On the Stability of Quadratic Double Centralizers and Quadratic Multipliers: A Fixed Point Approach

Volume 2011, Article ID 957541,9pages doi:10.1155/2011/957541

Research Article

On the Stability of Quadratic Double Centralizers and Quadratic Multipliers: A Fixed Point Approach

Abasalt Bodaghi,

Idham Arif Alias,

and Madjid Eshaghi Gordji

1Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

2Laboratory of Theoretical Studies, Institute for Mathematical Research, University Putra Malaysia UPM, 43400 Serdang, Selangor Darul Ehsan, Malaysia

3Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

Correspondence should be addressed to Abasalt Bodaghi,[email protected] Received 3 December 2010; Revised 11 January 2011; Accepted 18 January 2011 Academic Editor: Michel Chipot

Copyrightq2011 Abasalt Bodaghi et al. 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.

We prove the superstability of quadratic double centralizers and of quadratic multipliers on Banach algebras by fixed point methods. These results show that we can remove the conditions of being weakly commutative and weakly without order which are used in the work of M. E.

Gordji et al.2011for Banach algebras.

1. Introduction

In 1940, Ulam 1 raised the following question concerning stability of group homomor- phisms: under what condition does there exist an additive mapping near an approximately additive mapping? Hyers2answered the problem of Ulam for Banach spaces. He showed that for two Banach spacesXandY, if >0 andf:X → Ysuch that

f xy

−fx−f

y ≤, 1.1

for allx, y∈ X, then there exist a unique additive mappingT :X → Ysuch that

fx−Tx≤, x∈ X. 1.2

The work has been extended to quadratic functional equations. Considerf : X → Yto be a mapping such that ftx is continuous int ∈ R, for allx ∈ X. Assume that there exist constants≥0 andp∈0,1such that

f xy

−fx−f

y≤

x^py^p

, x∈ X. 1.3

Th. M. Rassias in3showed with the above conditions forf, there exists a uniqueR-linear mappingT :X → Ysuch that

fx−Tx≤ 2

2−2^px^p, x∈ X. 1.4

G˘avrut¸a then generalized the Rassias’s result in4.

A square norm on an inner product space satisfies the important parallelogram equality

xy²x−y² 2

x²y²

. 1.5

Recall that the functional equation

f xy

f x−y

2fx 2f y

1.6

is called quadratic functional equation. In addition, every solution of functional eqaution 1.6 is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof5for mappingsf :X → Y, whereXis a normed space andY is a Banach space. Cholewa 6 noticed that the theorem of Skof is still true if the relevant domainX is replaced by an abelian group. Indeed, Czerwik in7proved the Cauchy-Rassias stability of the quadratic functional equation. Since then, the stability problems of various functional equation have been extensively investigated by a number of authorse.g,8–13.

One should remember that the functional equation is called stable if any approximately solution to the functional equation is near to a true solution of that functional equation, and is super superstable if every approximately solution is an exact solution of itsee14. Recently, the first and third authors in15investigated the stability of quadratic double centralizer:

the maps which are quadratic and double centralizer. Later, Eshaghi Gordji et al. introduced a new concept of the quadratic double centralizer and the quadratic multipliers in16, and established the stability of quadratic double centralizer and quadratic multipliers on Banach algebras. They also established the superstability for those which are weakly commutative and weakly without order. In this paper, we show that the hypothesis on Banach algebras being weakly commutative and weakly without order in16can be eliminated, and prove the superstability of quadratic double centralizers and quadratic multipliers on a Banach algebra by a method of fixed point.

2. Stability of Quadratic Double Centralizers

A linear mappingL:A → Ais said to be left centralizer onAifLab Lab, for alla, b∈ A.

Similarly, a linear mappingR : A → AsatisfyingRab aRb, for alla, b ∈ Ais called right centralizer onA. A double centralizer onAis a pairL, R, whereLis a left centralizer,R is a right centralizer andaLb Rab, for alla, b∈ A. An operatorT :A → Ais said to be a multiplier ifaTb Tab, for alla, b∈ A.

Throughout this paper, let A be a complex Banach algebra. Recall that a mapping L:A → Ais a quadratic left centralizer ifLis a quadratic homogeneous mapping, that isL is quadratic andLλa λ²La, for alla∈ Aandλ∈C, andLab Lab², for alla, b∈ A.

A mappingR : A → A is a quadratic right centralizer ifR is a quadratic homogeneous mapping and Rab a²Rb, for all a, b ∈ A. Also, a quadratic double centralizer of an algebraA is a pairL, R whereL is a quadratic left centralizer, R is a quadratic right centralizer anda²Lb Rab², for alla, b∈ Asee16for details.

It is proven in8; that for the vector spacesXandYand the fixed positive integerk, the mapf:X → Yis quadratic if and only if the following equality holds:

2f

kxky 2

2f

kx−ky 2

k²fx k²f y

. 2.1

We thus can show thatfis quadratic if and only if for a fixed positive integerk, the following equality holds:

f

kxky f

kx−ky

2k²fx 2k²f y

. 2.2

Before proceeding to the main results, we will state the following theorem which is useful to our purpose.

Theorem 2.1 The alternative of fixed point 17. Suppose that we are given a complete generalized metric space X, d and a strictly contractive mapping T : X → X with Lipschitz constantL. Then for each givenx ∈ X, eitherdTⁿx, Tⁿ¹x ∞, for alln ≥ 0, or else exists a natural numbern0such that

1dTⁿx, Tⁿ¹x<∞, for alln≥n₀,

2the sequence{Tⁿx}is convergent to a fixed pointy^∗ofT,

3y^∗is the unique fixed point ofT in the setΛ {y∈X:dTⁿ⁰x, y<∞}, 4dy, y^∗≤1/1−Ldy, Ty, for ally∈Λ.

Theorem 2.2. Letf_j:A → Abe continuous mappings withf_j0 0 (j 0,1), and letφ:A⁶ → 0,∞be continuous in the first and second variables such that

fjλaλbcd fjλa−λbcd−2λ² fja fjb

−2 1−j

f_jcd²_1−j j

c²f_jd_j

u²f₀v−f₁uv²

≤a, b, c, d, u, v, 2.3

for allλ∈T{λ∈C:|λ|1}and, for alla, b, c, d, u, v∈ A, j0,1. If there exists a constantm, 0< m <1 such that

φa, b, c, d, u, v≤4mMin

φ a

2,b 2,c

2, d,u 2,v

2

, φ a

2,b 2, c,d

2,u 2,v

2

, 2.4

for all a, b, c, d, u, v ∈ A, then there exists a unique double quadratic centralizer L, R on A satisfying

f0a−La≤ 1

41−m φa, a,0,0,0,0, 2.5

f₁a−Ra≤ 1

41−mφa, a,0,0,0,0, 2.6

for alla∈ A.

Proof. From2.4, it follows that

limi 4⁻ⁱφ

2ⁱa,2ⁱb,2ⁱc, d,2ⁱu,2ⁱv

0, 2.7

for alla, b, c, d, u, v∈ A. Puttingj 0, λ1, ab, cduv0 and replacingaby 2a in2.3, we get

f₀2a−4f₀a≤φa, a,0,0,0,0, 2.8

for alla∈ A. By the above inequality, we have 1

4f₀2a−f₀a ≤ 1

4φa, a,0,0,0,0, 2.9

for alla∈ A. Consider the setX :{g :A → A |g0 0}and introduce the generalized metric onX:

d h, g

:inf

C∈R:ga−ha≤Cφa, a,0,0,0,0, ∀a∈ A

. 2.10

It is easy to show thatX, dis complete. Now, we define the linear mappingQ:X → Xby

Qha 1

4h2a, 2.11

for alla∈ A. Giveng, h∈X, letC∈Rbe an arbitrary constant withdg, h≤C, that is

ga−ha≤Cφa, a,0,0,0,0, 2.12

for alla∈ A. Substitutingaby 2ain the inequality2.12and using2.4and2.11, we have Qg

a−Qha 1

4g2a−h2a

≤ 1

4Cφ2a,2a,0,0,0,0

≤Cmφa, a,0,0,0,0,

2.13

for alla∈ A. Hence,dQg, Qh≤ Cm. Therefore, we conclude thatdQg, Qh≤ mdg, h, for allg, h∈X. It follows from2.9that

d

Qf₀, f₀

≤ 1

4. 2.14

By Theorem 2.1, Q has a unique fixed point L : A → A in the set X1 {h ∈ X, df0, h<∞}. On the other hand,

nlim→ ∞

f₀2ⁿa

4ⁿ La, 2.15

for alla∈ A. ByTheorem 2.1and2.14, we obtain

d f₀, L

≤ 1 1−md

Qf₀, L

≤ 1

41−m, 2.16

that is, the inequality2.5is true, for alla ∈ A. Now, substitute 2ⁿaand 2ⁿbbyaand b respectively, putc d u v 0 andj 0 in2.15. Dividing both sides of the resulting inequality by 2ⁿ, and lettingngoes to infinity, it follows from2.7and2.3that

Lλaλb Lλa−λb 2λ²La 2λ²Lb, 2.17

for alla, b∈ Aandλ∈T. Puttingλ1 in2.17we have

Lab La−b 2La 2Lb, 2.18

for alla, b∈ A. HenceLis a quadratic mapping.

Lettingb0 in2.17, we getLλa λ²La, for alla, b∈ Aandλ∈T. We can show from2.18thatLra r²Lafor any rational numberr. It follows from the continuity off0

andφthat for eachλ∈R,Lλa λ²La. So,

Lλa L λ

|λ||λ|a

λ²

|λ|²L|λ|a λ²

|λ|²|λ|²La λ²La, 2.19

for alla∈ Aandλ∈Cλ /0. Therefore,Lis quadratic homogeneous. Puttingj0,ab uv0 in2.3and replacing 2ⁿcbyc, we obtain

f₀2ⁿcd

4ⁿ − f₀2ⁿc 4ⁿ d²

≤ 1

24⁻ⁿφ0,0,2ⁿc, d,0,0. 2.20

By2.7, the right hand side of the above inequality tends to zero asn → ∞. It follows from 2.15thatLcd Lcd², for all c, d ∈ A. ThereforeLis a quadratic left centralizer. Also, one can show that there exists a unique mappingR:A → Awhich satisfies

nlim→ ∞

f₁2ⁿa

4ⁿ Ra, 2.21

for alla∈ A. The same manner could be used to show thatRis a quadratic right centralizer.

If we substituteuandvby 2ⁿuand 2ⁿvin2.3respectively, and putabcd0, and divide both sides of the obtained inequality by 8ⁿ, then we get

u²f₀2ⁿv

2ⁿ −f₁2ⁿu 2ⁿ v²

≤8⁻ⁿφ0,0,0,0,2ⁿu,2ⁿv. 2.22

Passing to the limit asn → ∞, and again from2.7, we conclude thatu²Lv Ruv², for allu, v∈ A. ThereforeL, Ris a quadratic double centralizer onA. This completes the proof of this theorem.

Now, we establish the superstability of double quadratic centralizers on Banach algebras as follows.

Corollary 2.3. Let 0< m <1, p <2 with 2^p−2≤m, letfj:A → Abe continuous mappings with f_j0 0 (j0,1), and let

fjλaλbcd fjλa−λbcd−2λ² fja fjb

−2 1−j

f_jcd²_1−j j

c²f_jd_j

u²f₀v−f₁uv²

≤

a^pb^pc^pu^pv^p d^p,

2.23

for allλ ∈T {λ ∈C:|λ| 1}and, for alla, b, c, d, u, v ∈ A, j 0,1. Thenf0, f₁is a double quadratic centralizer onA.

Proof. The result follows fromTheorem 2.2by puttingφa, b, c, d, u, v a^pb^pc^p u^pv^pd^p.

3. Stability of Quadratic Multipliers

Assume thatAis a complex Banach algebra. Recall that a mappingT :A → Ais a quadratic multiplier ifTis a quadratic homogeneous mapping, anda²Tb Tab², for alla, b∈ Asee 16. We investigate the stability of quadratic multipliers.

Theorem 3.1. Letf :A → Abe a continuous mapping withf0 0 and letφ:A⁴ → 0,∞be a function which is continuous in the first and second variables such that

fλaλb fλa−λb−2λ² fa fb

c²fd−fcd²≤φa, b, c, d, 3.1

for allλ∈Tand alla, b, c, d∈ A. Suppose exists a constantm, 0< m <1, such that

φ2a,2b,2c,2d≤4mφa, b, c, d, 3.2

for alla, b, c, d∈ A. Then there exists a unique multiplierT onAsatisfying fa−Ta≤ 1

41−mφa, a,0,0, 3.3

for alla∈ A.

Proof. It follows from3.2that

lim_{n→ ∞}φ2ⁿa,2ⁿb,2ⁿc,2ⁿd

4ⁿ 0, 3.4

for alla, b, c, d∈ A. Puttingλ1,ab, cd0 in3.1, we obtain

f2a−4fa≤φa, a,0,0, 3.5

for alla∈ A. Thus

fa−1 4f2a

≤ 1

4φa, a,0,0, 3.6

for alla∈ A. Now we setX :{h:A → A |h0 0}and introduce the generalized metric onXas

d g, h

:inf

C∈R:ga−ha≤Cφa, a,0,0, ∀a∈ A

. 3.7

It is easy to show thatX, dis complete. Consider the mappingΦ : X → X defined by Φha 1/4h2a, for alla∈ A. By the same reasoning as in the proof ofTheorem 2.2,Φis strictly contractive onX. It follows from3.6thatdΦf, f≤1/4. ByTheorem 2.1,Φhas a unique fixed point in the setX₁ :{h∈X :df, h<∞}. LetT be the fixed point ofΦ. Then

T is the unique mapping withT2a 4Ta, for alla∈ Asuch that there existsC∈0,∞ satisfying

Tx−fx≤Cφa, a,0,0, 3.8

for alla∈ A. On the other hand, we have limn→ ∞dΦⁿf, T 0. Thus

lim_{n→ ∞} 1

4ⁿf2ⁿx Tx, 3.9

for alla∈ A. Hence

d f, T

≤ 1 1−md

T,Φ f

≤ 1

41−m. 3.10

This implies the inequality3.3. It follows from3.1,3.4and3.9that Tλaλb Tλa−λb−2λ²Ta−2λ²Tb

lim_{n→ ∞} 1 4ⁿ

T2ⁿλaλb T2ⁿλa−λb−2λ²T2ⁿa−2λ²T2ⁿb

≤limn→ ∞ 1

4ⁿφ2ⁿa,2ⁿb,0,0 0,

3.11

for alla, b∈ A. Thus

Lλaλb Lλa−λb 2λ²La 2λ²Lb, 3.12

for alla, b ∈ Aandλ ∈T. Lettingb 0 in3.14, we haveLλa λ²La, for alla, b ∈ A andλ∈T. Now, it follows from the proof ofTheorem 2.1and continuity offandφthatT is C-linear. If we substitutecanddby 2ⁿcand 2ⁿdin3.1, respectively, and putab0 and we divide the both sides of the obtained inequality by 8ⁿ, we get

c²f2ⁿd

4ⁿ −f2ⁿc 4ⁿ d²

≤ φ0,0,2ⁿc,2ⁿd

8ⁿ . 3.13

Passing to the limit as n → ∞, and from 3.4 we conclude thatc²Td Tcd², for all c, d∈ A.

UsingTheorem 3.1, we establish the superstability of quadratic multipliers on Banach algebras.

Corollary 3.2. Let 0< m <1, p <2/3 with 2^3p−2 ≤m, andf :A → Abe a continuous mapping withf0 0, and let

fλaλb fλa−λb−2λ² fa fb

c²fd−fcd²≤

a^pab^p

c^pd^p, 3.14

for allλ∈T{λ∈C:|λ|1}and, for alla, b, c, d∈ A. Thenfis a quadratic multiplier onA.

Proof. The results follows fromTheorem 3.1by puttingφa, b, c, d a^pb^pc^pd^p.

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