Fuzzy Stability of Jensen-Type Quadratic Functional Equations

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2009, Article ID 535678, 17 pagesdoi:10.1155/2009/535678

Research ArticleFuzzy Stability of Jensen-Type QuadraticFunctional Equations

Sun-Young Jang,1 Jung Rye Lee,2 Choonkil Park,3and Dong Yun Shin4

1 Department of Mathematics, University of Ulsan, Ulsan 680-749, South Korea2 Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea3 Department of Mathematics, Hanyang University, Seoul 133-791, South Korea4 Department of Mathematics, University of Seoul, Seoul 130-743, South Korea

Correspondence should be addressed to Dong Yun Shin, [email protected]

Received 29 December 2008; Revised 26 March 2009; Accepted 10 April 2009

Recommended by John Rassias

We prove the generalized Hyers-Ulam stability of the following quadratic functional equations2f((x + y)/2) + 2f((x − y)/2) = f(x) + f(y) and f(ax + ay) + (ax − ay) = 2a2f(x) + 2a2f(y) infuzzy Banach spaces for a nonzero real number awith a/= ± 1/2.

Copyright q 2009 Sun-Young Jang et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concern-ing the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer tothe question of Ulam for Banach spaces. Hyers’ theoremwas generalized byAoki [3] for addi-tive mappings and by Th. M. Rassias [4] for linear mappings by considering an unboundedCauchy difference. The work of Th. M. Rassias [4] has provided a lot of influence in the devel-opment of what we call generalized Hyers-Ulam stability of functional equations. A generaliza-tion of the Th. M. Rassias theorem was obtained by Gavruta [5] by replacing the unboundedCauchy difference by a general control function in the spirit of Th. M. Rassias’ approach.

J. M. Rassias [6] proved a similar stability theorem in which he replaced the factor‖x‖p + ‖y‖p by ‖x‖p · ‖y‖q for p, q ∈ R with p + q /= 1 (see also [7, 8] for a number of other newresults). The papers of J. M. Rassias [6–8] introduced the Ulam- Gavruta-Rassias stability offunctional equations. See also [9–11].

The functional equation

f(x + y

)+ f(x − y

)= 2f(x) + 2f

(y)

(1.1)

2 Abstract and Applied Analysis

is called a quadratic functional equation. In particular, every solution of the quadratic functionalequation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for thequadratic functional equation was proved by Skof [12] for mappings f : X → Y , where Xis a normed space and Y is a Banach space. Cholewa [13] noticed that the theorem of Skof isstill true if the relevant domain X is replaced by an Abelian group. In [14], Czerwik provedthe generalized Hyers-Ulam stability of the quadratic functional equation.

J. M. Rassias [15] introduced and solved the stability problem of Ulam for the Euler-Lagrange-type quadratic functional equation

f(rx + sy

)+ f(sx − ry

)=(r2 + s2

)[f(x) + f

(y)], (1.2)

motivated from the following pertinent algebraic equation

|ax + by|2 + |bx − ay|2 =(a2 + b2

)(|x|2 + |y|2

). (1.3)

The solution of the functional equation (1.2) is called a Euler-Lagrange-type quadratic mapping.J. M. Rassias [16, 17] introduced and investigated the relative functional equations. Inaddition, J. M. Rassias [18] generalized the algebraic equation (1.3) to the following equation

mn|ax + by|2 + |nbx −may|2 =(ma2 + nb2

)(n|x|2 +m|y|2

), (1.4)

and introduced and investigated the general pertinent Euler-Lagrange quadratic mappings.Analogous quadratic mappings were introduced and investigated in [19, 20].

These Euler-Lagrange mappings are named Euler-Lagrange-Rassias mappings and thecorresponding Euler-Lagrange equations are called Euler-Lagrange-Rassias equations. Before1992, these mappings and equations were not known at all in functional equations andinequalities. However, a completely different kind of Euler-Lagrange partial differentialequations are known in calculus of variations. Therefore, we think that J. M. Rassias’introduction of Euler-Lagrange mappings and equations in functional equations andinequalities provides an interesting cornerstone in analysis. Already some mathematicianshave employed these Euler-Lagrange mappings.

Recently, Jun and Kim [21] solved the stability problem of Ulam for another Euler-Lagrange-Rassias-type quadratic functional equation. Jun and Kim [22] introduced andinvestigated the following quadratic functional equation of Euler-Lagrange-Rassias type:

n∑

i=1

riQ

⎛

⎝n∑

j=1

rj(xi − xj

)⎞

⎠ +

(n∑

i=1

ri

)

Q

(n∑

i=1

rixi

)

=

(n∑

i=1

ri

)2 n∑

i=1

riQ(xi), (1.5)

whose solution is said to be a generalized quadratic mapping of Euler-Lagrange-Rassias type.During the last two decades a number of papers and research monographs have

been published on various generalizations and applications of the generalized Hyers-Ulamstability to a number of functional equations and mappings (see [9, 23–26]).

Katsaras [27] defined a fuzzy norm on a vector space to construct a fuzzy vectortopological structure on the space. Some mathematicians have defined fuzzy norms on

Abstract and Applied Analysis 3

a vector space from various points of view [28–30]. In particular, Bag and Samanta [31],following Cheng and Mordeson [32], gave an idea of fuzzy norm in such a manner thatthe corresponding fuzzy metric is of Kramosil and Michalek type [33]. They established adecomposition theorem of a fuzzy norm into a family of crisp norms and investigated someproperties of fuzzy normed spaces [34].

We use the definition of fuzzy normed spaces given in [31] and [35–38] to investigate afuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations

2f(x + y

2

)+ 2f(x − y

2

)= f(x) + f

(y), (1.6)

f(ax + ay

)+ f(ax − ay

)= 2a2f(x) + 2a2f

(y)

(1.7)

in the fuzzy normed vector space setting.

Definition 1.1 (see [31, 35–38]). Let X be a real vector space. A function N : X × R → [0, 1] iscalled 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 ifN(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.The properties of fuzzy normed vector spaces and examples of fuzzy norms are given

in [35–38].

Definition 1.2 (see [31, 35–38]). Let (X,N) be a fuzzy normed vector space. A sequence {xn} inX is said to be convergent or converge if there exists an x ∈ X such that limn→∞N(xn − x, t) = 1for all t > 0. In this case, x is called the limit of the sequence {xn} and we denote it by N-limn→∞xn = x.

Definition 1.3 (see [31, 35–38]). 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 that for alln ≥ 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 Yis continuous 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, then f : X →Y is said to be continuous on X (see [34]).

In this paper, we prove the generalized Hyers-Ulam stability of the quadraticfunctional equations (1.6) and (1.7) in fuzzy Banach spaces.

Throughout this paper, assume that X is a vector space and that (Y,N) is a fuzzyBanach space. Let a be a nonzero real number with a/= (±1/2).


2. Fuzzy Stability of Quadratic Functional Equations

We prove the fuzzy stability of the quadratic functional equation (1.6).

Theorem 2.1. Let f : X → Y be an even mapping with f(0) = 0. Suppose that ϕ is a mapping fromX to a fuzzy normed space (Z,N

′) such that

N

(2f(x + y

2

)+ 2f(x − y

2

)− f(x) − f

(y), t + s

)≥ min

{N

′(ϕ(x), t

),N

′(ϕ(y), s)}

(2.1)

for all x, y ∈ X \ {0} and all positive real numbers t, s. If ϕ(3x) = αϕ(x) for some positive realnumber α with α < 9, then there is a unique quadratic mapping Q : X → Y such that Q(x) = N-limn→∞

f(3nx)/9n and

N(Q(x) − f(x), t

) ≥ M

(x,

(9 − α)t18

), (2.2)

where

M(x, t) := min{N

′(ϕ(x),

32t

),N

′(ϕ(2x),

32t

),N

′(ϕ(3x),

32t

),N

′(ϕ(0),

32t

)}. (2.3)

Proof. Putting y = 3x and s = t in (2.1), we get

N(2f(2x) + 2f(−x) − f(x) − f(3x), 2t

) ≥ min{N

′(ϕ(x), t

),N

′(ϕ(3x), t

)}(2.4)

for all x ∈ X and all t > 0. Replacing x by 2x, y by 0, and s by t in (2.1), we obtain

N(4f(x) − f(2x), 2t

) ≥ min{N

′(ϕ(2x), t

),N

′(ϕ(0), t

)}. (2.5)

Thus

N(9f(x) − f(3x), 6t

) ≥ min{N

′(ϕ(x), t

),N

′(ϕ(2x), t

),N

′(ϕ(3x), t

),N

′(ϕ(0), t

)}, (2.6)

and so

N

(f(x) − f(3x)

9, t

)≥ min

{N

′(ϕ(x),

32t

),N

′(ϕ(2x),

32t

)N

′(ϕ(3x),

32t

),N

′(ϕ(0),

32t

)}.

(2.7)

Then by the assumption,

M(3x, t) = M

(x,

t

α

). (2.8)


Replacing x by 3nx in (2.7) and applying (2.8), we get

N

(f(3nx)

9n− f(3n+1x

)

9n+1,αnt

9n

)

= N

(

f(3nx) − f(3n+1x

)

9, αnt

)

≥ M(3nx, αnt)

= M(x, t).

(2.9)

Thus for each n > m we have

N

(f(3mx)

9m− f(3nx)

9n,n−1∑

k=m

αkt

9k

)

= N

(n−1∑

k=m

(f(3kx)

9k− f(3k+1x

)

9k+1

)

,n−1∑

k=m

αkt

9k

)

≥ min

{n−1⋃

k=m

{

N

(f(3kx)

9k− f(3k+1x

)

9k+1,αkt

9k

)}}

≥ M(x, t).

(2.10)

Let ε > 0 and δ > 0 be given. Since limt→∞M(x, t) = 1, there is some t0 > 0 such thatM(x, t0) > 1 − ε. Since

∑∞k=0α

kt0/9k < ∞, there is some n0 ∈ N such that∑n−1

k=mαkt0/9k < δ for

n > m ≥ n0. It follows that

N

(f(3mx)

9m− f(3nx)

9n, δ

)≥ N

(f(3mx)

9m− f(3nx)

9n,n−1∑

k=m

αkt

9k

)

≥ M(x, t0)

≥ 1 − ε

(2.11)

for all t ≥ t0. This shows that the sequence {f(3nx)/9n} is Cauchy in (Y,N). Since (Y,N) iscomplete, {f(3nx)/9n} converges to someQ(x) ∈ Y . Thus we can define a mappingQ : X →Y by Q(x) := N − limt→∞f(3nx)/9n. Moreover, if we put m = 0 in (2.10), then we observethat

N

(f(3nx)

9n− f(x),

n−1∑

k=0

αkt

9k

)

≥ M(x, t). (2.12)

Thus

N

(f(3nx)

9n− f(x), t

)≥ M

(

x,t

∑n−1k=0 (α/9)

k

)

. (2.13)


Next we show that Q is quadratic. Let x, y ∈ X. Then we have

N

(2Q(x + y

2

)+ 2Q

(x − y

2

)−Q(x) −Q

(y), t

)

≥ min

{

N

(

2Q(x + y

2

)− 2f

(3n(x + y

)/2)

9n,t

5

)

,

N

(

2Q(x − y

2

)− 2f

(3n(x − y

)/2)

9n,t

5

)

,

N

(f(3nx)

9n−Q(x),

t

5

),N

(f(3ny)

9n−Q(y),t

5

)

,

N

(2f(3n(x + y

)/2)

9n+2f(3n(x − y

)/2)

9n− f(3nx)

9n− f(3ny)

9n,t

5

)}

.

(2.14)

The first four terms on the right-hand side of the above inequality tend to 1 as n → ∞ andthe fifth term, by (2.1), is greater than or equal to

min{N

′(ϕ(3nx),

9nt10

),N

′(ϕ(3ny),9nt10

)}=min

{N

′(ϕ(x),

(9α

)n t

10

),

(ϕ(y),

(9α

)n t

10

)},

(2.15)

which tends to 1 as n → ∞. Hence

N

(2Q(x + y

2

)+ 2Q

(x − y

2

)−Q(x) −Q

(y), t

)= 1 (2.16)

for all x, y ∈ X and all t > 0. This means that Q satisfies the Jensen quadratic functionalequation and so it is quadratic.

Next, we approximate the difference between f and Q in a fuzzy sense. For everyx ∈ X and t > 0, by (2.13), for large enough n, we have

N(Q(x) − f(x), t

) ≥ min

{

N

(

Q(x) − f(3ny)

9n,t

2

)

,N

(f(3ny)

9n− f(x),

t

2

)}

≥ M

(

x,t

2∑∞

k=0 (α/9)k

)

= M

(x,

(9 − α)t18

).

(2.17)


The uniqueness assertion can be proved by a standard fashion; cf. [36]: Let Q′ be anotherquadratic mapping fromX into Y , which satisfies the required inequality. Then for each x ∈ Xand t > 0,

N(Q(x) −Q

′(x), t

)≥ min

{N

(Q(x) − f(x),

t

2

),N

(Q

′(x) − f(x),

t

2

)}

≥ M

(x,

(9 − α)t36

).

(2.18)

Since Q and Q′are quadratic,

N(Q(x) −Q

′(x), t

)= N(Q(3nx) −Q

′(3nx), 9nt

)

≥ M

(x,

(9/α)n(9 − α)t36

).

(2.19)

for all x ∈ X, all t > 0 and all n ∈ N.Since 0 < α < 9, limn→∞(9/α)

n = ∞. Hence the right-hand side of the above inequalitytends to 1 as n → ∞. It follows that Q(x) = Q

′(x) for all x ∈ X.

Theorem 2.2. Let f : X → Y be an even mapping with f(0) = 0. Suppose that ϕ is a mapping fromX to a fuzzy normed space (Z,N

′) satisfying (2.1). If ϕ(3x) = αϕ(x) for some real number αwith α >

9, then there is a unique quadratic mapping Q : X → Y such that Q(x) = N-limn→∞9nf(x/3n)and

N(Q(x) − f(x), t

) ≥ M

(x,

(α − 9)t2α

), (2.20)

where

M(x, t) := min{N

′(ϕ

(x

3

),t

6

),N

′(ϕ

(2x3

),t

6

),N

′(ϕ(x),

t

6

),N

′(ϕ(0),

t

6

)}. (2.21)

Proof. It follows from (2.7) that

N

(f(x) − 9f

(x

3

), t

)≥min

{N

′(ϕ

(x

3

),t

6

),N

′(ϕ

(2x3

),t

6

),N

′(ϕ(x),

t

6

),N

′(ϕ(0),

t

6

)}.

(2.22)


M

(x

3, t

)= M(x, αt). (2.23)


Replacing x by x/3n in (2.22) and applying (2.23), we get

N

(9nf(

x

3n

)− 9n+1f

(x

3n+1

),9ntαn

)= N

(f

(x

3n

)− 9f(

x

3n+1

),t

αn

)

≥ M

(x

3n,t

αn

)

= M(x, t).

(2.24)


N

(

9mf(

x

3m

)− 9nf

(x

3n

),n−1∑

k=m

9ktαk

)

= N

(n−1∑

k=m

(9kf(

x

3k

)− 9k+1f

(x

3k+1

)),n−1∑

k=m

9ktαk

)

≥ min

{n−1⋃

k=m

{

N

(

9kf(

x

3k

)− 9k+1f

(x

3k+1

),9ktαk

)}}

≥ M(x, t).

(2.25)


∑∞k=09

kt0/αk < ∞, there is some n0 ∈ N such that

∑n−1k=m9

kt0/αk < δ for


N

(9mf(

x

3m

)− 9nf

(x

3n

), δ

)≥ N

(

9mf(

x

3m

)− 9nf

(x

3n

),n−1∑

k=m

9ktαk

)

≥ M(x, t0)

≥ 1 − ε

(2.26)

for all t ≥ t0. This shows that the sequence {9nf(x/3n)} is Cauchy in (Y,N). Since (Y,N)is complete, {9nf(x/3n)} converges to some Q(x) ∈ Y . Thus we can define a mappingQ : X → Y by Q(x) := N-limt→∞9nf(x/3n). Moreover, if we put m = 0 in (2.8), then weobserve that

N

(

9nf(

x

3n

)− f(x),

n−1∑

k=0

9ktαk

)

≥ M(x, t). (2.27)

Thus

N

(9nf(

x

3n

)− f(x), t

)≥ M

(

x,t

∑n−1k=0 (9/α)

k

)

. (2.28)

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


Theorem 2.3. Let f : X → Y be a mapping with f(0) = 0. Suppose that ϕ is a mapping from Xto a fuzzy normed space (Z,N

′) satisfying (2.1). If ϕ(2x) = αϕ(x) for some positive real number

α with α < 4, then there is a unique quadratic mapping Q : X → Y such that Q(x) = N-limn→∞f(2nx)/4n and

N(Q(x) − f(x), t

) ≥ M

(x,

(4 − α)t8

)(2.29)

whereM(x, t) := min{N ′(ϕ(2x), 2t),N

′(ϕ(0), 2t)}.

Proof. Letting y = 0 and replacing x by 2x and s by t in (2.1), we obtain

N(4f(x) − f(2x), 2t

) ≥ min{N

′(ϕ(2x), t

),N

′(ϕ(0), t

)}. (2.30)

Thus

N

(f(x) − f(2x)

4, t

)≥ min

{N

′(ϕ(2x), 2t

),N

′(ϕ(0), 2t

)}. (2.31)


M(2x, t) = M

(x,

t

α

). (2.32)

Replacing x by 2nx in (2.31) and applying (2.32), we get

N

(f(2nx)

4n− f(2n+1x

)

4n+1,αnt

4n

)

= N

(

f(2nx) − f(4n+1x

)

4, αnt

)

≥ M(2nx, αnt)

= M(x, t).

(2.33)


N

(f(2mx)

4m− f(2nx)

4n,n−1∑

k=m

αkt

4k

)

= N

(n−1∑

k=m

(f(2kx)

4k− f(2k+1x

)

4k+1

)

,n−1∑

k=m

αkt

4k

)

≥ min

{n−1⋃

k=m

{

N

(f(2kx)

4k− f(2k+1x

)

4k+1,αkt

4k

)}}

≥ M(x, t).

(2.34)



∑∞k=0α

kt0/4k < ∞, there is some n0 ∈ N such that∑n−1

k=mαkt0/4k < δ for


N

(f(2mx)

4m− f(2nx)

4n, δ

)≥ N

(f(2mx)

4m− f(2nx)

4n,n−1∑

k=m

αkt

4k

)

≥ M(x, t0)

≥ 1 − ε

(2.35)

for all t ≥ t0. This shows that the sequence {f(2nx)/4n} is Cauchy in (Y,N). Since (Y,N) iscomplete, {f(2nx)/4n} converges to someQ(x) ∈ Y . Thus we can define a mappingQ : X →Y by Q(x) := N-limt→∞f(2nx)/4n. Moreover, if we put m = 0 in (2.34), then we observe that

N

(f(2nx)

4n− f(x),

n−1∑

k=0

αkt

4k

)

≥ M(x, t). (2.36)

Thus

N

(f(2nx)

4n− f(x), t

)≥ M

(

x,t

∑n−1k=0 (α/4)

k

)

. (2.37)


Theorem 2.4. Let f : X → Y be a mapping with f(0) = 0. Suppose that ϕ is a mapping from X toa fuzzy normed space (Z,N

′) satisfying (2.1). If ϕ(2x) = αϕ(x) for some real number α with α > 4,

then there is a unique quadratic mapping Q : X → Y such that Q(x) = N-limn→∞4nf(x/2n) and

N(Q(x) − f(x), t

) ≥ M

(x,

(α − 4)t2α

), (2.38)

whereM(x, t) := min{N ′(ϕ(x), t/2),N

′(ϕ(0), t/2)}.


N(f(x) − 4f

(x2

), t)≥ min

{N

′(ϕ(x),

t

2

),N

′(ϕ(0),

t

2

)}. (2.39)


M(x2, t)= M(x, αt). (2.40)


Replacing x by x/2n in (2.39) and applying (2.40), we get

N

(4nf( x

2n)− 4n+1f

(x

2n+1

),4ntαn

)= N

(f( x

2n)− 4f(

x

2n+1

),t

αn

)

≥ M

(x

2n,t

αn

)

= M(x, t).

(2.41)


N

(

4mf( x

2m)− 4nf

( x

2n),n−1∑

k=m

4ktαk

)

= N

(n−1∑

k=m

(4kf(

x

2k

)− 4k+1f

(x

2k+1

)),n−1∑

k=m

4ktαk

)

≥ min

{n−1⋃

k=m

{

N

(

4kf(

x

2k

)− 4k+1f

(x

2k+1

),4ktαk

)}}

≥ M(x, t).

(2.42)


∑∞k=04

kt0/αk < ∞, there is some n0 ∈ N such that

∑n−1k=m4

kt0/αk < δ for


N(4mf( x

2m)− 4nf

( x

2n), δ)≥ N

(

4mf( x

2m)− 4nf

( x

2n),n−1∑

k=m

4ktαk

)

≥ M(x, t0)

≥ 1 − ε

(2.43)

for all t ≥ t0. This shows that the sequence {4nf(x/2n)} is Cauchy in (Y,N). Since (Y,N) iscomplete, {4nf(x/2n)} converges to someQ(x) ∈ Y . Thus we can define a mappingQ : X →Y by Q(x) := N-limt→∞4nf(x/2n). Moreover, if we put m = 0 in (2.42), then we observe that

N

(

4nf( x

2n)− f(x),

n−1∑

k=0

4ktαk

)

≥ M(x, t). (2.44)

Thus

N(4nf( x

2n)− f(x), t

)≥ M

(

x,t

∑n−1k=0 (4/α)

k

)

. (2.45)



Now we prove the fuzzy stability of the quadratic functional equation (1.7) for thecase a/= (±1/2).

Theorem 2.5. Let |2a| > 1 and f : X → Y a mapping with f(0) = 0. Suppose that ϕ is a mappingfrom X to a fuzzy normed space (Z,N

′) such that

N(f(ax + ay

)+ f(ax − ay

) − 2a2f(x) − 2a2f(y), t + s

)≥ min

{N

′(ϕ(x), t

),N

′(ϕ(y), s)}

(2.46)

for all x, y ∈ X \ {0} and all positive real numbers t, s. If ϕ(2ax) = αϕ(x) for some positive realnumber α with 0 < α < 4a2, then there is a unique quadratic mapping Q : X → Y such thatQ(x) = N-limn→∞f((2a)

nx)/(2a)2n and

N(Q(x) − f(x), t

) ≥ N′(

ϕ(x),

(4a2 − α

)t

4

)

(2.47)

for all x ∈ X and all t > 0.

Proof. Putting y = x and s = t in (2.46), we get

N(f(2ax) − 4a2f(x), 2t

)≥ N

′(ϕ(x), t

)(2.48)

for all x ∈ X and all t > 0. Thus

N

(f(x) − f(2ax)

4a2,

t

2a2

)≥ N

′(ϕ(x), t

)(2.49)

and so

N

(f(x) − f(2ax)

4a2, t

)≥ N

′(ϕ(x), 2a2t

). (2.50)

Replacing x by (2a)nx in (2.50), we get

N

⎛

⎜⎝

f((2a)nx

)

(2a)2n−f((2a)n+1x

)

(2a)2n+2,

αnt

(2a)2n

⎞

⎟⎠ = N

⎛

⎜⎝f((2a)nx

) −f((2a)n+1x

)

4a2, αnt

⎞

⎟⎠

≥ N′(ϕ(x), 2a2t

).

(2.51)



N

(f((2a)mx

)

(2a)2m− f((2a)nx

)

(2a)2n,n−1∑

k=m

αkt

(2a)2k

)

= N

⎛

⎜⎝

n−1∑

k=m

⎛

⎜⎝

f((2a)kx

)

(2a)2k−f((2a)k+1x

)

(2a)2k+2

⎞

⎟⎠,

n−1∑

k=m

αkt

(2a)2k

⎞

⎟⎠

≥ min

⎧⎪⎨

⎪⎩

n−1⋃

k=m

⎧⎪⎨

⎪⎩N

⎛

⎜⎝

f((2a)kx

)

(2a)2k−f((2a)k+1x

)

(2a)2k+2,

αkt

(2a)2k

⎞

⎟⎠

⎫⎪⎬

⎪⎭

⎫⎪⎬

⎪⎭

≥ N′(ϕ(x), 2a2t

).

(2.52)

Let ε > 0 and δ > 0 be given. Since limt→∞N′(ϕ(x), 2a2t) = 1, there is some t0 > 0

such that N′(ϕ(x), 2a2t0) > 1 − ε. Since

∑∞k=0α

kt0/(2a)2k < ∞, there is some n0 ∈ N such that

∑n−1k=mα

kt0/(2a)2k < δ for n > m ≥ n0. It follows that

N

(f((2a)mx

)

(2a)2m− f((2a)nx

)

(2a)2n, δ

)

≥ N

(f((2a)mx

)

(2a)2m− f((2a)nx

)

(2a)2n,n−1∑

k=m

αkt

(2a)2k

)

≥ N′(ϕ(x), 2a2t0

)

≥ 1 − ε

(2.53)

for all t ≥ t0. This shows that the sequence {f((2a)nx)/(2a)2n} is Cauchy in (Y,N). Since(Y,N) is complete, {f((2a)nx)/(2a)2n} converges to some Q(x) ∈ Y . Thus we can define amapping Q : X → Y by Q(x) := N-limt→∞f((2a)

nx)/(2a)2n. Moreover, if we put m = 0 in(2.52), then we observe that

N

(f((2a)nx

)

(2a)2n− f(x),

n−1∑

k=0

αkt

(2a)2k

)

≥ N′(ϕ(x), 2a2t

). (2.54)

Thus

N

(f((2a)nx

)

(2a)2n− f(x), t

)

≥ N′

⎛

⎝ϕ(x),2a2t

∑n−1k=0 (α/(2a)

2)k

⎞

⎠. (2.55)


Theorem 2.6. Let |2a| < 1 and f : X → Y a mapping with f(0) = 0. Suppose that ϕ is a mappingfrom X to a fuzzy normed space (Z,N

′) satisfying (2.46). If ϕ(2ax) = αϕ(x) for some real number


α with α > 4a2, then there is a unique quadratic mapping Q : X → Y such that Q(x) = N-limn→∞(2a)

2nf(x/(2a)n) and

N(Q(x) − f(x), t

) ≥ M

(

x,

(α − 4a2)t

4

)

(2.56)

for all x ∈ X and all t > 0.


N(f(x) − (2a)2f

( x

2a

), 2t)≥ N

′(ϕ( x

2a

), t)

(2.57)

for all x ∈ X and all t > 0. Thus

N(f(x) − 4a2f

( x

2a

), t)≥ N

′(ϕ( x

2a

),t

2

)= N

′(ϕ(x),

α

2t). (2.58)

Replacing x by x/(2a)n in (2.58), we get

N

(

(2a)2nf(

x

(2a)n

)− (2a)2n+2f

(x

(2a)n+1

)

,(2a)2ntαn

)

= N

(

f

(x

(2a)n

)− 4a2f

(x

(2a)n+1

)

, αnt

)

≥ N′(ϕ(x),

α

2t).

(2.59)


N

(

(2a)2mf(

x

(2a)m

)− (2a)2nf

(x

(2a)n

),n−1∑

k=m

(2a)2ktαk

)

= N

(n−1∑

k=m

(

(2a)2kf

(x

(2a)k

)

− (2a)2k+2f

(x

(2a)k+1

))

,n−1∑

k=m

(2a)2ktαk

)

≥ min

{n−1⋃

k=m

{

N

(

(2a)2kf

(x

(2a)k

)

− (2a)2k+2f

(x

(2a)k+1

)

,(2a)2ktαk

)}}

≥ N′(ϕ(x),

α

2t).

(2.60)


Let ε > 0 and δ > 0 be given. Since limt→∞N′(ϕ(x), (α/2)t) = 1, there is some t0 > 0 such

that N′(ϕ(x), (α/2)t0) > 1 − ε. Since

∑∞k=0(2a)

2kt0/αk < ∞, there is some n0 ∈ N such that

∑n−1k=m(2a)

2kt0/αk < δ for n > m ≥ n0. It follows that

N

((2a)2mf

(x

(2a)m

)− (2a)2nf

(x

(2a)n

), δ

)

≥ N

(

(2a)2mf(

x

(2a)m

)− (2a)2nf

(x

(2a)n

),n−1∑

k=m

(2a)2ktαk

)

≥ N′(ϕ(x),

α

2t0)

≥ 1 − ε

(2.61)

for all t ≥ t0. This shows that the sequence {(2a)2nf(x/(2a)n)} is Cauchy in (Y,N). Since(Y,N) is complete, {(2a)2nf(x/(2a)n)} converges to some Q(x) ∈ Y . Thus we can define amapping Q : X → Y by Q(x) := N-limt→∞(2a)

2nf(x/(2a)n). Moreover, if we put m = 0 in(2.60), then we observe that

N

(

(2a)2nf(

x

(2a)n

)− f(x),

n−1∑

k=0

(2a)2ktαk

)

≥ N′(ϕ(x),

α

2t). (2.62)

Thus

N

((2a)2nf

(x

(2a)n

)− f(x), t

)≥ N

′

⎛

⎝ϕ(x),αt

2∑n−1

k=0 ((2a)2/α)

k

⎞

⎠. (2.63)


Acknowledgment

Dr. Sun-Young Jang was supported by the Research Fund of University of Ulsan in 2008, andDr. Choonkil Park was supported by National Research Foundation of Korea (NRF-2009-0070788).

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Fuzzy Stability of Jensen-Type Quadratic Functional Equations

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