g

(x2

)− 2h

(x2

), Lεt

)= N

(g(x2

)− h

(x2

),L

2εt

)≥

Lt2

Lt2 + φ(x2 ,

x2 ,−x, 0, . . . , 0︸︷︷︸

m−3 times

)≥

Lt2

Lt2 + L

2φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)

=t

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

d(Jg, Jh) ≤ Ld(g, h)

for all g, h ∈ S.It follows from (3.9) that

N

(f(x)− 2f

(x2

),L

4mt

)≥ t

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)

for all x ∈ X and all t > 0. So d(f, Jf) ≤ L4m .

By Theorem 1.1, there exists a mapping A : X → Y satisfying the following:

(1) A is a fixed point of J , i.e.,

A(x2

)=

1

2A(x) (3.10)

for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A isa unique fixed point of J in the set

M = {g ∈ S : d(f, g) <∞}.

This implies that A is a unique mapping satisfying (3.10) such that there exists a µ ∈ (0,∞)satisfying

N(f(x)−A(x), µt) ≥ t

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)

for all x ∈ X;

(2) d(Jnf,A) → 0 as n→ ∞. This implies the equality

N - limn→∞

2nf( x2n

)= A(x)

for all x ∈ X;


(3) d(f,A) ≤ 11−Ld(f, Jf), which implies the inequality

d(f,A) ≤ L

4m− 4mL.

This implies that the inequality (3.8) holds true.By (2.2),

N(2kDf

(x12k, . . . ,

xm2k

), 2kt

)≥ t

t+ φ(x1

2k, . . . , xm

2k

)for all x1, . . . , xm ∈ X, all t > 0 and all k ∈ N. So

N(2kDf

(x12k, . . . ,

xm2k

), t)≥

t2k

t2k

+ Lk

2kφ (x1, . . . , xm)

for all x1, . . . , xm ∈ X, all t > 0 and all n ∈ N. Since limk→∞t

2k

t

2k+Lk

2kφ(x1,...,xm)

= 1 for all

x1, . . . xm ∈ X and all t > 0,

N (DA(x1, . . . , xm), t) = 1

for all x1, . . . , xm ∈ X and all t > 0. Thus A(x1, . . . , xm) = 0. Since A is odd, it followsfrom [9, Lemma 2.1] that the mapping A : X → Y is additive, as desired.

Corollary 3.1. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normedvector space with norm ∥ · ∥. Let f : X → Y be an odd mapping satisfying (2.6). ThenA(x) := N − limk→∞ 2kf

(x2k



N (f(x)−A(x), t) ≥ 2m(2p − 2)t

2m(2p − 2)t+ (2 + 2p)θ∥x∥p



φ(x1, . . . , xm) :=

m∑i=1

θ(∥xi∥p)

for all x1, . . . , xm ∈ X. Then we can choose L = 21−p and we get the desired result.


φ(x1, . . . , xm) ≤ 2Lφ(x12, . . . ,

xm2

)for all x1, . . . , xm ∈ X. Let f : X → Y be an odd mapping satisfying (2.2). ThenA(x) := N − limk→∞

12kf(2kx



N (f(x)−A(x), t) ≥ (4m− 4mL)t

(4m− 4mL)t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)(3.11)



Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.1.Consider the linear mapping J : S → S such that

Jg(x) :=1

2g (2x)

for all x ∈ X.It follows from (3.9) that

N

(f(x)− 1

4mf(2x),

1

4mt

)≥ t

t+ φ(x, x,−2x, 0, . . . , 0︸︷︷︸m−3 times

)

for all x ∈ X and all t > 0. So d(f, Jf) ≤ 14m .

The rest of the proof is similar to the proofs of Theorems 2.1 and 3.1.

Corollary 3.2. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normedvector space with norm ∥ · ∥. Let f : X → Y be an odd mapping satisfying (2.6). ThenA(x) := N − limk→∞

12kf(2kx



N (f(x)−A(x), t) ≥ 2m(2− 2p)t

2m(2− 2p)t+ (2 + 2p)θ∥x∥p



φ (x1, . . . , xm) := θ

m∑i=1

∥xi∥p

for all x1 . . . , xm ∈ X. Then we can choose L = 2p−1 and we get the desired result.

In this paper, we have proved the Hyers-Ulam stability of a functional equation, orig-inated from an inner product, in fuzzy Banach spaces for an even case and for an oddcase.

Acknowledgements

The first author was supported by University of Ulsan, 2009 Research Fund and had beenwritten during visiting the Research Institute of Mathematics, Seoul National University.The third author was supported by Basic Science Research Program through the Na-tional Research Foundation of Korea funded by the Ministry of Education, Science andTechnology (NRF-2009-0070788).

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Fixed Points and Fuzzy Stability of Functional Equations ... · It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is