Ovidius Constanta defined the notion of generalized homomorphisms in quasi–Banach algebras

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 423–430 DOI: 10.18514/MMN.2018.2056

A CORRECTION TO APPROXIMATION OF GENERALIZED HOMOMORPHISMS IN QUASI–BANACH ALGEBRAS

ISMAIL NIKOUFAR Received 23 June, 2016

Abstract. Eshaghi et. al [Approximation of generalized homomorphisms in quasi–Banach algeb- ras, An. St. Univ. Ovidius Constanta, 17(2), (2009), 203–214] defined the notion of generalized homomorphisms in quasi–Banach algebras. They investigated generalized homomorphisms from quasi–Banach algebras top–Banach algebras and proved the generalized Hyers–Ulam–Rassias stability. In this paper, we show that their results only hold for Banach algebras and then we correct and prove the results forp–Banach algebras.

2010Mathematics Subject Classification: 46B03; 47Jxx; 47B48; 39B52

Keywords: Hyers–Ulam–Rassias stability, quasi–Banach algebra,p–Banach algebra, general- ized homomorphism

1. INTRODUCTION AND PRELIMINARIES

We know that aC-linear mappingf WA!Bis called a homomorphism iff .xy/D f .x/f .y/for allx; y2Aso that every homomorphism is a generalized homomorph- ism, but the converse is false, in general.

Eshaghi et. al [1] investigated generalized homomorphisms from quasi–Banach algebras top–Banach algebras and proved the generalized Hyers–Ulam–Rassias sta- bility. In this paper, we verify that the results presented in [1] hold for Banach al- gebras. Then, we correct their results and prove the results for p–Banach algeb- ras. We remark that the presenting results in this paper hold forp–Banach algebras, where0 < p1, in general. The stability problems of several functional equations have been extensively investigated by a number of authors inp–Banach algebras and there are many interesting results concerning this problem (see [2,3,5] and references therein).

LetX be a real linear space. A quasi–norm is a real–valued function onX satis- fying the following conditions:

(i) jjxjj 0for allx2X andjjxjj D0if and only ifxD0, (ii) jjxjj D jjjjxjjfor all2Rand allx2X,

(iii) there is a constantK1such thatjjxCyjj K.jjxjjCjjyjj/for allx; y2X.

c 2018 Miskolc University Press

The pair.X;jj jj/is called a quasi–normed space ifjj jjis a quasi–norm onX. The smallest possibleKis called the modulus of concavity ofjj jj. A quasi–Banach space is a complete quasi–normed space. Indeed, by a quasi–Banach space we mean a quasi–normed space in which every jj jj–Cauchy sequence inX converges. This class includes Banach spaces and the most significant class of quasi–Banach spaces which are not Banach spaces. A quasi–normjj jjis called ap–norm.0 < p1/if

jjxCyjj^p jjxjj^pC jjyjj^p (1.1) for allx; y2X. In this case, a quasi–Banach space is called ap–Banach space.

Let.A;jjjj/be a quasi–normed space. The quasi–normed space.A;jjjj/is called a quasi–normed algebra if Ais an algebra and there is a constant K > 0 such that jjxyjj Kjjxjjjjyjjfor allx; y2A. A quasi–Banach algebra is a complete quasi–

normed algebra. If the quasi–normjj jjis ap–norm, then the quasi–Banach algebra is called ap–Banach algebra. Eshaghi et. al [1] defined the notion of generalized homomorphisms in quasi–Banach algebra as follows:

Definition 1. LetAbe a quasi–Banach algebra with quasi–normjj jjAand letB be ap-Banach algebra withp–normjj jj^B. AC-linear mappingf WA!Bis called a generalized homomorphism if there exists a homomorphismhWA!B such that f .xy/Df .x/h.y/for allx; y2A.

Then, they investigated generalized homomorphisms from quasi–Banach algebras top–Banach algebras associated with the following functional equation

rf .xCy

r /Df .x/Cf .y/

and proved the generalized Hyers–Ulam–Rassias stability and superstability of gen- eralized homomorphisms in quasi–Banach algebras. In this paper, we prove that their results only hold for Banach algebras and then we correct their results and confirm the results forp–Banach algebras.

2. MAIN PROBLEMS

Following [1] throughout this paper, assume thatAis a quasi–Banach algebra with quasi–normjj jj^Aand thatBis ap–Banach algebra withp–normjj jj^B. In addition, we assumerto be a constant positive integer.

We will use the following lemma in this section.

Lemma 1([4]). LetX andY be linear spaces and letf WX !Y be an additive mapping such thatf .x/Df .x/for allx2X and2T¹WD f´2CW j´j D1g. Then the mappingf isC-linear.

The following two theorems proved in [1, Theorems 2.2 and 2.5]. In these the- orems, the authors want to prove the generalized Hyers–Ulam–Rassias stability of generalized homomorphisms from quasi–Banach algebras top–Banach algebras.

Theorem 1. Supposef WA!Bis a mapping withf .0/D0for which there exist a mappinggWA!Bwithg.0/D0,g.1/D1and a function'WA⁴!R^Csuch that

jjrf .aCbCcd

r / f .a/ f .b/ f .c/g.d /jj^B'.a; b; c; d /; (2.1) jjg.abCcd / g.a/g.b/ g.c/g.d /jj^B'.a; b; c; d /; (2.2) and

Q

'.a; b; c; d /WD

1

X

iD0

'.2ⁱa; 2ⁱb; 2ⁱc; 2ⁱd / 2ⁱ <1

for alla; b; c; d 2Aand all2T¹. Then, there exists a unique generalized homo- morphismhWA!Bsuch that

jjf .a/ h.a/jj^B 1

2'.a; a; 0; 0/Q for alla2A.

Theorem 2. Supposef WA!Bis a mapping withf .0/D0for which there exist a mappinggWA!Bwithg.0/D0,g.1/D1and a function'WA⁴!R^Csatisfying the inequalities(2.1),(2.2)and

Q

'.a; b; c; d /WD

1

X

iD1

2ⁱ'.a 2ⁱ; b

2ⁱ; c 2ⁱ;d

2ⁱ/ <1

for alla; b; c; d 2Aand all2T¹. Then, there exists a unique generalized homo- morphismhWA!Bsuch that

jjf .a/ h.a/jjB 1

2'.a; a; 0; 0/Q for alla2A.

In the following theorems we correct the results and prove the generalized Hyers–

Ulam–Rassias stability of generalized homomorphisms from quasi–Banach algebras top–Banach algebras.

Theorem 3. Supposef WA!Bis a mapping withf .0/D0for which there exist a mappinggWA!Bwithg.0/D0,g.1/D1and a function'WA⁴!R^Csuch that

jjrf .aCbCcd

r / f .a/ f .b/ f .c/g.d /jjB'.a; b; c; d /; (2.3) jjg.abCcd / g.a/g.b/ g.c/g.d /jjB'.a; b; c; d /; (2.4) and

Q

'.a; b; c; d /WDX¹

iD0

'.2ⁱa; 2ⁱb; 2ⁱc; 2ⁱd /^p 2^ip

_p¹

<1 (2.5)

for alla; b; c; d 2Aand all2T¹. Then, there exists a unique generalized homo- morphismhWA!Bsuch that

jjf .a/ h.a/jjB 1

2'.a; a; 0; 0/Q (2.6) for alla2A.

Proof. SettingaDb,cDd D0andrDD1in (2.3) and dividing both sides of the resulting inequality by2, we obtain

jjf .2a/

2 f .a/jjB'.a; a; 0; 0/

2 (2.7)

for alla2A. Then, we have jjf .2a/

2 f .a/jj_B^p '.a; a; 0; 0/^p

2^p (2.8)

for all a2A. Replacing a in (2.7) by 2a and dividing both sides of the resulting inequality by2, we get

jjf .2²a/

2²

f .2a/

2 jj^B'.2a; 2a; 0; 0/

2² (2.9)

for alla2A. Then, we have jjf .2²a/

2²

f .2a/

2 jjB^p '.2a; 2a; 0; 0/^p

2^2p (2.10)

for alla2A. Applying (2.8), (2.10), and (1.1) we detect that jjf .2²a/

2² f .a/jj_B^p '.a; a; 0; 0/^p

2^p C'.2a; 2a; 0; 0/^p

2^2p (2.11)

for alla2A. By induction onnwe conclude that jjf .2ⁿa/

2ⁿ f .a/jjB^p 1 2^p

n 1

X

iD0

'.2ⁱa; 2ⁱa; 0; 0/^p

2^ip (2.12)

for alla2Aand all non-negative integersn. Consequently, jjf .2^nCma/

2^nCm

f .2^ma/

2^m jjB1 2

^nCX^{m 1}

iDm

'.2ⁱa; 2ⁱa; 0; 0/^p 2^ip

_p¹

(2.13) for all non-negative integers n and m with nm and alla2A. It follows from (2.5) and (2.13) that the sequence f^{f .2}₂^nna/gis Cauchy inB for alla2A. SinceB is ap–Banach algebras, this sequence is convergent inB for all a2A. Define the mapping

h.a/WD lim

n!1

f .2ⁿa/

2ⁿ : (2.14)

Setting cDd D0; r D1 and replacinga; b by2ⁿa; 2ⁿb, respectively, in (2.3) and dividing both sides of (2.3) by2ⁿand taking the limit asn! 1we deduce

h.aCb/Dh.a/Ch.b/ (2.15)

for alla; b2Aand2T¹. So the mappinghisC-linear by Lemma1. Note that inequality (2.6) follows from (2.12) and (2.14). To show that his unique, let kbe anotherC-linear mapping satisfying (2.6). From (2.6) we conclude that

jjh.a/ k.a/jjB^p D 1

2^npjjh.2ⁿa/ k.2ⁿa/jjB^p

1

2^np.jjh.2ⁿa/ f .2ⁿa/jj_B^pC jjf .2ⁿa/ k.2ⁿa/jj^p_B

1

2^np 2

2^p'.2Q ⁿa; 2ⁿa; 0; 0/^p D 2

2^p

1

X

iDn

'.2ⁱa; 2ⁱa; 0; 0/^p 2^ip

for alla2A. The right hand side tends to zero as n! 1. The rest of the proof is

similar to that of [1, Theorem 2.2] and we omit it.

Theorem 4. Supposef WA!Bis a mapping withf .0/D0for which there exist a mappinggWA!Bwithg.0/D0,g.1/D1and a function'WA⁴!R^Csatisfying the inequality(2.3)and(2.4)and

Q

'.a; b; c; d /WDX¹

iD1

2^ip'.a 2ⁱ; b

2ⁱ; c 2ⁱ;d

2ⁱ/^p _p¹

<1 (2.16) for alla; b; c; d 2Aand all2T¹. Then, there exists a unique generalized homo- morphismhWA!Bsuch that

jjf .a/ h.a/jjB 1

2'.a; a; 0; 0/Q (2.17) for alla2A.

Proof. SettingaDb,cDdD0andrDD1in (2.3), we obtain

jjf .2a/ 2f .a/jjB'.a; a; 0; 0/ (2.18) for alla2A. Replacingain (2.18) by ^a₂, we get

jjf .a/ 2f .a

2/jjB'.a 2;a

2; 0; 0/ (2.19)

for alla2A. Replacingain (2.19) by ^a₂ and multiplying both sides of the resulting inequality by2, we detect

jj2f .a

2/ 2²f .a

2²/jj^B2'. a 2²; a

2²; 0; 0/ (2.20)

for alla2A. Using (2.19), (2.20), and (1.1) we conclude that jjf .a/ 2²f .a

2²/jj^p_B'.a 2;a

2; 0; 0/^pC2^p'.a 2²; a

2²; 0; 0/ (2.21) for alla2A. By induction onnwe deduce that

jjf .a/ 2ⁿf .a

2ⁿ/jj^pB 1 2^p

n

X

iD1

2^ip'.a 2ⁱ; a

2ⁱ; 0; 0/^p (2.22) for alla2Aand all non-negative integersn. Hence,

jj2^mf . a

2^m/ 2^nCmf . a

2^nCm/jjB^p 1 2^p

mCn

X

iDmC1

2^ip'. a 2^iCm; a

2^iCm; 0; 0/^p (2.23) for all non-negative integers n and m with nm and alla2A. It follows from (2.16) and (2.23) that the sequencef2ⁿf .₂^an/gis Cauchy inB for all a2Aso that this sequence is convergent inB. Define the mapping

h.a/WD lim

n!12ⁿf .a

2ⁿ/: (2.24)

The rest of the proof is similar to that of Theorem3and we omit it.

Corollary 1. Supposef WA!B is a mapping with f .0/D0 for which there exist constants > 0; ˛¤1and a mappinggWA!Bwithg.0/D0,g.1/D1such that

jjrf .aCbCcd

r / f .a/ f .b/ f .c/g.d /jj^B .jjajjA^˛C jjbjjA^˛C jjcjjA^˛C jjdjjA^˛/;

jjg.abCcd / g.a/g.b/ g.c/g.d /jjB.jjajjA^˛C jjbjjA^˛C jjcjjA^˛C jjdjjA^˛/ for alla; b; c; d 2Aand all2T¹. Then, there exists a unique generalized homo- morphismhWA!Bsuch that

jjf .a/ h.a/jj^B

j1 2^{p.˛ 1/}j^p1jjajjA^˛: (2.25) for alla2A.

Proof. Define'.a; b; c; d /WD.jjajj_A^˛C jjbjj_A^˛C jjcjj_A^˛C jjdjj_A^˛/. If 0˛ < 1, then Theorem3entails that

Q

'.a; a; 0; 0/D 2

.1 2^{p.˛ 1/}/^p1jjajjA^˛: Consequently,

jjf .a/ h.a/jj^B

.1 2^{p.˛ 1/}/^p1 jjajjA^˛: (2.26)

If˛ > 1, then by applying Theorem4we find that Q

'.a; a; 0; 0/D 2

.2^{p.˛ 1/} 1/^p1jjajjA^˛: Hence,

jjf .a/ h.a/jjB

.2^{p.˛ 1/} 1/^p1 jjajjA^˛: (2.27) From inequalities (2.26) and (2.27) we conclude inequality (2.25).

3. CONCLUSIONS

We conclude that

(i) in Theorems3,4, if we take pD1, then we obtain [1, Theorems 2.2, 2.5], respectively,

(ii) in Corollary1, if we takepD1, then we deduce [1, Corollary 2.3],

(iii) in Corollary1, if we take pD1, D ^ı2, and ˛ D0, then we recover [1, Corollary 2.4].

Indeed, the results presented in [1] hold for1–Banach algebras. We know that1–

Banach algebras exactly coincide with Banach algebras and so the results of [1] only hold for Banach algebras. Our presented results in this paper hold forp–Banach al- gebras where0 < p1.

ACKNOWLEDGMENTS

This research was supported by a grant from Payame Noor University with the same title. The author would like to thank the referee for the careful reading of the paper.

REFERENCES

[1] M. Eshaghi Gordji and M. B. Savadkouhi, “Approximation of generalized homomorphisms in quasi–Banach algebras,”An. St. Univ. Ovidius Constanta, vol. 17, no. 2, pp. 203–214, 2009.

[2] A. Najati, “Homomorphisms in quasi–Banach algebras associated with a Pexiderized Cauchy–

Jensen functional equation,”Acta Math. Sinica, vol. 25, no. 9, pp. 1529–1542, 2009.

[3] A. Najati and C. Park, “Hyers–Ulam–Rassias stability of homomorphisms in quasi–Banach algeb- ras associated to the Pexiderized Cauchy functional equation,”J. Math. Anal. Appl., vol. 335, pp.

763–778, 2007.

[4] C. Park, “Homomorphisms between Poisson JC-algebras,”Bull. Braz. Math. Soc., vol. 36, pp.

79–97, 2005.

[5] C. Park, “Hyers–Ulam–Rassias stability of homomorphisms in quasi–Banach algebras,”Bull. Sci.

Math., vol. 132, pp. 87–96, 2008.

Author’s address

Ismail Nikoufar

Department of Mathematics, Payame Noor University, P.O. BOX 19395-3697 Tehran, Iran E-mail address:nikoufar@pnu.ac.ir