Fuzzy Stability of the Pexiderized Quadratic Functional Equation: A Fixed Point Approach

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Volume 2009, Article ID 460912,10pages doi:10.1155/2009/460912

Research Article

Fuzzy Stability of the Pexiderized Quadratic

Functional Equation: A Fixed Point Approach

Zhihua Wang

1, 2

_{and Wanxiong Zhang}

3

1_{Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China} 2_{School of Science, Hubei University of Technology, Wuhan, Hubei 430068, China}

3_{College of Mathematics and Physics, Chongqing University, Chongqing 400044, China}

Correspondence should be addressed to Wanxiong Zhang,[email protected]

Received 25 April 2009; Revised 31 July 2009; Accepted 16 August 2009

Recommended by Massimo Furi

The fixed point alternative methods are implemented to give generalized Hyers-Ulam-Rassias stability for the Pexiderized quadratic functional equation in the fuzzy version. This method introduces a metrical context and shows that the stability is related to some fixed point of a suitable operator.

Copyrightq2009 Z. Wang and W. Zhang. 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.

1. Introduction

The aim of this article is to extend the applications of the fixed point alternative method to provide a fuzzy version of Hyers-Ulam-Rassias stability for the functional equation:

fxyfx−y2gx 2hy, 1.1

which is said to be a Pexiderized quadratic functional equation or called a quadratic functional equation forf g h. During the last two decades, the Hyers-Ulam-Rassias stability of1.1has been investigated extensively by several mathematicians for the mapping

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As we see, the powerful method for studying the stability of functional equation was first suggested by Hyers17while he was trying to answer the question originated from the problem of Ulam18, and it is called a direct method because it allows us to construct the additive function directly from the given functionf. In 2003, Radu19proposed the fixed point alternative method for obtaining the existence of exact solutions and error estimations. Subsequently, Mihet¸20applied the fixed alternative method to study the fuzzy stability of the Jensen functional equation on the fuzzy space which is defined in14.

Practically, the application of the two methods is successfully extended to obtain a fuzzy approximate solutions to functional equations 14, 20. A comparison between the direct method and fixed alternative method for functional equations is given in19. The fixed alternative method can be considered as an advantage of this method over direct method in the fact that the range of approximate solutions is much more than the latter14.

2. Preliminaries

Before obtaining the main result, we firstly introduce some useful concepts: a fuzzy normed linear space is a pairX, N, whereXis a real linear space andNis a fuzzy norm onX, which is defined as follow.

Definition 2.1cf.6. A functionN :X×R → 0,1 the so-called fuzzy subsetis said to be a fuzzy norm onXif for allx, y∈Xand alls, t∈R,Nx,·is left continuous for everyx

and satisfies

N1Nx, c 0 forc≤0;

N2x0 if and only ifNx, c 1 for allc >0; N3Ncx, t Nx, t/|c|ifc /0;

N4Nxy, st≥min{Nx, s, Ny, t};

N5Nx,·is a nondecreasing function onRand limt→ ∞Nx, t 1.

Let X, N be a fuzzy normed linear space. A sequence {xn} in X is said to be convergent if there existsx ∈ X such that limn→ ∞Nxn−x, t 1t > 0. In that case, x is called the limit of the sequence{xn}and we writeN−limxnx.

A sequence{xn}in a fuzzy normed spaceX, N is called Cauchy if for eachε > 0 andδ > 0, there existsn0 ∈ N such thatNxm−xn, δ > 1−εm, n ≥ n0. 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.

We recall the following result by Margolis and Diaz.

Lemma 2.2cf.19,21. LetX, dbe a complete generalized metric space and letJ:X → Xbe a strictly contractive mapping, that is,

dJx, Jy≤ Ldx, y, ∀x, y∈X, 2.1

for someL≤1. Then, for each fixed elementx∈X, either

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or

dJnx, Jn1x<∞, ∀n≥n0, 2.3

for some natural numbern0. Moreover, if the second alternative holds, then:

ithe sequence{Jn_x_}_{is convergent to a fixed point}_y∗_of_J_;

iiy∗is the unique fixed point ofJin the setY :{y∈X|dJn0_{x, y}_<∞}_and_d_{y, y}∗≤ 1/1−Ldy, Jy,for allx, y∈Y.

3. Main Results

We start our works with a fuzzy generalized Hyers-Ulam-Rassias stability theorem for the Pexiderized quadratic functional equation 1.1. Due to some technical reasons, we first examine the stability for odd and even functions and then we apply our results to a general function.

The aim of this section is to give an alternative proof for that result in15, Section 3, based on the fixed point method. Also, our method even provides a better estimation.

Theorem 3.1. LetXbe a linear space and letZ, Nbe a fuzzy normed space. Letϕ:X×X → Z

be a function such that

ϕ2x,2yαϕx, y, ∀x, y∈X, t >0, 3.1

for some real numberαwith0<|α|<2. LetY, Nbe a fuzzy Banach space and letf, g,andhbe odd functions fromXtoY such that

Nfxyfx−y−2gx−2hy, t≥Nϕx, y, t, ∀x, y∈X, t >0. 3.2

Then there exists a unique additive mappingT :X → Y such that

NTx−fx, t≥M1

x,2− |α|

2 t

, 3.3

Ngx hx−Tx, t≥M1

x, 6−3|α|

10−2|α|t

, 3.4

whereM1x, t min{Nϕx, x,2/3t, Nϕx,0,2/3t, Nϕ0, x,2/3t}.

The nextLemma 3.2has been proved in15, Proposition 3.1.

Lemma 3.2. If α > 0, then Nfx−2−1_f₂_x_{, t} _≥ _M

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Proof ofTheorem 3.1. Without loss of generality we may assume thatα > 0. By changing the roles ofxandyin3.2, we obtain

Nfxy−fx−y−2gy−2hx, t≥Nϕy, x, t. 3.5

It follows from3.2,3.5, andN4that

Nfxy−gx−hy−gy−hx, t≥minNϕx, y, t, Nϕy, x, t . 3.6

Puttingy0 in3.6, we get

Nfx−gx−hx, t≥minNϕx,0, t, Nϕ0, x, t . 3.7

LetE:{φ |φ:X → Y, φ0 0}and introduce the generalized metricdM1,define it onEby

dM1

φ1, φ2

infε∈0,∞|Nφ1x−φ2x, εt

≥M1x, t,∀x∈X, t >0 . 3.8

Then, it is easy to verify thatdM1is a complete generalized metric onEsee the proof of22 or23. We now define a functionJ1:E → Eby

J1φx 1

2φ2x, ∀x∈X. 3.9

We assert that J1 is a strictly contractive mapping with the Lipschitz constant α/2. Given φ1, φ2∈E, letε∈0,∞be an arbitrary constant withdM1φ1, φ2≤ε. From the definition of

dM1, it follows that

Nφ1x−φ1x, εt

≥M1x, t, ∀x∈X, t >0. 3.10

Therefore,

NJ1φ1x−J1φ2x, α

2εt

N

1

2φ12x− 1 2φ22x,

α

2εt

Nφ12x−φ22x, αεt

≥M12x, αt M1x, t, ∀x∈X, t >0.

3.11

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Next, from Nfx − 2−1_f₂_x_{, t} _≥ _M

1x, t see Lemma 3.2, it follows that dM1f, J1f ≤ 1. From the fixed point alternative, we deduce the existence of a fixed point ofJ1, that is, the existence of a mappingT :X → Y such thatT2x 2Txfor eachx∈X.

Moreover, we havedM1J n

1f, T → 0, which implies

N− lim n→ ∞

f2n_x

2n Tx, ∀x∈X. 3.12

Also,dM1f, T≤1/1−LdM1f, J1fimplies the inequality

dM1

f, T≤ 1

1−α/2

2

2−α. 3.13

Ifεnis a decreasing sequence converging to 2/2−α, then

NTx−fx, εnt

≥M1x, t, ∀x∈X, t >0, n∈N. 3.14

Then implies that

x, 1 εn

t

, ∀x∈X, t >0, n∈N, 3.15

that is,asM1is left continuous

x,2−α

2 t

, ∀x∈X, t >0. 3.16

The additivity ofTcan be proved in a similar fashion as in the proof of Proposition 3.115. It follows from3.3and3.7that

N

gx hx−Tx,5−α

3 t

≥min

Nfx−Tx, t, N

gx hx−fx,2−α

3 t

≥min

M1

x,2−α

2 t

, N

ϕx,0,2−α

3 t

, N

ϕ0, x,2−α

3 t

≥M1

x,2−α

2 t

,

3.17

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The uniqueness ofT follows from the fact thatT is the unique fixed point ofJ1with

the property that there existsk∈0,∞such that

NTx−fx, kt≥M1x, t, ∀x∈X, t >0. 3.18

This completes the proof of the theorem.

ϕ2x,2yαϕx, y, ∀x, y∈X, t >0, 3.19

for some real numberαwith0 < |α| <4. LetY, Nbe a fuzzy Banach space and letf, g,andhbe even functions fromXtoY such thatf0 g0 h0 0and

Nfxyfx−y−2gx−2hy, t≥Nϕx, y, t, ∀x, y∈X, t >0. 3.20

Then there exists a unique quadratic mappingQ:X → Y such that

NQx−fx, t≥M1

x,4− |α|

2 t

,

NQx−gx, t≥M1

x,12−3|α|

10− |α|t

,

NQx−hx, t≥M1

x,12−3|α|

10− |α|t

,

3.21

whereM1x, t min{Nϕx, x,2/3t, Nϕx,0,2/3t, Nϕ0, x,2/3t}.

The followingLemma 3.4has been proved in15, Proposition 3.2.

Lemma 3.4. Ifα > 0, thenNfx−4−1_f₂_x_,_t _≥ _M

2x, tandM22x,t M2x,t/α,∀x ∈ X,t >0, whereM2x,t=min{Nϕx, x,4/3t,Nϕx,0,4/3t,Nϕ0, x,4/3t}.

Proof ofTheorem 3.3. Without loss of generality we may assume thatα > 0. By changing the roles ofxandyin3.20, we obtain

Nfxyfx−y−2gy−2hx, t≥Nϕy, x, t. 3.22

Puttingyxin3.20, we get

Nf2x−2gx−2hx, t≥Nϕx, x, t. 3.23

Puttingx0 in3.20, we get

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Similarly, puty0 in3.20to obtain

N2fx−2gx, t≥Nϕx,0, t. 3.25

LetE:{ψ |ψ :X → Y, ψ0 0}and introduce the generalized metricdM2,define it onEby

dM2

ψ1, ψ2

infε∈0,∞|Nψ1x−ψ2x, εt

≥M2x, t,∀x∈X, t >0 . 3.26

Then, it is easy to verify thatdM2is a complete generalized metric onEsee the proof of22 or23. We now define a functionJ2:E → Eby

J2ψx

1

4ψ2x, ∀x∈X. 3.27

We assert that J2 is a strictly contractive mapping with the Lipschitz constant α/4. Given ψ1, ψ2∈E, letε∈0,∞be an arbitrary constant withdM2ψ1, ψ2≤ε. From the definition of

dM2, it follows that

Nψ1x−ψ2x, εt

≥M2x, t, ∀x∈X, t >0. 3.28

Therefore,

NJ2ψ1x−J2ψ2x,α

4εt

N

1

4ψ12x− 1 4ψ22x,

α

4εt

Nψ12x−ψ22x, αεt

≥M22x, αt M2x, t, ∀x∈X, t >0.

3.29

Hence, it holds that dM2J2ψ1, J2ψ2 ≤ α/4ε, that is, dM2J2ψ1, J2ψ2 ≤ α/4dM2ψ1, ψ2,

∀ψ2, ψ2∈E.

Next, from Nfx − 4−1_f₂_x_{, t} _≥ _M

2x, t see Lemma 3.4, it follows that dM2f, J2f ≤ 1. From the fixed alternative, we deduce the existence of a fixed point ofJ2, that is, the existence of a mapping Q : X → Y such thatQ2x 4Qx for eachx ∈ X. Moreover, we havedM2J

n

2f, Q → 0, which implies that

N− lim n→ ∞

f2n_x

4n Qx, ∀x∈X. 3.30

Also,dM2f, Q≤1/1−LdM2f, J2fimplies the inequality

dM2

f, Q≤ 1

1−α/4 4

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Ifεnis a decreasing sequence converging to 4/4−α, then

NQx−fx, εnt

≥M2x, t, ∀x∈X, t >0, n∈N. 3.32

Then implies that

x, 1 εn

t

, ∀x∈X, t >0, n∈N, 3.33

that is,asM2is left continuous

x,4−α

4 t

M1

x,4−α

2 t

, ∀x∈X, t >0.

3.34

The quadratic ofQcan be proved in a similar fashion as in the proof of Proposition 3.215. It follows from3.25and3.34that

N

Qx−gx,10−α

6 t

≥min

NQx−fx, t, N

fx−gx,4−α

6 t

≥min

M2

x,4−α

4 t

, N

ϕx,0,4−α

3 t

≥M2

x,4−α

4 t

M1

x,4−α

2 t

,

3.35

whence

NQx−gx, t≥M1

x,12−3α

10−αt

. 3.36

A similar inequality holds forh. The rest of the proof is similar to the proof ofTheorem 3.1.

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for some real numberαwith0<|α|<2. LetY, Nbe a fuzzy Banach space and letfbe a mapping fromXtoY such thatf0 0and

Nfxyfx−y−2fx−2fy, t≥Nϕx, y, t, ∀x, y∈X, t >0. 3.38

Then there exist unique mappingT andQfromXtoYsuch thatTis additive,Qis quadratic, and

Nfx−Tx−Qx, t≥M

x,2− |α|

8 t

, 3.39

whereMx, t=min{Nϕx,x,2/3t,Nϕ−x,−x,2/3t,Nϕx,0,2/3t,

Nϕ0,x,2/3t,Nϕ−x,0,2/3t,Nϕ0,−x,2/3t}.

Proof. Letf0x 1/2fx−f−xfor allx∈X, thenf00 0, f0−x −f0xand

Nf0

xyf0

x−y−2f0x−2f0

y, t≥minNϕx, y, t, Nϕ−x,−y, t .

3.40

Letfex 1/2fx f−xfor allx∈X, thenfe0 0, fe−x fexand

Nfe

xyfe

x−y−2fex−2fe

y, t≥minNϕx, y, t, Nϕ−x,−y, t .

3.41

Using the proofs of Theorems3.1and3.3, we get unique an additive mappingT and unique quadratic mappingQsatisfying

Nf0x−Tx, t

≥M

x,2− |α|

4 t

,

Nfex−Qx, t≥M

x,4− |α|

4 t

.

3.42

Therefore,

Nfx−Tx−Qx, t≥min

N

f0x−Tx, t

2

, N

fex−Qx,t

2 ≥min M

x,2− |α|

8 t

, M

x,4− |α|

8 t

M

x,2− |α|

8 t

.

3.43

This completes the proof of the theorem.

Acknowledgment

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