CAUCHY RASSIAS STABILITY OF HOMOMORPHISMS ASSOCIATED TO A PEXIDERIZED CAUCHY JENSEN TYPE FUNCTIONAL EQUATION ABBAS NAJATI. 1.

Transcription

1 Journal of Mathematical Inequalitie Volume 3, Number 009, CAUCHY RASSIAS STABILITY OF HOMOMORPHISMS ASSOCIATED TO A PEXIDERIZED CAUCHY JENSEN TYPE FUNCTIONAL EQUATION ABBAS NAJATI Communicated by Th. Raia Abtract. We ue a fixed point method to prove the Cauchy Raia tability of homomorphim aociated to the Pexiderized Cauchy Jenen type functional equation x + y x y rf + g = hx, r, R \{0} r in Banach algebra. 1. Introduction The tability problem of functional equation originated from a quetion of S. M. Ulam [8] concerning the tability of group homomorphim : Let G 1, be a group and let G,,d be a metric group with the metric d,. Given ε > 0, doe there exit δε > 0 uch that if a mapping h : G 1 G atifie the inequality dhx y,hx hy < δ for all x,y G 1, then there i a homomorphim H : G 1 G with dhx,hx < ε for all x G 1? In other word, we are looking for ituation where homomorphim are table, i.e., if a mapping i almot a homomorphim, then there exit a homomorphim near it. D. H. Hyer [6] gave a firt affirmative anwer to the quetion of Ulam for Banach pace. Let X and Y be Banach pace: Aume that f : X Y atifie f x + y f x f y ε for ome ε 0 and all x,y X. Then there exit a unique additive mapping T : X Y uch that f x Tx ε for all x X. Mathematic ubject claification 000: Primary: 39B7; Secondary 47H15, 47H09. Keyword and phrae: Generalized metric pace, fixed point, tability, Banach module, C -algebra. c D l,zagreb Paper JMI

2 58 A. NAJATI T. Aoki [] and Th. M. Raia [5] provided a generalization of the Hyer theorem for additive and linear mapping, repectively, by allowing the Cauchy difference to be unbounded. The following theorem which i called the Cauchy Raia tability i a generalized olution to the tability problem. THEOREM 1.1. Th. M. Raia. Letf: E E be a mapping from a normed vector pace E into a Banach pace E ubject to the inequality f x + y f x f y ε x p + y p 1.1 for all x,y E,where ε and p are contant with ε > 0 and p < 1. Then the limit Lx= lim n f n x n exit for all x E and L : E E i the unique additive mapping which atifie f x Lx ε p x p 1. for all x E. If p< 0 then inequality 1.1 hold for x,y 0 and 1. for x 0. Alo, if for each x E the mapping t f tx i continuou in t R, then L i linear. The inequality 1.1 i called Cauchy Raia inequality and the tability of the functional equation i called Cauchy Raia tability [5, 1, 7]. The inequality 1.1 ha provided a lot of influence in the development of what i now known a a generalized Hyer Ulam tability or HyerUlamRaia tability of functional equation. A generalization of the Th.M. Raia theorem wa obtained by P. Găvruta [5]. We refer the reader to [8], [9], [14], [16] [4] and reference therein for more detailed reult on the tability problem of variou functional equation and mapping and their Pexider type. We alo refer the reader to the book [4], [7], [10] and [6]. Let E be a et. A function d : E E [0, ] i called a generalized metric on E if d atifie i dx,y=0 if and only if x = y; ii dx,y=dy,x for all x,y E ; iii dx,z dx,y+dy,z for all x,y,z E. We recall the following theorem by Margoli and Diaz. THEOREM 1.. [13] Let E,d be a complete generalized metric pace and let J : E E be a trictly contractive mapping with Lipchitz contant L < 1. Then for each given element x E, either dj n x,j n+1 x= for all non-negative integer n or there exit a non-negative integer n 0 uch that

3 CAUCHY-RASSIAS STABILITY OF HOMOMORPHISMS dj n x,j n+1 x < for all n n 0 ;. the equence {J n x} converge to a fixed point y of J ; 3. y i the unique fixed point of J in the et Y = {y E : dj n 0x,y < }; 4. dy,y 1 L 1 dy,jy for all y Y. Throughout thi paper, A denote a complex normed algebra and B repreent a complex Banach algebra. In addition, we aume r, to be fixed non-zero real number. In thi paper uing the fixed point method ee [1, 3, 11, 15], we prove the Cauchy Raia tability of homomorphim aociated to the Pexiderized Cauchy Jenen type functional equation x + y rf + g r y = hx, r, R \{0} in Banach algebra. For convenience, we ue the following abbreviation for given mapping f,g,h : X Y, μx + μy D μ f,g,hx,y := rf r + g μx μy μhx for all x,y X and all μ T 1,whereX and Y are linear pace and T 1 := { μ C : μ = 1}.. Main Reult We will ue the following Lemma in thi paper: LEMMA.1. [3] Let f : A B be an additive mapping uch that f μx=μ f x for all x A and all μ T 1. Then the mapping f i C-linear. PROPOSITION.. Let f,g,h : X Y be mapping with f 0=g0=0 uch that D μ f,g,hx,y=0.1 for all x,y X and all μ T 1. Then the mapping f,g,h arec-linear and f = g = h. Proof. Letting μ = 1andy = x in.1, we get rf x r = hx for all x X. So rf = h. r for all x X. Similarly, letting μ = 1andy = x in.1, we get g = h.3

4 60 A. NAJATI for all x X. Hence we get from.1,. and.3 that x + y x y h + h = hx.4 for all x,y X. Therefore h i additive, i.e., hx + y=hx+hy for all x,y X. Replacing x by μx in. and.3 and uing.1, we get that μx + μy μx μy h + h = μhx.5 for all x,y X and all μ T 1. It follow from.4 and.5 that hμx =μhx for all x X and all μ T 1. By Lemma.1 the mapping h i C-linear. Since h i C-linear, we get from. and.3 that f x= rx r h = hx, gx= h = hx for all x X. So f = g = h. Now we prove the Cauchy Raia tability of homomorphiim in Banach alebra. THEOREM.3. Let f,g,h : A B be mapping with f 0=g0=0 for which there exit function ϕ,φ : A A [0, uch that lim n n ϕ n, y n = 0, lim 4 n φ n n, y n = 0,.6 D μ f,g,hx,y ϕx,y,.7 f y g hy φx,y.8 for all x,y A and all μ T 1. If there exit a contant L < 1 uch that the function x ψx := ϕx,0+ϕ, x + ϕ, x ha the property ψx Lψx for all x A, then there exit a unique homomorphim H : A B uch that hx Hx 1 L ψx, f x Hx 1 rx r ϕ, rx + 1 rx r rl ψ, gx Hx 1 ϕ, x + 1 L ψ for all x A.

5 CAUCHY-RASSIAS STABILITY OF HOMOMORPHISMS 61 Proof. Letting μ = 1andy = 0,±x in.7, we get the following inequalitie rf + g hx ϕx,0,.9 r rf h ϕ r, x,.10 g h ϕ, x.11 for all x A. So it follow from.9,.10 and.11 that 1 hx h ψx.1 for all x A. Let X := {F : A B F0=0}. We introduce a generalized metric on X a follow: df,g := inf{c [0, ] : Fx Gx Cψx for all x A }. It i eay to how that X,d i a generalized complete metric pace [3]. Now we conider the mapping Λ : X X defined by ΛFx=F, for all F X and x A. Let F,G X and let C [0, ] be an arbitrary contant with df,g C. Fromthe definition of d,wehave Fx Gx Cψx forall x A. By the aumption and lat inequality, we have ΛFx ΛGx = F G Cψ CLψx for all x A.So dλf,λg LdF,G for any F,G X. It follow from.1 that dλh,h 1. Therefore according to Theorem 1., the equence {Λ n h} converge to a fixed point H of Λ, i.e., H : A B, Hx= lim Λ n hx= lim n h n n n and Hx=Hx for all x A.AloH i the unique fixed point of Λ in the et X = {F X : dh,f < } and dh,h 1 1 dλh,h 1 L L, i.e., the inequality hx Hx 1 ψx.13 L

6 6 A. NAJATI hold true for all x A. It follow from the definition of H,.6,.10 and.11 that lim n n rf n = lim n g r n n = Hx.14 for all x A. Hence we get get from.6 and.7 that Hμx + μy+hμx μy=μhx for all x,y A and all μ T 1. By Propoition. the mapping H : A B i C-linear. So we get from.10 and.13 that f x Hx f x rx r h rx rx + h H r 1 rx r ϕ, rx + 1 rx r rl ψ for all x A. In a imilar way we obtain the following inequality gx Hx 1 ϕ, x + 1 L ψ for all x A.SinceH i C-linear, it follow from.14 that lim n n f n = lim n g n n = Hx for all x A. Hence we get from.6 and.8 that Hxy =HxHy for all x,y A. So the mapping H : A B i a homomorphim. Finally it remain to prove the uniquene of H. Let P : A B be another homomorphim atifying dh,p 1 L. So P X and ΛPx =Px/=Px for all x A, i.e., P i a fixed point of Λ. Since H i the unique fixed point of Λ in X, we infer that P = H. COROLLARY.4. Let p > 1,q > and θ be non-negative real number and let f,g,h : A B be mapping with f 0=g0=0 atifying the inequalitie D μ f,g,hx,y θ x p + y p, f xy gxhy θ x q + y q for all x,y A and all μ T 1. Then there exit a unique homomorphim H : A B uch that hx Hx 4 + p p θ x p, for all x A. f x Hx gx Hx 3rp p r θ x p, 3p p θ x p

7 CAUCHY-RASSIAS STABILITY OF HOMOMORPHISMS 63 Proof. The proof follow from Theorem.3 by taking ϕx,y := θ x p + y p, φx,y := θ x q + y q for all x,y A. Then we can chooe L = 1 p and we get the deired reult. THEOREM.5. Let f,g,h : A B be mapping with f 0=g0=0 for which there exit function Φ,Ψ : A A [0, uch that 1 lim n n Ψn x, n 1 y=0, lim n 4 n Φn x, n y=0, D μ f,g,hx,y Ψx,y, f xy gxhy Φx,y for all x,y A and all μ T 1. If there exit a contant L < 1 uch that the function x ψx := Ψx,0+Ψ, x + Ψ, x ha the property ψx Lψx for all x A, then there exit a unique homomorphim H : A B uch that hx Hx L L ψx, f x Hx 1 rx r Ψ, rx + L rx r rl ψ,.15 gx Hx 1 Ψ, x + L L ψ for all x A. Proof. Uing the ame method a in the proof of Theorem.3, we have 1 hx hx 1 4 ψx L ψx.16 for all x A. We introduce the ame definition for X and d a in the proof of Theorem.3 uch that X,d become a generalized complete metric pace. Let Λ : X X be the mapping defined by ΛFx= 1 Fx, for all F X and x A. One can how that dλf,λg LdF,G for any F,G X. It follow from.16 that dλh,h L. Due to Theorem 1., the equence {Λn h} converge to a fixed point H of Λ, i.e., H : A B, Hx= lim n Λ n hx= lim n 1 n hn x

8 64 A. NAJATI and Hx=Hx for all x A.Alo dh,h 1 L dλh,h 1 L L, i.e., hx Hx L L ψx hold true for all x A. Similar to the proof of Theorem.3 we obtain the inequalitie.15. The ret of the proof i imilar to the proof of Theorem.3 and we omit the detail. COROLLARY.6. Let 0 < p < 1,0 < q < and θ,δ be non-negative real number and let f,g,h : A B be mapping with f 0=g0=h0=0 atifying the inequalitie D μ f,g,hx,y δ + θ x p + y p, f xy gxhy δ + θ x q + y q for all x,y A and all μ T 1. Then there exit a unique homomorphim H : A B uch that hx Hx 3 p p δ p p θ x p, f x Hx 1 + p p r δ + 8 p r p p p r θ x p, gx Hx 1 + p p δ + 8 p p p p θ x p for all x A. Proof. The proof follow from Theorem.5 by taking Ψx,y := δ + θ x p + y p, Φx,y := δ + θ x q + y q for all x,y A. Then we can chooe L = p 1 and we get the deired reult. REFERENCES [1] M. AMYARI AND M.S. MOSLEHIAN, Hyer-Ulam-Raia tability of derivation on Hilbert C - module, Contemporary Math., , [] T. AOKI, On the tability of the linear tranformationin Banach pace, J. Math. Soc. Japan, 1950, [3] L. CĂDARIU AND V. RADU, On the tability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber., , [4] P. CZERWIK, Functional Equation and Inequalitie in Several Variable, World Scientific Publihing Company, New Jerey, Hong Kong, Singapore and London, 00. [5] P. GǍVRUTA, A generalization of the Hyer Ulam Raia tability of approximately additive mapping, J. Math. Anal. Appl., ,

9 CAUCHY-RASSIAS STABILITY OF HOMOMORPHISMS 65 [6] D. H. HYERS, On the tability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., , 4. [7] D. H. HYERS, G. ISAC AND TH. M. RASSIAS, Stability of Functional Equation in Several Variable, Birkhä uer, Bael, [8] G. ISAC AND TH. M. RASSIAS, Stability of Ψ-additive mapping: application to nonlinear analyi, Internat. J. Math. Math. Sci., , [9] K.-W. JUN,Y.-H. LEE, On the Hyer Ulam Raia tability of a Pexiderized quadratic inequality, Math. Ineq. Appl., 4 001, [10] S.-M. JUNG, Hyer Ulam Raia Stability of Functional Equation in Mathematical Analyi, Hadronic Pre lnc., Palm Harbor, Florida, 001. [11] S.-M. JUNG AND T.-S. KIM, A fixed point approach to tability of cubic functional equation, Bol. Soc. Mat. Mexicana, 1 006, [1] R. V. KADISON AND G. PEDERSEN, Mean and convex combination of unitary operator, Math. Scand., , [13] B. MARGOLIS,J.B.DIAZ, A fixed point theorem of the alternative for contraction on a generalized complete metric pace, Bull. Amer. Math. Soc., , [14] Y. H. LEE, K. W. JUN, A generalization of the Hyer Ulam Raia tability of Pexider equation, J. Math. Anal. Appl., , [15] M. MIRZAVAZIRI AND M. S. MOSLEHIAN, A fixed point approach to tability of a quadratic equation, Bull. Braz. Math. Soc., , [16] M. S. MOSLEHIAN AND TH. M. RASSIAS, Stability of functional equation in non-archimedian pace, Appl. Anal. Dic. Math., 1 007, [17] A. NAJATI, Hyer Ulam tability of an n-apolloniu type quadratic mapping, Bulletin of the Belgian Mathematical Society-Simon Stevin, , [18] A. NAJATI, On the tability of a quartic functional equation, J. Math. Anal. Appl., , [19] A. NAJATI AND M. B. MOGHIMI, Stability of a functional equation deriving from quadratic and additive function in quai-banach pace, J. Math. Anal. Appl., , [0] A. NAJATI AND C. PARK, Hyer Ulam-Raia tability of homomorphim in quai-banach algebra aociated to the Pexiderized Cauchy functional equation, J. Math. Anal. Appl., , [1] C. PARK, Modified Trif functional equation in Banach module over a C -algebra and approximate algebra homomorphim, J.Math.Anal.Appl.,78 003, [] C. PARK, On the tability of the linear mapping in Banach module, J. Math. Anal. Appl., 75 00, [3] C. PARK, Homomorphim between Poion JC -algebra, Bull. Braz. Math. Soc., , [4] D.-W. PARK AND Y.-H. L EE, The Hyer Ulam Raia tability of the pexiderized equation, Nonlinear Analyi, , e503 e513. [5] TH. M. RASSIAS, On the tability of the linear mapping in Banach pace, Proc. Amer. Math. Soc., , [6] TH. M.RASSIAS, Functional Equation, Inequalitie and Application, Kluwer Academic Publiher Co., Dordrecht, Boton, London, 003. [7] T. TRIF, On the tability of a functional equation deriving from an inequality of Popoviciu for convex function, J. Math. Anal. Appl., 7 00, [8] S. M. ULAM, A Collection of the Mathematical Problem, Intercience Publ. New York, Received July 1, 008 Abba Najati Department of Mathematic Faculty of Science Univerity of Mohaghegh Ardabili Ardabil Iran a.nejati@yahoo.com Journal of Mathematical Inequalitie jmi@ele-math.com