Hyers-Ulam-Rassias Stability of n-apollonius Type

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1 It. Joural of Math. Aalysis, Vol. 6, 2012, o. 3, Hyers-Ulam-Rassias Stability of -Apolloius Type Additive Mappig ad Isomorphisms i C -Algebras Fridou Moradlou Departmet of Mathematics Sahad Uiversity of Techology, Tabriz, Ira moradlou@sut.ac.ir Chookil Park Departmet of Mathematics, Hayag Uiversity Seoul, , Republic of Korea baak@hayag.ac.kr Jug Rye Lee Departmet of Mathematics, Daeji Uiversity Kyeoggi , Republic of Korea jrlee@daeji.ac.kr Abstract. I this paper, we prove Hyers-Ulam-Rassias stability of the followig fuctioal equatio i Baach modules over a uital C -algebra: fz x i )+ 1 i=1 1 i<j fx i + x j )=f z 1 2 x i ), which is fixed iteger ad 2. As a applicatio, we show that every almost liear bijectio h : A B of a uital C -algebra A oto a uital C - algebra B is a C -algebra isomorphism whe h ) 2 1 duy ) = h ) 2 1 du ) 2 hy) 2 for all uitaries u A, all y A, ad all d Z. i=1

2 112 F. Moradlou, C. Park ad J. Lee Mathematics Subject Classificatio: Primary 39B72, 47H10, 46L05, 46B03, 47Jxx Keywords: Hyers-Ulam-Rassias stability, -Apolloius type additive mappig, C -algebra homomorphism, geeralized derivatio 1. Itroductio ad prelimiaries The stability problem of fuctioal equatios origiated from a questio of Ulam [36] cocerig the stability of group homomorphisms: Let G 1, ) bea group ad let G 2,,d) be a metric group with the metric d, ). Give ɛ>0, does there exist a δɛ) > 0 such that if a mappig h : G 1 G 2 satisfies the iequality dhx y),hx) hy)) <δ for all x, y G 1, the there is a homomorphism H : G 1 G 2 with dhx),hx)) <ɛ for all x G 1? If the aswer is affirmative, we would say that the equatio of homomorphism Hx y) =Hx) Hy) is stable. The cocept of stability for a fuctioal equatio arises whe we replace the fuctioal equatio by a iequality which acts as a perturbatio of the equatio. Thus the stability questio of fuctioal equatios is that how do the solutios of the iequality differ from those of the give fuctioal equatio? Hyers [9] gave a first affirmative aswer to the questio of Ulam for Baach spaces. Let X ad Y be Baach spaces. Assume that f : X Y satisfies fx + y) fx) fy) ε for all x, y X ad some ε 0. The there exists a uique additive mappig T : X Y such that fx) T x) ε for all x X. Th.M. Rassias [30] provided a geeralizatio of Hyers Theorem which allows the Cauchy differece to be ubouded. Theorem 1.1. Th.M. Rassias). Letf : E E be a mappig from a ormed vector space E ito a Baach space E subject to the iequality 1.1) fx + y) fx) fy) ɛ x p + y p )

3 Stability of the -Apolloius type additive mappig 113 for all x, y E, where ɛ ad p are costats with ɛ>0 ad p<1. The the limit f2 x) Lx) = lim 2 exists for all x E ad L : E E is the uique additive mappig which satisfies fx) Lx) 2ɛ 2 2 p x p for all x E. Also, if for each x E the mappig ftx) is cotiuous i t R, the L is R-liear. The above iequality 1.1) has provided a lot of ifluece i the developmet of what is ow kow as a Hyers-Ulam-Rassias stability of fuctioal equatios. Begiig aroud the year 1980 the topic of approximate homomorphisms, or the stability of the equatio of homomorphism, was studied by a umber of mathematicias. Găvruta [8] geeralized the Rassias result. The stability problems of several fuctioal equatios have bee extesively ivestigated by a umber of authors ad there are may iterestig results cocerig this problem see [2], [5], [7], [13] [24], [31] [33]). J.M. Rassias [28] followig the spirit of the iovative approach of Th.M. Rassias [30] for the ubouded Cauchy differece proved a similar stability theorem i which he replaced the factor x p + y p by x p y q for p, q R with p + q 1 see also [29] for a umber of other ew results). Theorem 1.2. [27, 28, 29] Let X be a real ormed liear space ad Y areal complete ormed liear space. Assume that f : X Y is a approximately additive mappig for which there exist costats θ 0 ad p R {1} such that f satisfies iequality fx + y) fx) fy) θ x p 2 y p 2 for all x, y X. The there exists a uique additive mappig L : X Y satisfyig θ fx) Lx) 2 p 2 x p for all x X. If, i additio, f : X Y is a mappig such that the trasformatio t ftx) is cotiuous i t R for each fixed x X, the L is a R-liear mappig.

4 114 F. Moradlou, C. Park ad J. Lee The followig fuctioal equatio 1.2) Qx + y)+qx y) =2Qx)+2Qy), is called a quadratic fuctioal equatio, ad every solutio of equatio 1.2) is said to be a quadratic mappig. F. Skof [35] proved the Hyers-Ulam-Rassias stability of the quadratic fuctioal equatio 1.2) for mappigs f : E 1 E 2, where E 1 is a ormed space ad E 2 is a Baach space. I [6], S. Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic fuctioal equatio. C. Borelli ad G.L. Forti [4] geeralized the stability result as follows: let G be a abelia group, E a Baach space. Assume that a mappig f : G E satisfies the fuctioal iequality fx + y)+fx y) 2fx) 2fy) ϕx, y) for all x, y G, ad ϕ : G G [0, ) is a fuctio such that 1 φx, y) := 4 i+1 ϕ2i x, 2 i y) < i=0 for all x, y G. The there exists a uique quadratic mappig Q : G E with the properties fx) Qx) φx, x) for all x G. Ju ad Lee [11] proved the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equatio fx + y)+gx y) =2hx)+2ky) for mappigs f,g,h ad k. I a ier product space, the equality z x 2 + z y 2 = 1 2 x y 2 +2 z x + y 1.3) 2 2 holds, ad is called the Apolloius idetity. The followig fuctioal equatio, which was motivated by this equatio, Qz x)+qz y) = 1 2 Qx y)+2q z x + y ) 1.4), 2 is quadratic see [25]). For this reaso, the fuctioal equatio 1.4) is called a quadratic fuctioal equatio of Apolloius type, ad each solutio of the fuctioal equatio 1.4) is said to be a quadratic mappig of Apolloius type. The quadratic fuctioal equatio ad several other fuctioal equatios are useful to characterize ier product spaces [1].

5 Stability of the -Apolloius type additive mappig 115 I [25], C. Park ad Th.M. Rassias itroduced ad ivestigated a fuctioal equatio, which is called the geeralized Apolloius type quadratic fuctioal equatio. Recetly i [26], C. Park ad Th.M. Rassias itroduced ad ivestigated the followig fuctioal equatio fz x)+fz y) = 1 2 fx + y)+2f z x + y ) 1.5) 4 which is called Apolloius type additive fuctioal equatio, ad whose solutio is called a Apolloius type additive mappig. I [23], C. Park itroduced ad ivestigated a fuctioal equatio, which is called the geeralized Apolloius- Jese type additive fuctioal equatio ad whose solutio of the fuctioal equatio is said to be a geeralized Apolloius-Jese type additive mappig. I this paper, for a fixed iteger 2, we itroduce the ew fuctioal equatio, which is called the additive fuctioal equatio of -Apolloius type ad whose solutio is called a additive mappig of -Apolloius type, 1.6) fz x i )+ 1 i=1 1 i<j fx i + x j )=f z 1 2 x i ). i=1 This paper is orgaized as follows: I Sectio 2, we ivestigate the Hyers- Ulam-Rassias stability of additive fuctioal equatio of -Apolloius type i Baach modules over C -algebras. I Sectio 3, we ivestigate C -algebra isomorphisims i uital C -algebras associated with additive fuctioal equatio of -Apolloius type. 2. Hyers-Ulam-Rassias Stability of -Apolloius type additive mappig i Baach modules over a C -algebra Throughout this sectio, assume that A is a uital C -algebra with orm ad uitary group UA), ad that X ad Y are Baach modules over a uital C -algebra A with orms X ad Y, respectively. Lemma 2.1. A fuctio f : X Y satisfies 1.6) for all z, x 1,,x if ad oly if the fuctio f is additive. Proof. Lettig x 1 = = x = z = 0 i 1.6), we get that f0) = 0. Let j ad k be fixed itegers with 1 j<k. Settig x i = 0 for all 1 i, i j, k

6 116 F. Moradlou, C. Park ad J. Lee i 1.6), we have 2.1) fz x j )+fz x k )+ 2)fz) = 1 fx j + x k ) 2 fxj )+fx k ) ) + f z 1 x 2 j + x k ) ) for all x j,x k,z X. Replacig x j by x j ad x k by x j i 2.1), respectively, we get 2.2) fz + x j )+fz x j )= 2 f xj )+fx j ) ) +2fz) for all x j,z X. Puttig z = 0 i 2.2), we coclude that f x j )= fx j ) for all x j X. This meas that f is a odd fuctio. Lettig x k = z =0i 2.1) ad usig the oddess of f, we obtai that 2.3) f 1 2 x j)= 1 2 fx j), f 2 x j )= 2 fx j ) for all x j X. Lettig z = 0 i 2.1), usig the oddess of f ad 2.3) we have fx j + x k )=fx j )+fx j ) for all x j,x k X. Therefore, f : X Y is a additive mappig. The coverse is obviously true. For a give mappig f : X Y ad a give u UA) ad a fixed iteger 2, we defie D u fz, x 1,,x ):= fuz ux i )+ 1 i=1 uf 1 i<j z 1 2 fux i + μux j ) ) x i i=1 for all z, x 1,,x X. We ivestigate the Hyers-Ulam-Rassias stability of a type additive mappig of -Apolloius type i Baach modules over a C -algebra.

7 Stability of the -Apolloius type additive mappig 117 Theorem 2.2. Let f : X Y be a mappig satisfyig f0) = 0 for which there exists a fuctio ϕ : A +1 [0, ) such that 2.4) ϕ i z, x) = 2.5) lim j ) jϕ 2 1 ) jz, 2 1 ) jx 0,, 0,,, 0) <, j=0 }{{} i th ) jϕ ) jz, ) jx1,, 2.6) D μ fz, x 1,,x ) Y ϕz, x 1,,x ) ) ) jx =0, for all u UA) ad all x, z, x 1,,x X. The there exists a uique A-Liear additive mappig L : X Y of -Apolloius type such that 2.7) fx) Lx) Y 2 1 ϕ ix, x) for all x X. Proof. Let u =1 UA). Lettig z = x i = x ad for each 1 j with j i, x j = 0 i 2.6), we get 2.8) 2 1 fx) f 2 1 x ) Y ϕx, 0,, 0, x 2 }{{}, 0,, 0) i th for all x X. For coveiece, set ϕ i z, x) =ϕz, 0,, 0, }{{} x, 0,, 0) i th for all x, z X ad all 1 i. Lettig = 2 1, we get fx) f x) 2.9) Y ϕ i x, x) for all x X. It follows from 2.9) that fx) f x) 2.10) Y 1 ϕ ix, x)

8 118 F. Moradlou, C. Park ad J. Lee ) kx for all x X. If we replace x i 2.10) by ad multiply both sides of ) k, 2.10) by the we have ) kf ) kx ) k+1f ) ) k+1x Y ) 2.11) 1 ) kϕi ) kx, ) ) kx for all x X ad all o-egative itegers k. So we obtai that for all oegative itegers m, l with l>m ) mf ) ) mx ) l+1f ) ) l+1x Y l ) if ) ) ix ) i+1f ) ) i+1x Y 2.12) i=m 1 l ) iϕi ) ix, ) ) ix i=m { ) d for all x X. So it follows from 2.4) ad 2.12) that the sequece f ) dx ) } is Cauchy for all x X, ad thus coverges by the completeess of Y. Thus we ca defie a mappig L : X Y by ) df ) dx ) Lx) = lim d for all x X. Lettig m = 0 i 2.12), we obtai ) l+1f ) ) l+1x Y fx) 2.13) 1 l ) jϕi ) jx, ) ) jx j=0 for all x X ad all l N. Takig the limit as l i 2.13), we obtai the iequality 2.7). It follows from 2.4) ad 2.6) that ) d D 1 Lz,x 1,,x ) Y = lim D 1 f d lim d ) dϕ ) dz, ) dz, ) dx1 ) ) dx,, ) dx1 ) ) dx,, =0.

9 Stability of the -Apolloius type additive mappig 119 Therefore, the mappig L : X Y satisfies the equatio 1.6) ad hece L is a geeralized additive mappig of -Apolloius type. To prove the uiqueess, let L be aother geeralized additive mappig of -Apolloius type satisfyig 2.7). The we have, for ay positive iteger k ) { k ) ) kx ) kx) Lx) L x) L f ) ) kx ) ) } kx j=0 f L ) k+jϕi ) k+jx, ) ) k+jx, which teds to zero as k. So we coclude that Lx) =L x) for all x X. By the assumptio, for each u UA), we get ) d ) ) dx, D u Lx, 0,, 0) = lim D }{{} u f 0,, 0 d }{{} times times dϕ ) dx, lim 0,, 0 )=0 d ) for all x X. Hece for all u UA) ad all x X. So Lux) =ulx) Lux) =ulx) } {{ } times for all u UA) ad all x X. By the same reasoig as i the proofs of [19] ad [22], Lax + by) = Lax)+Lby) = alx)+bly) for all a, b Aa, b 0) ad all x, y X. Ad L0x) =0=0Lx) for all x X. So the uique geeralized additive mappig L : X Y of -Apolloius type is a A-liear mappig. Corollary 2.3. Let ɛ 0 ad let p be a real umber with p>1. Assume that a mappig f : X Y satisfies the iequality D u fz, x 1,,x ) Y ɛ z p X + x j p ) 2.14) X

10 120 F. Moradlou, C. Park ad J. Lee for all u UA) ad all z, x 1,,x X. The there exists a uique A-liear additive mappig L : X Y of -Apolloius type such that for all x X. fx) Lx) Y 2 2 1) 1 p ɛ 2 1) 2 p) 2 1) 21 p) x p X Proof. It follows from 2.14) that f0) = 0. Defie ϕz, x 1,,x ):=ɛ z p X + x j X) p, ad apply Theorem 2.2 to get the result. Theorem 2.4. Let f : X Y be a mappig satisfyig f0) = 0 for which there exists a fuctio ϕ : X +1 [0, ) satisfyig 2.6) such that 2.15) 2 1 ) jϕ 2 ) jz, 2 ) ) jx ϕ i z, x) = 0,, 0,, 0,, 0 < }{{} i th 2.16) 2 1 lim j 2 ) jϕ 2 ) jz, 2 ) jx1 2 ) ) jx,, = for all x, z, x 1,,x X. The there exists a uique A-liear additive mappig L : X Y of -Apolloius type such that 2.17) fx) Lx) Y 2 1 ϕ ix, x) for all x X. Proof. Let u =1 UA). It follows from 2.9) that fx) f x) Y 1 ϕ i x, ) 2.18) x for all x X, where = 2 1 ) kx. If we replace x i 2.18) by ad ) k, multiply both sides of 2.18) by the we have ) kf ) kx ) k+1f ) ) k+1x Y ) 2.19) 1 ) kϕi ) k+1x, ) k+1x )

11 Stability of the -Apolloius type additive mappig 121 for all x X ad all o-egative itegers k. So we obtai that for all oegative itegers m, l with l>m 2.20) ) mf ) mx ) l+1f ) ) l+1x Y ) l ) jf ) jx ) 1 j=m ) j+1f ) ) j+1x Y l ) jϕi ) j+1x, ) j+1x ) j=m { d for all x X. So it follows from 2.15) ad 2.20) that the sequece ) f ) dx ) } is Cauchy for all x X, ad thus coverges by the completeess of Y. Thus we ca defie a mappig L : X Y by ) df ) dx ) Lx) = lim d for all x X. Lettig m = 0 i 2.20), we obtai ) l+1f ) ) l+1x Y fx) 1 l ) jϕi ) j+1x, ) j+1x 2.21) ) = 1 j=0 l+1 ) jϕi ) jx, ) jx ) for all x X ad all l N. Takig the limit as l i 2.21), we obtai the iequality 2.17). The rest of the proof is similar to the proof of Theorem 2.2. Corollary 2.5. Let ɛ 0 ad let p be a real umber with 0 <p<1. Assume that a mappig f : X Y satisfies the iequality 2.14) for all u UA) ad all z, x 1,,x X. The there exists a uique A-liear additive mappig L : X Y of -Apolloius type such that for all x X. fx) Lx) Y 2 2 1) 1 p ɛ 2 1) 21 p) 2 1) 2 p) x p X

12 122 F. Moradlou, C. Park ad J. Lee Proof. It follows from 2.14) that f0) = 0. Defie ϕz, x 1,,x ):=ɛ z p X + x j p X), ad apply Theorem 2.4 to get the result. 3. Isomorphisms i uital C -algebras Throughout this sectio, assume that A is a uital C -algebra with orm A ad uit e, ad that B is a uital C -algebra with orm B. Let UA) be the set of uitary elemets i A. We ivestigate C -algebra isomorphisms i uital C -algebras. Theorem 3.1. Let h : A B be a bijective mappig satisfyig h0) = 0 ad h duy) ) =h du)hy) ) for all u UA), ally A, ad all d Z, for which there exists a fuctio ϕ : A +1 [0, ) satisfyig 2.4) ad 2.5) such that 3.1) 3.2) D μ hz, x 1,,x ) B ϕz, x 1,,x ), ) ) d h u ) d B h u) ϕ ) d u,, ) d ) u } {{ } +1 times for all μ S 1 := {λ C λ =1}, allz, x 1,,x A, allu UA) ad ) d ) d all d Z. Assume that lim h e) is ivertible. The the bijective d mappig h : A B is a C -algebra isomorphism. Proof. Cosider the C -algebras A ad B as Baach modules over the uital C -algebra C. By Theorem 2.2, there exists a uique C-liear additive mappig H : A B of -Apolloius type such that 3.3) hx) Hx) 1 ϕ ix, x) for all x A. The additive mappig H : A B of -Apolloius type is give by ) d ) d Hx) = lim h x) d for all x A. By 2.5) ad 3.2), we get ) Hu d ) ) d ) ) = lim h u d ) d = lim h u) d d ) d ) d = lim h u) ) = Hu) d

13 Stability of the -Apolloius type additive mappig 123 for all u UA). Sice H is C-liear ad each x A is a fiite liear combiatio of uitary elemets see [12]), i.e., x = m λ ju j λ j C,u j UA)), m m m m Hx ) = H λ j u j)= λ j Hu j)= λ j Hu j ) = λ j Hu j )) = H m λ j u j ) = Hx) for all x A. ) d Sice h uy) = h d Z, ) d u) hy) for all u UA), all y A, ad all ) dh ) ) duy 3.4) Huy) = lim = lim d d for all u UA) ad all y A. By the additivity of H ad 3.4), ) d ) d Huy) =H uy) = H u ) d y )) ) d = Hu)h y) for all u UA) ad all y A. Hece Huy) = ) d 3.5) Hu)h ) dh ) ) du hy) =Hu)hy) ) d y) = Hu) ) d ) d h y) for all u UA) ad all y A. Takig the limit i 3.5) as d, we obtai 3.6) Huy) =Hu)Hy) for all u UA) ad all y A. Sice H is C-liear ad each x A is a fiite liear combiatio of uitary elemets, i.e., x = m λ ju j λ j C,u j UA)), it follows from 3.6) that m m m m Hxy) = H λ j u j y)= λ j Hu j y)= λ j Hu j )Hy) =H λ j u j )Hy) = Hx)Hy) for all x, y A. By 3.4) ad 3.6), He)Hy) =Hey) =He)hy) ) d ) d for all y A. Sice lim h e) = He) is ivertible, d Hy) =hy) for all y A.

14 124 F. Moradlou, C. Park ad J. Lee Therefore, the bijective mappig h : A B is a C -algebra isomorphism. Corollary 3.2. Let ɛ 0 ad let p be a real umber with p>1. Let h : A B be a bijective mappig satisfyig h duy) ) =h du)hy) ) for all u UA), ally A, ad all d Z, such that D μ hz, x 1,,x ) B ɛ 3.7) z p A + x j p A), ) ) d h u ) d B h u) +1) ) dp ɛ for all μ S 1,allu UA), alld Z, ad all z, x 1,,x A. Assume that lim d ) d h ) d e) is ivertible. The the bijective mappig h : A B is a C -algebra isomorphism. Proof. It follows from 3.7) that h0) = 0. Defie ϕz, x 1,,x ):=ɛ z p A + x j p A), ad apply Theorem 3.1 to get the result. The followig theorem is a alterative result of Theorem 3.1. Theorem 3.3. Let h : A B be a bijective mappig satisfyig h0) = 0 ad h ) d uy ) = h d ) u ) hy) for all u UA), ally A, ad all d Z, for which there exists a fuctio ϕ : A +1 [0, ) satisfyig 2.15), 2.16) ad 3.1) such that ) du h ) ) ) du B h ϕ ) du, ) du ) 3.8), } {{ } +1 times for all u UA). Assume that lim d ) d h ) d e ) is ivertible. The the bijective mappig h : A B is a C -algebra isomorphism. Proof. The proof is similar to the proofs of Theorems 2.2, 2.4 ad 3.1. Corollary 3.4. Let ɛ 0 ad let p be a real umber with 0 <p<1. Let h : A B be a bijective mappig satisfyig h ) duy ) = h ) du ) hy) for

15 Stability of the -Apolloius type additive mappig 125 all u UA), ally A, ad all d Z, such that D μ hz, x 1,,x ) B ɛ 3.9) z p A + x j A) p, ) du h ) ) ) du B h +1) ) dpɛ for all μ S 1,allu UA), alld Z, ad all z, x 1,,x A. Assume that lim dh de ) d ) ) is ivertible. The the bijective mappig h : A B is a C -algebra isomorphism. Proof. It follows from 3.9) that h0) = 0. Defie ϕz, x 1,,x ):=ɛ z p A + x j p A), ad apply Theorem 3.3 to get the result. Theorem 3.5. Let h : A B be a bijective mappig satisfyig h0) = 0 ad h duy ) ) = h du ) ) hy) for all u UA), ally A, ad all d Z, for which there exists a fuctio ϕ : A +1 [0, ) satisfyig 2.4), 2.5) ad 3.2) such that 3.10) D μ hz, x 1,,x ) B ϕz, x 1,,x ) for μ =1,i, ad all z, x 1,,x A. Assume that lim d d ) h ) d e ) is ivertible. If htx) is cotiuous i t R for each fixed x A, the the bijective mappig h : A B is a C -algebra isomorphism. Proof. Put μ = 1 i 3.7). By the same reasoig as i the proof of Theorem 2.2, there exists a uique additive mappig H : A B of -Apolloius type satisfyig 3.3). By the same reasoig as i the proof of [30], the additive mappig H : A B of -Apolloius type is R-liear. Put μ = i i 3.10). By the same method as i the proof of Theorem 2.2, oe ca obtai that Hix) = lim d ) dh ) dix ) ) = lim d i for all x A. For each elemet λ C, λ = s + it, where s, t R. So ) dh ) dx ) )=ihx) Hλx) = Hsx + itx) = shx)+thix) = shx)+ithx) =s + it)hx) = λhx)

16 126 F. Moradlou, C. Park ad J. Lee for all λ C ad all x A. So Hζx+ ηy) = Hζx)+Hηy) = ζhx)+ηhy) for all ζ,η C, ad all x, y A. Hece the geeralized Apolloius-Jese type additive mappig H : A B is C-liear. The rest of the proof is the same as i the proof of Theorem 3.1. The followig theorem is a alterative result of Theorem 3.5. Theorem 3.6. Let h : A B be a bijective mappig satisfyig h0) = 0 ad h duy ) ) = h du ) ) hy) for all u UA), ally A, ad all d Z, for which there exists a fuctio ϕ : A +1 [0, ) satisfyig 2.15), 2.16), 3.8) ad 3.10) for μ = 1,i, ad all z, x 1,,x A. Assume that lim d d ) h d ) e ) is ivertible. If htx) is cotiuous i t R for each fixed x A, the the bijective mappig h : A B is a C -algebra isomorphism. Proof. The proof is similar to the proofs of Theorems 2.2, 2.4, 3.1 ad 3.5. Ackowledgemets The third author was supported by Basic Sciece Research Program through the Natioal Research Foudatio of Korea fuded by the Miistry of Educatio, Sciece ad Techology NRF ). Refereces [1] J. Aczél ad J. Dhombres, Fuctioal Equatios i Several Variables, Cambridge Uiversity Press, [2] M. Amyari, C. Baak ad M. S. Mohammad, Nearly terary derivatios, Taiwaese J. Math ), [3] P. Ara ad M. Mathieu, Local Multipliers of C -Algebras, Spriger-Verlag, Lodo, [4] C. Borelli ad G.L. Forti, O a geeral Hyers-Ulam stability result, Iterat. J. Math. Math. Sci ), [5] H.-L. Chou ad J.-H Tzeg, O approximate isomorphisms betwee Baach -algebras or C -algebras, Taiwaese J. Math ), [6] S. Czerwik, The stability of the quadratic fuctioal equatio, i: Th.M. Rassias, J. Tabor Eds.), Stability of Mappigs of Hyers-Ulam Type, Hadroic Press, Florida, 1994, pp [7] M. Eshaghi Gordji, H. Khodaei, Solutio ad stability of geeralized mixed type cubic, quadratic ad additive fuctioal equatio i quasi Baach spaces, Noliear Aalysis.-TMA )

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