STABILITY OF DGC S QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH ALGEBRAS: DIRECT AND FIXED POINT METHODS

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1 Internatonal Journal of Pure and ppled Mathematcal Scence. ISSN Volume 0, Number (07), pp Reearch Inda Publcaton STILITY OF DGC S QUDRTIC FUNCTIONL EQUTION IN QUSI-NCH LGERS: DIRECT ND FIXED POINT METHODS John M. Raa, M. runumar and S. Kartheyan 3* Pedagogcal Department E.E., Secton of Mathematc and Informatc, Natonal and Capodtran Unverty of then, 4, gamemnono Str., gha Paraev, then 534, Greece. E-mal: jraa@prmedu.uoa.gr, URL: Department of Mathematc, Government rt College, Truvannamala , TamlNadu, Inda. E-mal: annarun00@yahoo.co.n 3* Department of Mathematc, R.M.K. Engneerng College, Kavarapetta , Taml Nadu, Inda. E-mal:arth.ma04@yahoo.com STRCT In th paper, we obtan the general oluton and the generalzed Ulam-Hyer tablty of Degen-Grave-Cayley -Eght Square (DGC ) quadratc functonal equaton of the form 8 8 = = f ( x ) f ( y ) = f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y

2 4 John M. Raa, M. runumar and S. Kartheyan n qua-anach algebra ung drect and fxed pont method. 00 Mathematc Subject Clafcaton: 395, 37, 38. Key word and phrae: quadratc functonal equaton, generalzed Ulam-Hyer tablty, qua-anach algebra, fxed pont.. Introducton The tablty problem of a functonal equaton wa frt poed by Ulam [4] concernng the tablty of group homomorphm whch wa anwered by Hyer [] for anach pace. Hyer theorem wa generalzed by T. o [] for addtve mappng and by Th.M. Raa [36] for lnear mappng by conderng an unbounded Cauchy dfference. The paper of Th.M. Raa [36] ha provded a lot of nfluence n the development of what we call generalzed Hyer-Ulam tablty of functonal equaton. generalzaton of the Th.M. Raa theorem wa obtaned by P. Gavruta [8] by replacng the unbounded Cauchy dfference by a general control functon n the prt of Raa approach. In 98, J.M. Raa [3] followed the nnovatve approach of the Th.M. Raa theorem [36] n whch he replaced the p p p q factor x y by x y for p, q wth p q. In 994, the above tablty reult were further extended by P. P. G a vruta [8] who condered a more general control functon n the real varable y for the unbounded Cauchy dfference n the prt of Th.M. Raa tablty approach. In 008, a pecal cae of G a vruta theorem for the unbounded Cauchy dfference wa obtaned n [37] by condereng the ummaton of both the um and product of two p norm. The quadratc functon f ( x) = cx atfe the functonal equaton f ( x y) f ( x y) = f ( x) f ( y) (.) and therefore the equaton () called quadratc functonal equaton. In 04, M. runumar and S. Kartheyan [4] ntroduced and nvetgated the Ulam-Hyer tablty of rahmagupta quadratc functonal equaton f ( x ) nf ( x ) f ( x ) nf ( x ) f x x nx x nf x x x x (.) 3 4 = on non-rchmedean anach algebra ung drect and fxed pont method. 3 In 05, John M. Raa, M. runumar and S. Kartheyan [33] ntroduced and proved the generalzed Ulam-Hyer tablty of Lagrange quadratc functonal equaton 4 4 3

3 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 43 n = n n f ( x ) f ( y ) = f x y f x y j x j y (.3) = = < jn where n a potve nteger n Le method. * C -algebra ung drect and fxed pont Recently, John M. Raa, M. runumar and S. Kartheyan [34] obtan the general oluton and the generalzed Ulam-Hyer tablty of Euler quadratc functonal equaton of the form 4 4 f ( x) f ( y) = f x y x y x3 y3 x4 y4 f x y xy x3y4 x4y3 = = (.4) f x y x y x y x y f x y x y x y x y * n JC -algebra ung drect and fxed pont method. n applcaton of th functonal equaton alo tuded. In th paper, we obtan the general oluton and the generalzed Ulam-Hyer tablty of Degen-Grave-Cayley -Eght Square (DGC ) quadratc functonal equaton of the form 8 8 = = f ( x ) f ( y ) = f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y (.5) n qua-anach algebra ung drect and fxed pont method.. Prelmnare We now gve ome bac defnton concernng qua-anach pace and ome prelmnary reult. Defnton. Let X be a real lnear pace. qua-norm a real-valued functon on X atfyng the followng:

4 44 John M. Raa, M. runumar and S. Kartheyan () x > 0 x X and x = 0 f and only f x = 0. () x =. x and x X. (3) There a contant K uch that x y K x y y X. The par,. called a qua-normed pace f. a qua norm on X. The mallet poble K called the modulu of concavty of.. qua-anach pace a complete qua-normed pace. Defnton. all qua-norm. called a p -norm (0 < p ) f p x y x y for y X. In th cae, a qua-anach pace called a p -anach pace. p p Gven a p -norm, the formula p d( y) := x y gve u a tranlaton nvarant metrc on X. y the o-rolewcz theorem [38] (ee alo [8]), each quanorm equvalent to ome p -norm. Snce t much eaer to wor wth p -norm than qua-norm, henceforth we retrct our attenton manly to p -norm. Defnton.3 Let,. be a qua-normed pace. The qua-normed pace,. called a qua-normed algebra f an algebra and there a contant C > 0 uch that xy C x y y. Defnton.4 qua-anach algebra a complete qua-normed algebra. If the quanorm. a p -norm, then the qua-anach algebra called a p -anach algebra. ourgn [9] proved the tablty of rng homomorphm n two untal anach algebra. adora [5] gave a generalzaton of the ourgn reult. The tablty reult concernng dervaton on operator algebra wa frt obtaned by Ŝ emrl [39]. In [6], adora proved the tablty of functonal equaton f ( xy) = xf ( y) f ( x) y, where f a mappng on normed algebra wth unt. Defnton.5 Let, be two algebra. mappng f : called a quadratc homomorphm f f a quadratc mappng atfyng f ( xy) = f ( x) f ( y) xy,. For ntance, let be commutatve. Then the mappng f :, defned

5 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 45 by f ( x) = x ( x ), a quadratc homomorphm. Defnton.6 mappng f : called a quadratc dervaton f f a quadratc mappng atfyng f ( xy) = x f ( y) f ( x) y, xy. We note that quadratc dervaton and rng dervaton are dfferent. 3 General Soluton of the Functonal Equaton (.5) In th ecton, the author nvetgate the general oluton of the functonal equaton (.5). Through out th ecton, let u conder X and Y be real vector pace. Theorem 3. If the mappng f : X Y atfe the functonal equaton (.5) x, y, x, y,, x y X then f : X Y atfyng the functonal equaton (.) for 8, all y X. 8 Proof. Settng x y = = x = = 0 n (.5), we get 3 = 3 8 y8 f x ) f ( x ) f ( y ) f ( y ) = f ( x y x y ) f ( x y x ) ( y (3.) x, y, x y X. Replacng x, y, x, ) by ( x,0, 0) n (3.), we obtan, x = f ( x) ( y f (3.) x X. Settng x, y, x, ) by ( x,0, y,0) n (3.), we get xf y f x y ( y f = (3.3) y X. Replacng x, y, x, ) by ( x, x) n (3.) and ung (3.) and ung (3.3), we arrve x = f ( x) ( y f (3.4) x X. Lettng x, y, x, ) by ( x,0, 0) n (3.), we obtan ( y f x = f ( x ) x X. It can be rewrtten a f ( x) = f ( x ) (3.5)

6 46 John M. Raa, M. runumar and S. Kartheyan x X. Replacng x by x n (3.5), we get f ( x) = f ( x) an even functon. Settng (( x, y, y) by ( y, y) n () and ung (3.), (3.4), we get xf y = f ( x ) f ( x) f ( y) f y (3.6) y X. Lettng x, y, x, ) by ( x, y, y) n (3.) and ung (3.), (3.3), (3.4), we obtan ( y xf y = f ( x ) f ( x) f ( y) f y (3.7) y X. ddng (3.6) and (3.7), we derve (.) y X. Hence the proof complete. Hereafter throughout th paper, aume that a qua-normed algebra wth qua-norm. and that a p -anach algebra wth p -norm.. For convenence, we defne a mappng f : by 8 8 = = F( y, y,..., x8, y8) f ( x ) f ( y ) f x y x y x y x y x y x y x y x y x, y,, x8, y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y Stablty of Quadratc Homomorphm and Dervaton of (.5): Drect Method In th ecton, the author preent the generalzed Ulam-Hyer tablty of the functonal equaton (.5). Defnton 4. -lnear mappng H: called a quadratc homomorphm n qua-banach algebra f H xy = HxH y (4.)

7 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 47 y. Theorem 4. 6 Let j {,}. Let : [0, ) be a functon uch that x, y,, x, y j j j j 8 8 converge to (4.) j =0 j j x, y,, lm j x, 8 8 and <, j and f : be a mappng atfe the nequalty j y (4.3) F( x, y,, x, y ) x, y,, x, y (4.4) x, y,, x8, y8 and f ( xy) f ( x) f ( y) y,0,0,,0,0 (4.5) xy,. Then there ext a unque quadratc homomorphm H: whch atfe (.5) and j = j, j x f ( x) H( x) (4.6) j 8 x. The mappng H( x ) defned by j f( x) H( x) = lm j (4.7) x. Proof. ume j =. (4.4), we get Replacng, y,, x, ) 8 ( 8 y8 x by x, x, and dvdng by 8 n f( x) f ( x), x (4.8) x. Replacng x by x n (4.8) and dvded by, we have

8 48 John M. Raa, M. runumar and S. Kartheyan f ( 4 x) f (x), x (4.9).8 x. Combnng (4.8) and (4.9), we obtan f ( 4 x) f ( x) 8, x, x (4.0) f ( x by x. Ung nducton on a potve nteger, we obtan that x) f ( x), x (4.).7 =0,.7 =0 x. In order to prove the convergence of the equence m x and dvdng by m n (4.), for any, > 0 x m, we arrve f ( x ), replace.7 =0 m f ( x) f ( x) f ( x) m m m = m m m f ( m m m, ( m).7 m =0, ( m) m m x m x) m x (4.) x. Snce the rght hand de of the nequalty (4.) tend to 0 a m, f ( x) the equence a Cauchy equence. Snce complete, there ext a mappng H: uch that Lettng f( x) H( x) = lm, x. n (4.), we ee that (4.6) hold x. Now, we need to prove H atfe (.5), replacng x, y,, x, ) by ( y,, x8, y8) and dvdng by n (4.4), we arrve ( 8 y8

9 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 49 F,,,,,,,, x y x8 y8 x y x8 y8 x, y,, x8, y8. Lettng n the above nequalte, we arrve H x, y,, x, y = Hence H atfe (.5) x, y,, x8, y8. Th how that H quadratc. lo H( x y ) H( x )H( y ) = lm f x y f x f y 4 lm x, y,0,0,,0,0 = 0 4 y. Therefore, H a quadratc homomorphm. In order to prove H unque, let H'( x ) be another quadratc homomorphm atfyng (4.6) and (.5). Then H( x) H'( x) = H( x) H'( x) ( x ) f ( x) f ( x) '( x), ( ).7 =0 0 a x. Hence unque. Thu the mappng H: a unque quadratc homomorphm mappng atfyng (4.6). of the theorem. For j =, we can prove the mlar tablty reult. Th complete the proof The followng corollary an mmedate conequence of Theorem 4. concernng the tablty of (.5). Corollary 4.3 Let and be nonnegatve real number. If a functon f : atfe the nequalty x

10 50 John M. Raa, M. runumar and S. Kartheyan F( x, y,, x, y ) 8 8, 8 x y, =, x y x y = = 8 x y = (4.3) x, y,, x8, y8 and f y f ( x) f ( y), x y, x y, x y x y (4.4) xy,. Then there ext a unque quadratc homomorphm H: uch that, / 6 x, 4; 7 f ( x) H( x) 8 x, ; x, (4.5) x. Defnton 4.4 -lnear mappng D: called a quadratc dervaton n quabanach algebra f xy x y x y D = D D (4.6) xy,. Theorem 4.5

11 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 5 6 Let j {,}. Let : [0, ) be a functon uch that x, y,, x, y j j j j 8 8 converge to and (4.7) j =0 j j j j y,, x8, y8 <, lm j and f : be a functon atfe the nequalty (4.8) F( x, y,, x, y ) x, y,, x, y (4.9) x, y,, x8, y8 and f ( xy) x f ( y) f ( x) y y,0,0,,0,0 (4.0) xy,. Then there ext a unque quadratc dervaton D: whch atfe (.5) and j, j x f ( x) H( x) (4.).7 j j = x. The mappng D( x ) defned by j f( x) D( x) = lm j (4.) x. Proof. ume j =. y the ame reaonng a that n the proof of the Theorem 4., there ext a unque quadratc mappng D: atfyng (4.). The mappng j f( x) D: gven by D( x) = lm. It follow from (4.9) that j D( x y ) x D( y ) D( x ) y = f x y x f y f x y lm 4 j lm j x, y,0,0,,0,0 = 0 4 y. Therefore D: a quadratc dervaton atfyng (4.). The followng corollary an mmedate conequence of Theorem 4.5 concernng the tablty of (.5).

12 5 John M. Raa, M. runumar and S. Kartheyan Corollary 4.6 Let and be non-negatve real number. If a functon f : atfe the nequalty F( x, y,, x, y ) 8 8, 8 x y, =, x y x y, = = 8 x y = (4.3) x, y,, x8, y8 and f ( xy) x f ( y) f ( x) y, x y, x y, x y x y (4.4) xy,. Then there ext a unque quadratc dervaton D: uch that ; / 6 x, 4; 7 f ( x) D( x) 8 x, ; x, (4.5) x. 5. Stablty of Quadratc Homomorphm and Quadratc Dervaton of (.5): Fxed Pont Method In th ecton, the author preented the generalzed Ulam-Hyer tablty of the functonal equaton (.5) n qua-anach algebra ung fxed pont method. Now we wll recall the fundamental reult n fxed pont theory.

13 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 53 Theorem 5. (anach Contracton Prncple) Let (, d) a non-rchmedean generalzed complete metrc pace and conder a mappng L <. Then, T : whch trctly contractve mappng, that () d( T Ty) Ld( y), y T for ome (Lpchtz contant) () The mappng T ha one and only fxed pont x = T( x ); ()The fxed pont for each gven element n () lm = n T x x, for any tartng pont x ; () One ha the followng etmaton nequalte: n n n (3) d( T x ) d( T T x), n 0, x; L (4) d ( x ) d( x ), x.. L Theorem 5. [3](The alternatve of fxed pont) x globally attractve, that Suppoe that for a complete generalzed metrc pace (, d) and a trctly contractve mappng T : wth Lpchtz contant L. Then, for each gven element x, ether ( ) d( T n T n x) = n 0, or ( ) there ext a natural number n 0 uch that: n n () d( T T x) < n n0 ; () The equence ( T n x) convergent to a fxed pont y of T () n 0 y the unque fxed pont of T n the et Y = { y : d( T y) < }; (v) d( y, y) d( y, Ty) L Theorem 5.3 y Y. 6 Let f : be a mappng for whch there ext a functon : [0, ) wth the condton

14 54 John M. Raa, M. runumar and S. Kartheyan lm ( y,, x8, y8) = 0 (5.) where = f = 0 and = f = uch that the functonal nequalty wth F( x, y,, x, y ) ( x, y,, x, y ) (5.) x, y,, x8, y8 and f ( xy) f ( x) f ( y) y,0,0,,0,0 (5.3) xy,. If there ext L = L( ) < uch that the functon x x x x x ( x) =,,,, 7 (5.4) ha the property ( x) = L (5.5) then there ext a unque quadratc homomorphm H: atfyng the functonal equaton (.5) and L f ( x) H( x) ( x) L (5.6) x. Proof. Conder the et and ntroduce the generalzed metrc on, It eay to ee that (, d) complete. ={ p/ p :, p(0) = 0} d( p, q) = nf{ K (0, ) : p( x) q( x) K ( x), x }. Defne T : by Tp( x) = p( x), x. Now p, q, we have d( p, q) K p( x) q( x) K ( x), x.

15 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 55 p( x) q( x) K ( x), x, p( x) q( x) LK ( x), x, Tp( x) Tq( x) LK ( x), x, d( p, q) LK. Th mple d( Tp, Tq) Ld( p, q), p, q..e., T a trctly contractve mappng on wth Lpchtz contant L. (5.), we get Replacng, y,, x, ) ( 8 y8 x by x, x, and dvdng by 8 n f (x) f ( x), x (5.7) 7 x. Hence from the above nequalty, we have f (x) f ( x), x (5.8).7 x. Ung (5.4) and (5.5) for the cae = 0, t reduce to f (x) f ( x) ( x) x.. e., d( f, Tf ) L L <. gan replacng x by x n (5.7), we get x x x x x f ( x) f,,,, 7 (5.9) x. Ung (5.4) and (5.5) for the cae = t reduce to f ( x) x f ( x) x,

16 56 John M. Raa, M. runumar and S. Kartheyan. e., d( f, Tf ) L 0 <. In both cae, we arrve d( f, Tf ) L. Therefore ( ) hold. y ( ), t follow that there ext a fxed pont H of T n uch that H( x) = lm f ( x) (5.0) x. To prove H: quadratc. Replacng x, y,, x, ) by y,, x8, 8 ( 8 y8 x, y n (5.) and dvdng by, t follow from (5.) that F( y,, x8, y8) H( x, y,, x, y ) = lm 8 8 y,, x8, y8 = 0 lm x, y,, x8, y8..e., H atfe the functonal equaton (.5). lo, H( x y ) H( x )H( y ) lm f x y f x f y 4 x,0,,0= 0 lm 4 y y. Therefore, H a quadratc homomorphm. y ( 3), H the unque fxed pont of T n the et ={H : d( f,h) < },H the unque functon uch that f ( x) H( x) K ( x) x and K > 0. Fnally by ( 4), we obtan th mple d( f,h) d( f, Tf ) L

17 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 57 whch yeld L d( f,h) L th complete the proof of the theorem. L f ( x) H( x) ( x) L The followng corollary an mmedate conequence of Theorem 5.3 concernng the tablty of (.5). Corollary 5.4 Let f : be a mappng and there ext real number and uch that F( x, y,, x, y ) 8 8, 8 x y, =, x y x y, = = 8 x y = (5.) x, y,, x8, y8 and f y f ( x) f ( y), x y, x y, x y x y (5.) xy,. Then there ext a unque quadratc homomorphm H: uch that

18 58 John M. Raa, M. runumar and S. Kartheyan, / 6 x, 4; 7 f ( x) H( x) 8 x, ; x, (5.3) x. Proof. Settng, 8 = 8 x y = x y, ( x, y,, x8, y8 ) =, x y x y, = = x, y,, x8, y8. Now,, 8, x y ( =,,, 8, 8) x y x y = 8, x y = x y x y, = =

19 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 59, 8 ( ) x y, = 8 = (6) x y, = 8 8 (6) 6 6 x y x y, = = Thu, (5.) hold. 0 a 0 a = 0 a 0 a,,,. x x x x ut we have ( x) =,,,, 7 ( x) = L x x. Hence ha the property Now,, 7 6 x, x x x x 7 ( x) =,,,, = 7 8 x, x.

20 60 John M. Raa, M. runumar and S. Kartheyan, 7 ( x), 6 x, 4 7 ( ), x ( ) = = x 8 8, ( x), 8 x ( x). 8 8 x 7 From (5.6), we prove the followng x cae: Cae : If = 0 then Cae : If = then L = 0 f ( x) H( x) ( x) = =. ( ) 7 L = f ( x) H( x) ( x) = =. 7 Cae 3: ( 4)/ L = for < 4 f = 0 ( 4)/ 0 f x x x 6 x ( ) H( ) ( ) =. ( 4)/ 7 Cae 4: L = (4)/ for > 4 f = (4 )/ f x x x 6 x ( ) H( ) ( ) =. (4 )/ 7 Cae 5: 8 L = for < f = (8) 8 ( ) H( ) ( ) =. (8) 8 f x x x 7 x

21 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 6 Cae 6: L 8 = for > f = 4 (8 ) 8 ( ) H( ) ( ) =. (8 ) 8 f x x x 7 x Hence the proof complete Theorem 5.5 Let f : be a mappng for whch there ext a functon 6 : [0, ) wth the condton lm ( y,, x8, y8) = 0 (5.4) where = f = 0 and = f = uch that the functonal nequalty wth F( x, y,, x, y ) ( x, y,, x, y ) (5.5) x, y,, x8, y8 and f ( xy) x f ( y) f ( x) y y,0,0,,0,0 (5.6) xy,. If there ext L = L( ) uch that the functon x x x x x ( x) =,,,,, ha the property (5.7) ( x) = L x (5.8) x, then there ext a unque quadratc dervaton D: atfyng the functonal equaton (.5) and L f ( x) D( x) ( x) L (5.9) x. Proof. y the ame reaonng a that n the proof of Theorem 5.3, there ext

22 6 John M. Raa, M. runumar and S. Kartheyan a unque quadratc mappng D: atfyng (5.9). The mappng D: f x gven by D( x) = lm x. It follow from (5.5) that D( x y ) x D( y ) D( x ) y lm f x y x f y f x y 4 lm x, y,0,0,,0,0= 0 4 y. Therefore, D: a quadratc dervaton atfyng. The ret of the proof mlar to that of Theorem 5.3 The followng corollary an mmedate conequence of Theorem 5.5 concernng the tablty of (.5). Corollary 5.6 Let and be non-negatve real number. If a functon f : atfe the nequalty F( x, y,, x, y ) 8 8, 8 x y, =, x y x y, = = 8 x y = (5.0) x, y,, x8, y8 and f ( xy) x f ( y) f ( x) y, x y, x y, x y x y (5.) xy,. Then there ext a unque quadratc dervaton D: uch that

23 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 63 ; / 6 x, 4; 7 f ( x) D( x) 8 x, ; x, (5.) x. 6. pplcaton of The Functonal Equaton (.5) Conder the functonal equaton (.5), that Snce 8 8 = = f ( x ) f ( y ) = f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f x y x y x y x y x y x y x y x y f ( x) = x the oluton of the above functonal equaton, then the above equaton can be wrtten a follow

24 64 John M. Raa, M. runumar and S. Kartheyan 8 8 ( x) ( y) = x y x y x y x y x y x y x y x y = = x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y xy 5 x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y The above dentty called Degen-Grave-Cayley-Eght Square dentty and how that the product of two um of eght quare agan a um of eght quare". REFERENCES [] J. czel and J. Dhombre, Functonal Equaton n Several Varable, Cambrdge Unv, Pre, 989. [] T. o, On the tablty of the lnear tranformaton n anach pace", J. Math. Soc. Japan, (950), [3] M. runumar, S. Jayanth, S. Hemalatha, Soluton Quadratc Dervaton of run -quadratc Functonal Equaton", Internatonal Journal of Mathematcal Scence and Engneerng pplcaton, Vol. 5, No.4, September 0, [4] M. runumar and S. Kartheyan, rahmagupta Quadratc Functonal Equaton Connected wth Homomorphm and Dervaton on Non- rchmedean lgebra: Drect and Fxed Pont Method", Proceedng of the Internatonal Conference on Mathematcal Scence (ICMS-04) Publhed by Elever, ISN: , pp, [5] R. adora, On approxmate rng homomorphm", J. Math. nal. ppl. 76, (00). [6] R. adora, On approxmate dervaton", Math. Inequal. ppl. 9, (006). [7] ae. Y.H, Jun. K. W, on the Hyer-Ulam-Raa tablty of a quadratc functonal equaton", ull.korean.math.soc., 38() (00), [8] enyamn Y. and Lndentrau J, Geometrc Nonlnear Functonal naly", vol., mer. Math. Soc. Colloq. Publ., vol. 48, mer. Math. Soc., Provdence, RI, 000.

25 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 65 [9] D.G. ourgn, pproxmately ometrc and multplcatve tranformaton on contnuou functon rng", Due Math. J. 6, (949). [0] P. W. Cholewa, Remar on the tablty of functonal equaton", equatone Mathematcae, vol. 7, no. -, pp , 984. [] S. Czerw, Functonal Equaton and Inequalte n Several Varable, World Scentfc, Rver Edge, NJ, 00. [] I.S. Chang, H.M. Km, On the Hyer-Ulam-Raa tablty of a quadratc functonal equaton", J. Ineq. ppl. Math, 33 (00), -. [3] J. Daz,. Margol, fxed pont theorem of the alternatve for contracton on the generalzed complete metrc pace", ull. m. Math. Soc. 6, (968). [4] M. Ehagh Gordj and M. avand Savadouh, pproxmaton of generalzed homomorphm n qua-anach algebra", 7(), 03 4, 009. [5] M. Ehagh Gordj, Nearly rng homomorphm and nearly rng dervaton on non-rchmedean anach algebra". btr. ppl. nal. 00, rtcle ID (00). [6] M. Ehagh Gordj, H. Khodae, On the generalzed Hyer-Ulam-Raa tablty of quadratc functonal equaton". btr. ppl. nal. 009, rtcle ID (009). [7] M. Ehagh Gordj, H. Khodae, R. Khodabahh, C. Par, Fxed pont and quadratc equaton connected wth homomorphm and dervaton on non- rchmedean algebra". dvance n Dfference Equaton, 0, 0:8. [8] P. G a vruta, generalzaton of the Hyer-Ulam-Raa tablty of approxmately addtve mappng", J. Math. nal. ppl., 84 (994), [9] P. G a vruta, L. G a vruta, new method for the generalzed Hyer-Ulam- Raa tablty". Int. J. Nonlnear nal. ppl. (), -8 (00). [0]. Grabec, The generalzed Hyer-Ulam tablty of a cla of functonal equaton", Publcatone Mathematcae Debrecen, vol. 48, no. 3-4, pp. 7-35, 996. [] D.H. Hyer, On the tablty of the lnear functonal equaton", Proc.Nat. cad.sc.,u.s..,7 (94) -4. [] S.M. Jung, On the Hyer-Ulam tablty of the functonal equaton that have the quadratc property", J. Math. nal. ppl. (998), [3] Pl. Kannappan, Quadratc functonal equaton nner product pace", Reult Math. 7, No.3-4, (995), [4] Lee Jung-Rye, Shn Dong-Yun, Iomorphm nd Dervaton In C*- lgebra", cta Mathematca Scenta 0,3():

26 66 John M. Raa, M. runumar and S. Kartheyan [5] K. Mrmotafaee, Hyer-Ulam tablty of cubc mappng n non- rchmedean normed pace". Kyungpoo Math. J. 50, (00). [6] C. Par and bba Najat, Homomorphm and Dervaton n C*- lgebra", Hndaw Publhng Corporaton btract and ppled naly, Volume 007, rtcle ID 80630, page, do:0.55/007/ [7] C. Par, Homomorphm between Le JC*-algebra and Cauchy-Raa tablty of Le JC*-algebra dervaton", Journal of Le Theory, vol. 5, no., pp , 005. [8] C. Par, J. C.Hou, and S. Q. Oh, Homomorphm between JC*-algebra and Le C*-algebra", cta Mathematca Snca, vol., no. 6, pp , 005. [9] C. Par, Homomorphm between Poon JC*-algebra", ulletn of the razlan Mathematcal Socety, vol. 36, no., pp , 005. [30] V. Radu, The fxed pont alternatve and the tablty of functonal equaton". Fxed Pont Theory 4, 9-96 (003). [3] J.M. Raa, On approxmately of approxmately lnear mappng by lnear mappng", J. Funct. nal. US, 46, (98) [3] J.M. Raa, On approxmately of approxmately lnear mappng by lnear mappng", ull. Sc. Math, 08, (984) [33] J. M. Raa, M. runumar and S. Kartheyan, Lagrange Quadratc Functonal Equaton Connected Wth Homomorphm and Dervaton On Le C*-lgebra: Drect nd Fxed Pont Method", Malaya Journal of Matemat, S(), 8â 4, 05. [34] J. M. Raa, M. runumar and S. Kartheyan, Eulerâ quadratc functonal equaton aocated to JC*-algebra omorphm and JC*-algebra dervaton between JC*-algebra", Global Journal of Pure and ppled Mathematc, (3), , 06. [35] Th. M. Raa, On the tablty of functonal equaton and a problem of Ulam", cta ppl. Math., 6()(000), [36] Th.M. Raa, On the tablty of the lnear mappng n anach pace", Proc.mer.Math. Soc., 7 (978), [37] K. Rav, M. runumar and J.M. Raa, On the Ulam tablty for the orthogonally general Euler-Lagrange type functonal equaton", Internatonal Journal of Mathematcal Scence, utumn 008 Vol.3, no. 08, [38] Rolewcz S, Metrc Lnear Space, PWN-Polh Sc. Publ., Warzawa, Redel, Dordrecht, 984. [39] P. Ŝ emrl, The functonal equaton of multplcatve dervaton upertable on tandard operator algebra", Integral Equ. Oper. Theory 8, 8- (994).

27 Stablty Of Dgc Quadratc Functonal Equaton In Qua-anach 67 [40] F. Sof, Local properte and approxmaton of operator", Rendcont del Semnaro Matematco e Fco d Mlano, vol. 53, pp. 3-9, 983. [4] S.M. Ulam, Problem n Modern Mathematc, Scence Edton, Wley, NewYor, 964.

28 68 John M. Raa, M. runumar and S. Kartheyan