J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC O Sabiliy of Quiic Fucioal Equaios i Radom Normed Spaces Afrah A.N. Abdou, Y. J. Cho,,, Liaqa A. Kha ad S. S. Kim3, Deparme of Mahemaics, Kig Abdulaziz Uiversiy Jeddah 589, Saudi Arabia E-mail: aabdou@kau.edu.sa; lkha@kau.edu.sa Deparme of Mahemaics Educaio ad he RINS Gyeogsag Naioal Uiversiy Jiju 660-70, Korea E-mail: yjcho@gu.ac.kr 3 Deparme of Mahemaics, Dogeui Uiversiy Busa 64-74, Korea E-mail: sskim@deu.ac.kr Absrac. I his paper, usig he direc ad fixed poi mehods, we ivesigae he geeralized Hyers-Ulam sabiliy of he quiic fucioal equaio: f x + y + f x y + f x + y + f x y = 0[f x + y + f x y] + 90f x i radom ormed spaces uder he miimum -orm.. Iroducio A classical quesio i sabiliy of fucioal equaios is as follows: Uder wha codiios, is i rue ha a mappig which approximaely saisfies a fucioal equaio ξ mus be somehow close o a exac soluio of ξ? We say he fucioal equaio ξ is sable if ay approximae soluio of ξ is ear o a rue soluio of ξ. The sudy of sabiliy problem for fucioal equaios is relaed o a quesio of Ulam [5] cocerig he sabiliy of group homomorphisms. The famous Ulam sabiliy problem was parially solved by Hyers [9] for liear fucioal equaio of Baach spaces. Subsequely, he resul of Hyers heorem was geeralized by Aoki [] for addiive mappigs ad by Rassias [] for liear mappigs by cosiderig a ubouded Cauchy differece. Ca dariu ad Radu [3] applied he fixed poi mehod o ivesigaio of he Jese fucioal equaio. They could prese a shor ad a simple proof differe from he direc mehod iiiaed by Hyers i 94 for he geeralized Hyers-Ulam sabiliy of Jese fucioal equaio ad for quadraic fucioal equaio. Their mehods are a powerful ool for sudyig he sabiliy of several fucioal equaios. 0 000 Mahemaics Subjec Classificaio: 39B5, 39B7, 47H09, 47H47. Keywords: Geeralized Hyers-Ulam sabiliy, quiic fucioal equaio, radom ormed spaces, fixed poi heorem. 0 *The correspodig auhor. 0 64 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC O Sabiliy of Quiic Fucioal Equaios O he oher had, he heory of radom ormed spaces briefly, RN -spaces is impora as a geeralizaio of deermiisic resul of ormed spaces ad also i he sudy of radom operaor equaios. The oio of a RN -space correspods o he siuaios whe we do o kow exacly he orm of he poi ad we kow oly probabiliies of passible values of his orm. The RN spaces may provide us he appropriae ools o sudy he geomery of uclear physics ad have usefully applicaio i quaum paricle physics. A umber of papers ad research moographs have bee published o geeralizaios of he sabiliy of differe fucioal equaios i RN spaces [5, 6, 0,, 6]. I he sequel, we use he defiiios ad oaios of a radom ormed space as i [, 3, 4]. A fucio F : R {, +} [0, ] is called a disribuio fucio if i is odecreasig ad lef-coiuous, wih F 0 = 0 ad F + =. The class of all probabiliy disribuio fucios F wih F 0 = 0 is deoed by Λ. D+ is a subse of Λ cosisig of all fucios F Λ for which F + =, where l F x = lim x F. For ay a 0, ϵa is he eleme of D+, which is defied by { 0, if a, ϵa =, if > a. Defiiio.. [3] A fucio T : [0, ] [0, ] [0, ] is a coiuous riagular orm briefly, a -orm if T saisfies he followig codiios: T is commuaive ad associaive; T is coiuous; 3 T a, = a for all a [0, ]; 4 T a, b T c, d wheever a c ad b d for all a, b, c, d [0, ]. Three ypical examples of coiuous -orms are as follows: TM a, b = mi{a, b}, TP a, b = ab, TL a, b = max{a + b, 0}. Recall ha, if T is a -orm ad {x } is a sequece of umbers i [0, ], he Ti= xi is defied recurrely by Ti= xi = x ad Ti= xi = T Ti= xi, x = T x,, x for each ad Ti= x is defied as Ti= x+i [8]. Defiiio.. [4] Le X be a real liear space, µ be a mappig from X io D+ for ay x X, µx is deoed by µx ad T be a coiuous -orm. The riple X, µ, T is called a radom ormed space briefly RN -space if µ saisfies he followig codiios: RN µx = ϵo for all > 0 if ad oly if x = 0; RN µαx = µx α for all x X, α = 0 ad all 0; RN3 µx+y + s T µx, µy s for all x, y X ad all, s 0. Example.. Every ormed space X, defies a RN -space X, µ, TM, where µx = + x for all > 0 ad TM is he miimum -orm. This space is called he iduced radom ormed space. 65 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC Afrah A.N. Abdou, Y. J. Cho, Liaqa A. Kha ad S. S. Kim 3 Defiiio.3. Le X, µ, T be a RN -space. A sequece {x } i X is said o be coverge o a poi x X if, for all > 0 ad λ > 0, here exiss a posiive ieger N such ha µx x > λ wheever N. I his case, x is called he limi of he sequece {x } ad we deoe i by lim µx x =. A sequece {x } i X is called a Cauchy sequece if, for all > 0 ad λ > 0, here exiss a posiive ieger N such ha µx xm > λ wheever m N. 3 The RN -space X, µ, T is said o be complee if every Cauchy sequece i X is coverge o a poi i X. Theorem.4. [3] If X, µ, T is a RN -space ad {x } is a sequece of X such ha x x, he lim µx = µx almos everywhere. Recely, Cho e. al. [4] was iroduced ad proved he Hyers-Ulam-Rassias sabiliy of he followig quiic fucioal equaios f x + y + f x y + f x + y + f x y = 0[f x + y + f x y] + 90f x. for fixed k Z+ wih k 3 i quasi-β-ormed spaces. Remark.. If we pu x = y = 0 i he equaio., he f 0 = 0. f x = 5 f x for all x X ad Z+. 3 f is a odd mappig. Throughou his paper, le X be a real liear space, Z, µ, TM be a RN -space ad Y, µ, TM be a complee RN -space. For ay mappig f : X Y, we defie Df x, y = f x + y + f x y + f x + y + f x y 0[f x + y + f x y] 90f x for all x, y X. I his paper, usig he direc ad fixed poi mehods, we ivesigae he geeralized Hyers-Ulam sabiliy of he quiic fucioal equaio: f x + y + f x y + f x + y + f x y = 0[f x + y + f x y] + 90f x i radom ormed spaces uder he miimum -orm.. Radom sabiliy of he fucioal equaio. I his secio, we ivesigae he geeralized Hyers-Ulam sabiliy problem of he quiic fucioal equaio. i RN -spaces i he sese of Schersev uder he miimum -orm TM. Theorem.. Le ϕ : X Z be a fucio such ha, for some 0 < α < 5, µϕx,y µαϕx,y 66. ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC O Sabiliy of Quiic Fucioal Equaios 4 ad lim µϕ x, y 5 = for all x, y X ad > 0. If f : X Y is a mappig wih f 0 = 0 such ha µdf x,y µϕx,y. for all x, y X ad > 0, he here exiss a uique quiic mappig Q : X Y such ha µf x Qx µϕx,0 5 α.3 for all x X ad > 0. Proof. Leig y = 0 i., we ge µ f x f x µϕx,0 8.4 5 for all x X ad > 0. Replacig x by x i.4, we ge 5 8 µ f + x f x µϕx,0 α 5 5+ j+ f x f j x x for all x X ad > 0. Sice f f x =, 5 5j 5j+ j=0 µ f x f x 5 α j TM j=0 µϕx,0 = µϕx,0 5 8 j=0 for all x X ad > 0. Subsiuig x by m x i.5, we ge µ f +m x f m x µϕx,0 +m 5+m 5m j=m.5.6 α5 j x for all x X ad m, Z wih > m 0. Sice α < k 3, he sequece { f 5 } is a Cauchy sequece i he complee RN -space Y, µ, TM ad so i coverges o some poi Qx Y. Fix x X ad pu m = 0 i.6. The we ge 8 µ f x f x µϕx,0 α, j 5 j=0 5 ad so, for ay δ > 0, µqx f x δ + TM µqx f x δ, µ f x f x 5 5 8 TM µqx f x δ, µϕx,0 α j 5 j=0 5.7 for all x X ad > 0. Takig he limi as i.7, we ge µqx f x δ + µϕx,0 5 α.8 Sice δ is arbirary, by akig δ 0 i.8, we have µqx f x µϕx,0 5 α.9 for all x X ad > 0. Therefore, we coclude ha he codiio.3 holds. Also, replacig x ad y by x ad y i., respecively, we have µ Df x, y µϕ x, y 5 5 67 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC Afrah A.N. Abdou, Y. J. Cho, Liaqa A. Kha ad S. S. Kim 5 for all x, y X ad > 0. I follows from lim µϕ x, y 5 = ha Q saisfies he equaio., which implies ha Q is a quiic mappig. To prove he uiqueess of he quiic mappig Q, le us assume ha here exiss aoher e : X Y which saisfies.3. Fix x X. The Q x = 5 Qx ad Q e x = mappig Q + 5 e Qx for all Z. Thus i follows from.3 ha µqx Qx e = µ Q x Q e x 5 5.0 TM µ Q x f x, µ f x Q e x 5 5 5 5 5 5 µϕx,0 α. α 5 = for all > 0. Thus he quiic Sice lim 5 α α =, we have µqx Qx e mappig Q is uique. This complees he proof. Theorem.. Le ϕ : X Z be a fucio such ha, for some 5 < α, µϕ x, y µϕx,y α. ad lim µ5 ϕ x, y = for all x, y X ad > 0. If f : X Y is a mappig wih f 0 = 0 which saisfies., he here exiss a uique cubic mappig Q : X Y such ha. µf x Qx µϕx,0 α 5 for all x X ad > 0. Proof. I follows from. ha µf x 5 f x µϕx,0 α.3 for all x X. Applyig he riagle iequaliy ad.3, we have µf x 5 f x µϕx,0 α +m 5 j j=m.4 α for all x X ad m, Z wih > m 0. The he sequece {5 f x } is a Cauchy sequece i he complee RN -space Y, µ, TM ad so i coverges o some poi Qx Y. We ca defie a mappig Q : X Y by x Qx = lim 5 f for all x X. The he mappig Q saisfies. ad.. The remaiig asserio follows he similar proof mehod i Theorem.. This complee he proof. Corollary.3. Le θ be a oegaive real umber ad z0 be a fixed ui poi of Z. If f : X Y is a mappig wih f 0 = 0 which saisfies µdf x,y µθz0.5 for all x, y X ad > 0, he here exiss a uique quiic mappig C : X Y such ha µf x Qx µθz0 4.6 68 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC O Sabiliy of Quiic Fucioal Equaios 6 for all x X ad > 0. Proof. Le ϕ : X Z be defied by ϕx, y = θz0. The, he proof follows from Theorem. by α =. This complees he proof. Corollary.4. Le p, q R be posiive real umbers wih p, q < 5 ad z0 be a fixed ui poi of Z. If f : X Y is a mappig wih f 0 = 0 which saisfies µdf x,y µ x p + y q z0.7 for all x, y X ad > 0, he here exiss a uique quiic mappig Q : X Y such ha µf x Qx µ x p z0 5 p.8 for all x X ad > 0. Proof. Le ϕ : X Z be defied by ϕx, y = x p + y q z0. The he proof follows from Theorem. by α = p. This complees he proof. Now, we give a example o illusrae ha he quiic fucioal equaio. is o sable for r = 5 i Corollary.4 Example.. Le ϕ : R R be defied by { x5, for x <, ϕx =, oherwise. Cosider he fucio f : R R defied by f x = ϕ x 5 =0 for all x R. The f saisfies he fucioal iequaliy f x + y + f x y + f x + y + f x y 0[f x + y + f x y] 90f x.9 36 3 5 x + y 5 3 for all x, y X, bu here do o exis a quiic mappig Q : R R ad a cosa d > 0 such ha f x Qx d x 5 for all x R. I fac, i is clear ha f is bouded by rivial. If x 5 + y 5 3, he 3 3 o R. If x 5 + y 5 = 0, he.9 is 36 3 5 36 3 x + y 5. 3 3 Now, suppose ha 0 < x 5 + y 5 < 3. The here exiss a posiive ieger k Z + such ha Df x, y 3k+ ad so x 5 + y 5 < 3k+, 3k y 5 <, 3 3 x + y, x y, x + y, x y, x y, x, 3k x 5 < 69 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC Afrah A.N. Abdou, Y. J. Cho, Liaqa A. Kha ad S. S. Kim 7 ad ϕ x + y + ϕ x y + ϕ x + y + ϕ x y 0[ϕ x + y + ϕ x y] 90ϕ x =0 for all = 0,,, k. Thus we obai Df x, y ϕ x + y + ϕ x y + ϕ x + y 5 =0 + ϕ x y 0[ϕ x + y + ϕ x y] 90ϕ x ϕ x + y + ϕ x y + ϕ x + y 5 =k + ϕ x y 0[ϕ x + y + ϕ x y] 90ϕ x 36 3 5 x + y 5. 3 Therefore, f saisfies.9. Now, we claim ha he quiic fucioal equaio. is o sable for r = 5 i Corollary.4. Suppose o he corary ha here exiss a quiic mappig Q : R R ad cosa d > 0 such ha f x Qx d x 5 for all x R. Sice f is bouded ad coiuous for all x R, Q is bouded o ay ope ierval coaiig he origi ad coiuous a he origi. I view of Theorem., Q mus have Qx = cx5 for all x R. So, we obai f x d + c x 5.0 for all x R. Le m Z+ such ha m + > d + c. If x is i 0, m, he x 0, for = 0,,, m. For his x, we have m ϕ x5 f x = = m + x5 > d + c x 5, 5 5 =0 =0 which coradicio.0. Remark.. I Corollary.4, if we assume ha ϕx, y = x r y r z0 or ϕx, y = x r y s + x r+s + y r+s z0, he we have Ulam-Gavua-Rassias produc sabiliy ad JMRassias mixed produc-sum sabiliy, respecively. Nex, we apply a fixed poi mehod for he geeralized Hyer-Ulam sabiliy of he fucioal equaio. i RN -spaces. The followig Theorem will be used i he proof of Theorem.6. 630 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC O Sabiliy of Quiic Fucioal Equaios 8 Theorem.5. [7] Suppose ha Ω, d is a complee geeralized meric space ad J : Ω Ω is a sricly coracive mappig wih Lipshiz cosa L <. The, for each x Ω, eiher dj x, J + x = for all oegaive iegers 0 or here exiss a aural umber 0 such ha dj x, J + x < for all 0 ; he sequece {J x} is coverge o a fixed poi y of J; 3 y is he uique fixed poi of J i he se Λ = {y Ω : dj 0 x, y < }; 4 dy, y L dy, Jy for all y Λ. Theorem.6. Le ϕ : X D+ be a fucio such ha, for some 0 < α < 5, µϕx,y µϕx,y α. for all x, y X ad > 0. If f : X Y is a mappig wih f 0 = 0 such ha µdx,y µϕx,y. for all x, y X ad > 0, he here exiss a uique quiic mappig Q : X Y such ha µf x Qx µϕx,y 5 α.3 for all x X ad > 0. Proof. I follows from. ha µf x f x µϕx,0 8.4 5 for all x X ad > 0. Le Ω = {g : X Y, gx = 0} ad he mappig d defied o Ω by dg, h = if{c [0, : µgx hx c µϕx,0, x X} where, as usual, if =. The Ω, d is a geeralized complee meric space see [0]. Now, le us cosider he mappig J : Ω Ω defied by Jgx = 5 gx for all g Ω ad x X. Le g, h i Ω ad c [0, be a arbirary cosa wih dg, h < c. The µgx hx c µϕx,0 for all x X ad > 0 ad so αc µjgx Jhx = µgx hx αc µϕx,0.5 5 for all x X ad > 0. Hece we have α αc djg, Jh 5 5 dg, h for all g, h Ω. The J is a coracive mappig o Ω wih he Lipschiz cosa L = α5 <. Thus i follows from Theorem.5 ha here exiss a mappig Q : X Y, which is a uique fixed poi of J i he se Ω = {g Ω : df, g < }, such ha f x 5 for all x X sice lim dj f, Q = 0. Also, from µf x f x µϕx,0 8, i follows Qx = lim ha df, Jf 8. 5 Therefore, usig Theorem.5 agai, we ge df, Q df, Jf 5. L α 63 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC Afrah A.N. Abdou, Y. J. Cho, Liaqa A. Kha ad S. S. Kim This meas ha 9 µf x Qx µϕx,0 5 α for all x X ad > 0. Also, replacig x ad y by x ad y i., respecively, we have 5 µdqx,y lim µϕ x, y 5 = lim µϕx,y = α for all x, y X ad > 0. By RN, he mappig Q is quiic. To prove he uiqueess, le us assume ha here exiss a quiic mappig Q : X Y which saisfies.3. The Q is a fixed poi of J i Ω. However, i follows from Theorem.5 ha J has oly oe fixed poi i Ω. Hece Q = Q. This complees he proof. Theorem.7. Le ϕ : X D+ be a fucio such ha, for some 0 < 5 < α, µϕx,y µϕ x, y α.6 for all x, y X ad > 0. If f : X Y is a mappig wih f 0 = 0 which saisfies., he here exiss a uique quiic mappig Q : X Y such ha µf x Qx µϕx,0 α 5.7 for all x X ad > 0. Proof. By a modificaio i he proofs of Theorem. ad.6, we ca easily obai he desired resuls. This complees he proof. Now, we prese a corollary ha is a applicaio of Theorem.6 ad.7 i he classical case. Corollary.8. Le X be a Baach space, ϵ ad p be posiive real umbers wih p = 5. Assume ha f : X X is a mappig wih f 0 = 0 which saisfies Df x, y ϵ x p + y p for all x, y X. The here exiss a uique quiic mappig Q : X Y such ha Qx f x ϵ x p 5 p for all x X ad > 0. Proof. Defie µ : X R R by { µx = + x, 0, if > 0, oherwise for all x X ad R. The X, µ, TM is a complee RN -space. Deoe ϕ : X X R by ϕx, y = ϵ x p + y p for all x, y X ad > 0. I follows from Df x, y θ x p + y p ha µdf x,y µϕx,y 63 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC 0 O Sabiliy of Quiic Fucioal Equaios for all x, y X ad > 0, where µ : R R R give by {, if > 0, µx = + x 0, oherwise, is a radom orm o R. The all he codiios of Theorems.6 ad.7 hold ad so here exiss a uique quiic mappig Q : X X such ha = µqx f x + Qx f x µϕx,0 5 α = 5 α. 5 α + ϵ x p Therefore, we obai he desired resul, where α = p. This complees he proof. Ackowledgmes This projec was fuded by he Deaship of Scieific Research DSR, Kig Abdulaziz Uiversiy, uder gra o. 8-30-36-HiCi. The auhors, herefore, ackowledge wih haks DSR echical ad fiacial suppor. Also, Yeol Je Cho was suppored by Basic Sciece Research Program hrough he Naioal Research Foudaio of Korea NRF fuded by he Miisry of Sciece, ICT ad fuure Plaig 04RAAA00000. Refereces [] C. Alsia, B. Schweizer, A. Sklar, O he defiiio of a probabiliic ormed spaces, Equal. Mah. 46993, 9 98. [] T. Aoki, O he sabiliy of he liear rasformaio i Baach spaces, J. Mah. Soc. Japa. 950, 64 66. [3] L. Ca dariu, V. Radu, Fixed pois ad he sabiliy of Jese s fucioal equaio, J. Iequal. Pure Appl. Mah. 4 003, No., Ar. 4. [4] I.G. Cho, D.S. Kag, H.J. Koh, Sabiliy problems of quiic mappigs i quasi-β-ormed spaces, J. Ieq. Appl. 00, Ar. ID 36898, 9 pp. [5] Y.J. Cho, C. Park, TM. Rassias, R. Saadai, Sabiliy of Fucioal Equaios i Baach Alegbras, Spriger Opimizaio ad Is Applicaio, Spriger New York, 05. [6] Y.J. Cho, TM. Rassias, R. Saadai, Sabiliy of Fucioal Equaios i Radom Normed Spaces, Spriger Opimizaio ad Is Applicaio 86, Spriger New York, 03. [7] J.B. Dias, B. Margolis, A fixed poi heorem of he aleraive for coraios o a geeralized complee meric space, Bull. Amer. Mah. Soc. 74 968, 305 309. [8] O. Hadz ic, E. Pap, M. Budicevic, Couable exesio of riagular orms ad heir applicaios o he fixed poi heory i probabilisic meric spaces, Kybereika 3800, 363 38. [9] D.H. Hyers, O he sabiliy of he liear fucioal equaio, Proc. Nal. Acad. Sci. USA 7 94, 4. [0] D. Mihe, V. Radu, O he sabiliy of he addiive Cauchy fucioal equaio i radom ormed spaces, J. Mah. Aal. Appl. 343008, 567 57. [] J.M. Rassias, R. Saadai, G. Sadeghi, J. Vahidi, O oliear sabiliy i various radom ormed spaces, J. Iequal. Appl. 0, 0:6. 633 ABDOU ET AL 64-634
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC Afrah A.N. Abdou, Y. J. Cho, Liaqa A. Kha ad S. S. Kim [] Th.M. Rassias, O he sabiliy of he liear mappig i Baach spaces, Proc. Amer. Mah. Soc. 7 978, 97 300. [3] B. Schweizer, A. Skar, Probabiliy Meric Spaces, Norh-Hollad Series i Probabiliy ad Applied Mah. New York, USA 983. [4] A.N. Shersev, O he oio of s radom ormed spaces, Dokl. Akad. Nauk SSSR 49, 80 83 i Russia. [5] S.M. Ulam, Problems i Moder Mahemaics, Sciece Ediios, Joh Wiley & Sos, New York, USA, 940. [6] T.Z. Xu, J.M. Rassias, W.X. Xu, O sabiliy of a geeral mixed addiive-cubic fucioal equaio i radom ormed spaces, J. Iequal. Appl. 00, Ar. ID 38473, 6 pp. [7] T.Z. Xu, J.M. Rassias, M.J. Rassias, W.X. Xu, A fixed poi approach o he sabiliy of quiic ad sexic fucioal equaios i quasi-β-ormed spaces, J. Iequal. Appl. 00, Ar. ID 433, 3 pp. 634 ABDOU ET AL 64-634