Research Article Fixed Points and Stability of an Additive Functional Equation of n-apollonius Type in C -Algebras

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1 Abstract ad Applied Aalysis Volume 008, Article ID 67618, 13 pages doi: /008/67618 Research Article Fixed Poits ad Stability of a Additive Fuctioal Equatio of -Apolloius Type i C -Algebras Fridou Moradlou, 1 Hamid Vaezi, 1 ad Chookil Park 1 Faculty of Mathematical Scieces, Uiversity of Tabriz, Tabriz, Ira Departmet of Mathematics, Hayag Uiversity, Seoul , South Korea Correspodece should be addressed to Chookil Park, baak@hayag.ac.kr Received April 008; Revised 11 Jue 008; Accepted 16 July 008 Recommeded by Joh Rassias Usig the fixed poit method, we prove the geeralized Hyers-Ulam stability of C -algebra homomorphisms ad of geeralized derivatios o C -algebras for the followig fuctioal equatio of Apolloius type f z x i 1/ 1 i<j f x i x j f z 1/ x i. Copyright q 008 Fridou Moradlou et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. 1. Itroductio ad prelimiaries A classical questio i the theory of fuctioal equatios is the followig: whe is it true that a fuctio, which approximately satisfies a fuctioal equatio E, must be close to a exact solutio of E? If the problem accepts a solutio, we say that the equatio E is stable. Such a problem was formulated by Ulam 1 i 1940 ad solved i the ext year for the Cauchy fuctioal equatio by Hyers. It gave rise to the stability theory for fuctioal equatios. The result of Hyers was exteded by Aoki 3 i 1950 by cosiderig the ubouded Cauchy differeces. I 1978, Rassias 4 proved that the additive mappig T, obtaied by Hyers or Aoki, is liear if, i additio, for each x E, the mappig f tx is cotiuous i t R. Găvruţa 5 geeralized the Rassias result. Followig the techiques of the proof of the corollary of Hyers, we observed that Hyers itroduced i 1941 the followig Hyers cotiuity coditio about the cotiuity of the mappig for each fixed poit ad the he proved homogeeity of degree oe ad, therefore, the famous liearity. This coditio has bee assumed further till ow, through the complete Hyers direct method, i order to prove liearity for geeralized Hyers-Ulam stability problem forms see 6. Begiig aroud 1980, the stability problems of several fuctioal equatios ad approximate homomorphisms have bee extesively ivestigated by a umber of authors ad there are may iterestig results cocerig this problem see 7 1.

2 Abstract ad Applied Aalysis Rassias, followig the spirit of the iovative approach of Hyers, Aoki 3, ad Rassias 4 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 3, 4 for a umber of other ew results. I 003, Cădariu ad Radu applied the fixed-poit method to the ivestigatio of the Jese fuctioal equatio 5 see also 8, They could preset a short ad a simple proof differet of the direct method, iitiated by Hyers i 1941 for the geeralized Hyers- Ulam stability of Jese fuctioal equatio 5, for Cauchy fuctioal equatio 8,adfor quadratic fuctioal equatio 6. The followig fuctioal equatio: Q x y Q x y Q x Q y 1.1 is called a quadratic fuctioal equatio, ad every solutio of 1.1 is said to be a quadratic mappig. Skof 31 proved the Hyers-Ulam stability of the quadratic fuctioal equatio 1.1 for mappigs f : E 1 E, where E 1 is a ormed space ad E is a Baach space. I 3, Czerwik proved the Hyers-Ulam stability of the quadratic fuctioal equatio 1.1. Borelli ad Forti 33 geeralized the stability result of the quadratic fuctioal equatio 1.1. Ju ad Lee 34 proved the Hyers-Ulam stability of the Pexiderized quadratic equatio f x y g x y h x k y 1. for mappigs f, g, h, ad k. The stability problem of the quadratic equatio has bee extesively ivestigated by some mathematicias 35. I a ier product space, the equality z x z y 1 x y z x y 1.3 holds, the it is called the Apolloius idetity. The followig fuctioal equatio, which was motivated by this equatio, Q z x Q z y 1 Q x y Q z x y ), 1.4 holds, the it is called quadratic see 36. 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 37. I 36, Park ad Rassias itroduced ad ivestigated a fuctioal equatio, which is called a geeralized Apolloius type quadratic fuctioal equatio. I 38, Najati itroduced ad ivestigated a fuctioal equatio, which is called a quadratic fuctioal equatio of - Apolloius type. Recetly i 39, Park ad Rassias itroduced ad ivestigated the followig fuctioal equatio: f z x f z y 1 f x y f z x y ) 4 1.5

3 Fridou Moradlou et al. 3 which is called a Apolloius type additive fuctioal equatio, ad whose solutio is called a Apolloius type additive mappig. I 40, Park itroduced ad ivestigated a fuctioal equatio, which is called a geeralized Apolloius-Jese type additive fuctioal equatio ad whose solutio is said to be a geeralized Apolloius-Jese type additive mappig. I this paper, employig the above equality 1.5, for a fixed positive iteger, we itroduce a ew fuctioal equatio, which is called a additive fuctioal equatio of - Apolloius type ad whose solutio is said to be a additive mappig of -Apolloius type; f z x i 1 f x i x j f z 1 x i ) i<j We will adopt the idea of Cădariu ad Radu 8, 5, 8 to prove the geeralized Hyers- Ulam stability results of C -algebra homomorphisms as well as to prove the geeralized Ulam-Hyers stability of geeralized derivatios o C -algebra for additive fuctioal equatio of -Apolloius type. We recall two fudametal results i fixed-poit theory. Theorem 1.1 see 5. Let X, d be a complete metric space ad let J : X X be strictly cotractive, that is, d Jx,Jy Lf x, y, x, y X 1.7 for some Lipschitz costat L<1. The, the followig hold: 1 the mappig J has a uique fixed poit x Jx ; the fixed poit x is globally attractive, that is, lim J x x 1.8 for ay startig poit x X; 3 oe has the followig estimatio iequalities: d J x, x L d x, x, d J x, x d x, x 1 1 L d J x, J 1 x, 1 d x, Jx 1 L 1.9 for all oegative itegers ad all x X. Let X be a set. A fuctio d : X X 0, is called a geeralized metric o X if d satisfies the followig: 1 d x, y 0ifad oly if x y; d x, y d y, x for all x, y X; 3 d x, z d x, y d y, z for all x, y, z X.

4 4 Abstract ad Applied Aalysis Theorem 1. see 41. Let X, d be a complete geeralized metric space ad let J : X X be a strictly cotractive mappig with Lipschitz costat L < 1. The for each give elemet x X, either d J x, J 1 x 1.10 for all oegative itegers or there exists a positive iteger 0 such that the followig hold: 1 d J x, J 1 x < for all 0 ; the sequece {J x} coverges to a fixed poit y of J; 3 y is the uique fixed poit of J i the set Y {y X d J 0 x, y < }; 4 d y, y 1/ 1 L d y, Jy for all y Y. Throughout this paper, assume that A is a C -algebra with orm A ad that B is a C -algebra with orm B.. Stability of C -algebra homomorphisms Lemma.1. Let X ad Y be real-vector spaces. A mappig f : X Y satisfies 1.6 for all x 1,...,x,zif ad oly if the mappig f is additive. Proof. Lettig x 1 x z 0i 1.6, wegetthatf 0 0. Let j ad k be fixed itegers with 1 j<k. Settig x i 0 for all 1 i, i / j, k i 1.6, we have f z x j f z x k f z 1 f x j x k f x j f x k f z 1 ) x j x k.1 for all x j,x k,z X. Replacig x j by x j ad x k by x j i.1, respectively, we get f z x j f z x j f x j f x j f z. for all x j,z X. Puttig z 0i., we coclude that f x j f x j for all x j X. This meas that f is a odd fuctio. Lettig x k z 0i.1 ad usig the oddess of f, we obtai that ) 1 f x j 1 f x j, f x j f x j.3 for all x j X. Lettig z 0i.1, usig the oddess of f ad.3, we have f x j x k f x j f x k.4 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 : A B ad for a fixed positive iteger, we defie C μ f z, x 1,...,x : μf z x i 1 f μx i μx j f μz 1 ) μx 1 i<j i.5 for all μ T 1 : {ν C : ν 1} ad all z, x 1,...,x A. We prove the geeralized Hyers-Ulam stability of C -algebra homomorphisms for the fuctioal equatio C μ f z, x 1,...,x 0.

5 Fridou Moradlou et al. 5 Theorem.. Let f : A B be a mappig satisfyig f 0 0 for which there exists a fuctio ϕ : A 1 0, such that ) j ) 1 j ) 1 j ) 1 j ) ϕ z, x 1 1,..., x <,.6 j 0 C μ f z, x 1,...,x B ϕ z, x 1,...,x, f xy f x f y B ϕ x, y, 0,...,0, 1 times f x f x B ϕ x,...,x 1 times for all μ T 1 ad all x, y, z, x 1,...,x A. Ifforsome1 j there exists a Lipschitz costat L<1such that ) Lϕ, 0,...,0.10 ϕ x, 0,...,0, x, 0,...,0 1 1 x, 0,...,0, 1 x for all x A, the there exists a uique C -algebra homomorphism H : A B such that Proof. Cosider the set f x H x B 1 1 L ϕ x, 0,...,0, x, 0,...,0.11 X : {g : A B, g 0 0}.1 ad itroduce the geeralized metric o X: { d g,h if C R : g x h x B Cϕ x, 0,...,0, It is easy to show that X, d is complete. For coveiece, set x }, 0,...,0 x A..13 ϕ j x, y : ϕ x, 0,...,0, y, 0,...,0.14 for all x, y A ad all 1 j. Now we cosider the liear mappig J : X X such that Jg x : ) α α g x.15 for all x A, where α 1 /.

6 6 Abstract ad Applied Aalysis For ay g,h X, we have d g,h <C g x h x B Cϕ j x, x x A ) α α g x ) α B α h x α α Cϕ j x, α ) x ) α α g x ) α B α h x LCϕ j x, x.16 Therefore, we see that d Jg,Jh LC. d Jg,Jh Ld g,h, g,h A..17 This meas J is a strictly cotractive self-mappig of X, with the Lipschitz costat L. Lettig μ 1, z x j x, ad for each 1 k with k / j, x k 0i.7, weget ) α B αf x f x ϕ j x, x.18 So f x ) α B α f x 1 α ϕ j x, x.19 Hece d f, Jf 1/α. By Theorem 1., there exists a mappig H : A B such that the followig hold: 1 H is a fixed poit of J,thatis, ) α H x α H x for all x A; the mappig H is a uique fixed poit of J i the set.0 Y {g X : d f, g < };.1 ad this implies that H is a uique mappig satisfyig.0 such that there exists C 0, satisfyig H x f x B Cϕ j x, x d J m f, H 0asm ; ad this implies the equality lim m ) m f α ) α m x) H x..3 for all x A;

7 Fridou Moradlou et al. 7 3 d f, H 1/ 1 L d f, Jf, which implies the iequality d f, H 1 α αl ;.4 ad this implies that the iequality.11 holds. It follows from.6,.7,ad.3 that H z x i 1 1 i<j ) m lim m α H x i x j H z 1 x i) B f ) α m z x i ) 1 1 i<j α ) m ) α m f z 1 x i) B ) m ) α m ) α m ) α m ) lim ϕ z, x 1,..., x m α ) m ) α m ) α m ) α m ) lim ϕ z, x 1,..., x 0 m α f ) α m x i x j ).5 for all x 1,...,x,z A.So H z x i 1 H x i x j H z 1 ) x 1 i<j i.6 for all x 1,...,x,z A. ByLemma.1, the mappig H : A B is Cauchy additive, that is, H x y H x H y for all x, y A. By a similar method to the proof of 14, oe ca show that the mappig H : A B is C-liear. It follows from.8 that ) m α H xy H x H y B lim f m α ) m ) α m lim ϕ x, m α for all x, y A.So ) m xy) f α ) α m x) f ) ) m y, 0,...,0 1 times 0 ) α m ) y B.7 H xy H x H y.8 for all x, y A.

8 8 Abstract ad Applied Aalysis It follows from.9 that ) m ) α m ) α H x H x B lim f x f m α ) m ) α m ) α m ) lim ϕ x,..., x m α 1 times ) α m ) α m ) x,..., x ) m lim ϕ m α } {{ } 1 times ) m ) x 0 B.9 So H x H x.30 Thus H : A B is a C -algebra homomorphism satisfyig.11 as desired. Corollary.3. Let r>ad θ be oegative real umbers, ad let f : A B be a mappig such that ) C μ f z, x 1,...,x B θ z r A x i r A,.31 f xy f x f y B θ x r A y r A,.3 f x f x B 1 θ x r A.33 for all μ T 1 ad all x, y, z A. The there exists a uique C -algebra homomorphism H : A B such that f x H x B 1 r θ 1 1 r 1 r x r A.34 Proof. The proof follows from Theorem. by takig ) ϕ z, x 1,...,x : θ z r A x i r A.35 for all x, y, z A. It follows from.31 that f 0 0. We ca choose L / 1 1 r to get the desired result. Theorem.4. Let f : A B be a mappig satisfyig f 0 0 for which there exists a fuctio ϕ : A 1 0, satisfyig.7,.8, ad.9 such that )j 1 ) j ) j ) j ) ϕ z, x 1 1,..., x 1 <.36 1 j 0

9 Fridou Moradlou et al. 9 for all z, x 1,...,x A. Ifforsome1 j there exists a Lipschitz costat L<1such that ϕ x, 0,...,0, x, 0,...,0 ) 1 Lϕ 1 x, 0,...,0, 1 x, 0,...,0.37 for all x A, the there exists a uique C -algebra homomorphism H : A B such that f x H x B L 1 1 L ϕ x, 0,...,0, x, 0,...,0.38 Proof. Similar to proof of Theorem., we cosider the liear mappig J : X X such that Jg x : α g α x ).39 for all x A, where α 1 /. We ca coclude that J is a strictly cotractive self mappig of X with the Lipschitz costat L. It follows from.18 that f x α ) B f α x 1 ϕ j α x, ) α x L α ϕ j x, x.40 Hece, d f, Jf L/α. By Theorem 1., there exists a mappig H : A B such that the followig hold: 1 H is a fixed poit of J,thatis, ) H α x α H x.41 for all x A; the mappig H is a uique fixed poit of J i the set Y {g X : d f, g < };.4 ad this implies that H is a uique mappig satisfyig.41 such that there exists C 0, satisfyig H x f x B Cϕ j x, x for all x A; d J m f, H 0asm ; ad this implies the equality ) α m ) α m lim f x) H x m for all x A; 3 d f, H 1/ 1 L d f, Jf, which implies the iequality L d f, H α αl, which implies that the iequality.38 holds. The rest of the proof is similar to the proof of Theorem

10 10 Abstract ad Applied Aalysis Corollary.5. Let r < 1 ad θ be oegative real umbers, ad let f : A B be a mappig satisfyig.31,.3, ad.33. The there exists a uique C -algebra homomorphism H : A B such that f x H x B 1 r Lθ 1 r 1 r 1 x r A.46 for all x A ad L / 1 r 1. Proof. The proof follows from Theorem.4 by takig ) ϕ z, x 1,...,x : θ z r A x i r A.47 for all z, x 1,...,x A. It follows from.31 that f 0 0. We ca choose L / 1 r 1 to get the desired result. 3. Stability of geeralized derivatios o C -algebras For a give mappig f : A A ad for a fixed positive iteger, we defie C μ f z, x 1,...,x : μf z x i 1 f μx i μx j f μz 1 ) μx i 1 i<j 3.1 for all μ T 1 ad all z, x 1,...,x A. Defiitio 3.1 see 4. A geeralized derivatio δ : A A is ivolutive C-liear ad fulfills δ xyz δ xy z xδ y z xδ yz for all x, y, z A. We prove the geeralized Hyers-Ulam stability of derivatios o C -algebras for the fuctioal equatio C μ f z, x 1,...,x 0. Theorem 3.. Let f : A A be a mappig satisfyig f 0 0 for which there exists a fuctio ϕ : A 1 0, such that ) 3j ) 1 j ) 1 j ) 1 j ) ϕ z, x 1 1,..., x <, 3. j 0 C μ f x 1,...,x,z A ϕ z, x 1,...,x, f xyz f xy z xf y z xf yz A ϕ x, y, z, 0,...,0, times f x f x A ϕ x,...,x 1 times

11 Fridou Moradlou et al. 11 for all μ T 1 ad all x, y, z, x 1,...,x A. Ifforsome1 j there exists a Lipschitz costat L<1such that ϕ x, 0,...,0, x, 0,...,0 1 Lϕ 1 x, 0,...,0, ) 1 x, 0,...,0 3.6 for all x A, the there exists a uique geeralized derivatio δ : A A such that f x δ x A 1 1 L ϕ x, 0,...,0, x, 0,...,0 3.7 Proof. By the same reasoig as i the proof of Theorem., there exists a uique ivolutive C-liear mappig δ : A A satisfyig 3.7. The mappig δ : A A is give by ) m ) m δ x f x) 3.8 α α It follows from 3.4 that δ xyz δ xy z xδ y z xδ yz A ) 3m ) α 3m ) α m ) α m lim f xyz) f xy) z m α ) α m ) α m ) α m ) α m ) α m ) xf y) z xf yz A ) 3m ) α m ) α m ) α m ) lim ϕ x, y, z, 0,...,0 0 m α times for all x, y, z A. So δ xyz δ xy z xδ y z xδ yz for all x, y, z A. Thusδ : A A is a geeralized derivatio satisfyig Theorem 3.3. Let f : A A be a mappig satisfyig f 0 0for which there exists a fuctio ϕ : A 1 0, satisfyig.36, 3.3, 3.4 ad 3.5 for all x, y, z, x 1,...,x A. Ifforsome 1 j there exists a Lipschitz costat L<1such that ) 1 ϕ x, 0,...,0, x, 0,...,0 1 Lϕ x, 0,...,0, 1 x, 0,...,0 } {{ } for all x A, the there exists a uique geeralized derivatio δ : A A such that f x δ x B L 1 1 L ϕ x, 0,...,0, x Proof. The proof is similar to the proofs of Theorems.4 ad , 0,...,0 3.1

12 1 Abstract ad Applied Aalysis Ackowledgmets This paper is based o fial report of the research project of the Ph.D. thesis i Uiversity of Tabriz ad the third author was supported by Grat o. F from the Korea Sciece ad Egieerig Foudatio. The authors would like to thak the referees for a umber of valuable suggestios regardig a previous versio of this paper. Refereces 1 S. M. Ulam, A Collectio of Mathematical Problems, Itersciece Tracts i Pure ad Applied Mathematics, o. 8, Itersciece Publishers, New York, NY, USA, D. H. Hyers, O the stability of the liear fuctioal equatio, Proceedigs of the Natioal Academy of Scieces of the Uited States of America, vol. 7, pp. 4, T. Aoki, O the stability of the liear trasformatio i Baach spaces, Joural of the Mathematical Society of Japa, vol., pp , Th. M. Rassias, O the stability of the liear mappig i Baach spaces, Proceedigs of the America Mathematical Society, vol. 7, o., pp , P. Găvruţa, A geeralizatio of the Hyers-Ulam-Rassias stability of approximately additive mappigs, Joural of Mathematical Aalysis ad Applicatios, vol. 184, o. 3, pp , L. Maligrada, A result of Tosio Aoki about a geeralizatio of Hyers-Ulam stability of additive fuctios a questio of priority, Aequatioes Mathematicae, vol. 75, o. 3, pp , D. G. Bourgi, Classes of trasformatios ad borderig trasformatios, Bulleti of the America Mathematical Society, vol. 57, pp. 3 37, L. Cădariu ad V. Radu, O the stability of the Cauchy fuctioal equatio: a fixed poit approach, Grazer Mathematische Berichte, vol. 346, pp. 43 5, H.-M. Kim ad J. M. Rassias, Geeralizatio of Ulam stability problem for Euler-Lagrage quadratic mappigs, Joural of Mathematical Aalysis ad Applicatios, vol. 336, o. 1, pp , Y.-S. Lee ad S.-Y. Chug, Stability of a Euler-Lagrage-Rassias equatio i the spaces of geeralized fuctios, Applied Mathematics Letters, vol. 1, o. 7, pp , P. Nakmahachalasit, O the geeralized Ulam-Gavruta-Rassias stability of mixed-type liear ad Euler-Lagrage-Rassias fuctioal equatios, Iteratioal Joural of Mathematics ad Mathematical Scieces, vol. 007, Article ID 6339, 10 pages, C.-G. Park, O the stability of the liear mappig i Baach modules, Joural of Mathematical Aalysis ad Applicatios, vol. 75, o., pp , C.-G. Park, O a approximate automorphism o a C -algebra, Proceedigs of the America Mathematical Society, vol. 13, o. 6, pp , C.-G. Park, Homomorphisms betwee Poisso JC -algebras, Bulleti of the Brazilia Mathematical Society, vol. 36, o. 1, pp , C. Park, Fixed poits ad Hyers-Ulam-Rassias stability of Cauchy-Jese fuctioal equatios i Baach algebras, Fixed Poit Theory ad Applicatios, vol. 007, Article ID 50175, 15 pages, C.-G. Park, Stability of a Euler-Lagrage-Rassias type additive mappig, Iteratioal Joural of Applied Mathematics & Statistics, vol. 7, pp , C. Park, Geeralized Hyers-Ulam stability of quadratic fuctioal equatios: a fixed poit approach, Fixed Poit Theory ad Applicatios, vol. 008, Article ID , 9 pages, A. Pietrzyk, Stability of the Euler-Lagrage-Rassias fuctioal equatio, Demostratio Mathematica, vol. 39, o. 3, pp , J. M. Rassias ad M. J. Rassias, Refied Ulam stability for Euler-Lagrage type mappigs i Hilbert spaces, Iteratioal Joural of Applied Mathematics & Statistics, vol. 7, pp , Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappigs, Joural of Mathematical Aalysis ad Applicatios, vol. 46, o., pp , Th. M. Rassias, O the stability of fuctioal equatios i Baach spaces, Joural of Mathematical Aalysis ad Applicatios, vol. 51, o. 1, pp , 000. J. M. Rassias, O approximatio of approximately liear mappigs by liear mappigs, Bulleti des Scieces Mathématiques, vol. 108, o. 4, pp , J. M. Rassias, O approximatio of approximately liear mappigs by liear mappigs, Joural of Fuctioal Aalysis, vol. 46, o. 1, pp , 198.

13 Fridou Moradlou et al J. M. Rassias, Solutio of a problem of Ulam, Joural of Approximatio Theory, vol. 57, o. 3, pp , L. Cădariu ad V. Radu, Fixed poits ad the stability of Jese s fuctioal equatio, Joural of Iequalities i Pure ad Applied Mathematics, vol. 4, o. 1, article 4, 7 pages, L. Cădariu ad V. Radu, Fixed poits ad the stability of quadratic fuctioal equatios, Aalele Uiversităţii de Vest di Timişoara, vol. 41, o. 1, pp. 5 48, S.-M. Jug ad J. M. Rassias, A fixed poit approach to the stability of a fuctioal equatio of the spiral of Theodorus, Fixed Poit Theory ad Applicatios. I press. 8 V. Radu, The fixed poit alterative ad the stability of fuctioal equatios, Fixed Poit Theory, vol. 4, o. 1, pp , J. M. Rassias, Alterative cotractio priciple ad Ulam stability problem, Mathematical Scieces Research Joural, vol. 9, o. 7, pp , J. M. Rassias, Alterative cotractio priciple ad alterative Jese ad Jese type mappigs, Iteratioal Joural of Applied Mathematics & Statistics, vol. 4, pp. 1 10, F. Skof, Local properties ad approximatio of operators, Redicoti del Semiario Matematico e Fisico di Milao, vol. 53, pp , S. Czerwik, The stability of the quadratic fuctioal equatio, i Stability of Mappigs of Hyers-Ulam Type, Th. M. Rassias ad J. Tabor, Eds., Hadroic Press Collectio of Origial Articles, pp , Hadroic Press, Palm Harbor, Fla, USA, C. Borelli ad G. L. Forti, O a geeral Hyers-Ulam stability result, Iteratioal Joural of Mathematics ad Mathematical Scieces, vol. 18, o., pp. 9 36, K.-W. Ju ad Y.-H. Lee, O the Hyers-Ulam-Rassias stability of a Pexiderized quadratic iequality, Mathematical Iequalities & Applicatios, vol. 4, o. 1, pp , G. H. Kim, O the stability of the quadratic mappig i ormed spaces, Iteratioal Joural of Mathematics ad Mathematical Scieces, vol. 5, o. 4, pp. 17 9, C.-G. Park ad Th. M. Rassias, Hyers-Ulam stability of a geeralized Apolloius type quadratic mappig, Joural of Mathematical Aalysis ad Applicatios, vol. 3, o. 1, pp , J. Aczél ad J. Dhombres, Fuctioal Equatios i Several Variables, vol. 31 of Ecyclopedia of Mathematics ad Its Applicatios, Cambridge Uiversity Press, Cambridge, UK, A. Najati, Hyers-Ulam stability of a -Apolloius type quadratic mappig, Bulleti of the Belgia Mathematical Society. Simo Stevi, vol. 14, o. 4, pp , C. Park ad Th. M. Rassias, Homomorphisms i C -terary algebras ad JB -triples, Joural of Mathematical Aalysis ad Applicatios, vol. 337, o. 1, pp. 13 0, C. Park, Hyers-Ulam-Rassias stability of a geeralized Apolloius-Jese type additive mappig ad isomorphisms betwee C -algebras, Mathematische Nachrichte, vol. 81, o. 3, pp , J. B. Diaz ad B. Margolis, A fixed poit theorem of the alterative, for cotractios o a geeralized complete metric space, Bulleti of the America Mathematical Society, vol. 74, pp , P. Ara ad M. Mathieu, Local Multipliers of C -Algebras, Spriger Moographs i Mathematics, Spriger, Lodo, UK, 003.

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