Approximately intertwining mappings

Transcription

1 J. Math. Anal. Appl. 332 (2007) Appoximately intetwining mapping Mohammad Sal Molehian a,b,1 a Depatment of Mathematic, Fedowi Univeity, PO Box 1159, Mahhad 91775, Ian b Functional Analyi Goup, School of Pue Mathematic, Univeity of Leed, Leed, LS2 9JT, UK Received 28 Apil 2006 Available online 7 Novembe 2006 Submitted by T. Kiztin Abtact Let A be a Banach algeba, and let E be a weak Banach A-bimodule. An appoximately intetwining mapping coeponding to a functional equation E(f ) = 0 i a mapping f : A E with f(0) = 0uchthat E(f ) ε, and fo each a A the mapping f a (x) = f(ax) af (x), af(x)= f(xa) f(x)a, ae continuou at a point. In thi pape, we how that evey appoximately intetwining mapping coeponding to Cauchy, genealized Jenen o Tif functional equation can be etimated by an intetwining mapping Elevie Inc. All ight eeved. Keywod: Stability; Intetwining mapping; Banach algeba; Cauchy equation; Genealized Jenen equation; Tif equation 1. Intoduction In 1940, S.M. Ulam popoed the following poblem duing hi talk befoe a Mathematical Colloquium at the Univeity of Wiconin [20]: adde: molehian@fedowi.um.ac.i. 1 The autho wa uppoted by Ian National Science Foundation (INSF) (No ) X/$ ee font matte 2006 Elevie Inc. All ight eeved. doi: /j.jmaa

2 172 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) Given a goup G 1, a metic goup (G 2,d), and a poitive numbe ε, doe thee exit δ>0 uch that, if a function f : G 1 G 2 atifie the inequality d(f(xy),f(x)f(y)) δ fo all x,y G 1, then thee exit a homomophim T : G 1 G 2 uch that d(f (x), T (x)) ε fo all x G 1? In 1941 D.H. Hye [9] patially olved thi poblem fo linea mapping in the context of Banach pace, a follow. Suppoe that E 1 and E 2 ae Banach pace, and f : E 1 E 2 atifie the condition that thee i ε>0 uch that f(x+y) f(x) f(y) ε fo all x,y E 1. Then thee i a unique additive mapping T : E 1 E 2 uch that f(x) T(x) ε fo all x E 1. In 1951, D.G. Bougin [5] teated the Ulam poblem fo additive mapping. In 1978, Th.M. Raia [16] extended the theoem of Hye by conideing the unbounded Cauchy diffeence f(x+ y) f(x) f(y) ε( x p + y p ), whee 0 p<1 and ɛ>0 ae fixed. Thi eult ha been ignificant in the development of what we now call Hye Ulam Raia tability of functional equation. In the lat decade, the topic of tability of functional equation wa extenively tudied in eveal way by a numbe of mathematician; ee [6,10 12,17,18] and efeence theein. Let A be a Banach algeba, and let E be a weak Banach A-bimodule. By a weak Banach A-module we mean a Banach pace E which i an A-bimodule uch that fo each a A, the mapping z az and z za defined on E ae continuou. A linea mapping S : A E i called intetwining if fo each a A, the mapping x S(ax) as(x) and x S(xa) S(x)a fom A into E ae continuou. Thee mapping wee fit intoduced by Bade and Cuti [3] and wee genealized by K.B. Lauen [13]. Evidently, evey A-module homomophim of A into E and any deivation of A into E ae intetwining; cf. [7,8]. By an appoximately intetwining mapping coeponding to a functional equation E(f ) = 0, we mean a mapping f : A E with f(0) = 0 uch that E(f ) ε, and fo each a A the mapping f a (x) = f(ax) af (x), af(x)= f(xa) f(x)a, (1.2) ae continuou at a point. In thi pape, by uing the diect method (cf. [9]), the tability of intetwining mapping on Banach algeba via Cauchy, genealized Jenen and Tif equation ae etablihed. Thoughout the pape, A denote a Banach algeba, and E i a weak Banach A-bimodule. 2. Stability via the Cauchy equation We tat ou wok with etablihing the tability of intetwining mapping via the Cauchy equation. Theoem 2.1. Fo each appoximately intetwining mapping f coeponding to the Cauchy inequality f(λx+ λy) λf (x) λf (y) ε, (2.1) (1.1)

3 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) whee x,y A and λ T ={z C: z =1}, thee i a unique intetwining mapping S uch that f(x) S(x) ε (2.2) fo all x A. Poof. Fix a A.Wehave f a (λx + λy) λf a (x) λf a (y) f(λax+ λay) λf (ax) λf (ay) + af (λx + λy) λaf (x) λaf (y) ( 1 + a ) ε (2.3) fo all x,y A and λ T. Putting λ = 1 and y = x in (2.3) we get f a (2x) 2f a (x) ( 1 + a ) ε (x A). (2.4) One can apply (2.4) and ue induction to how that f a (2 n x) 2 n f a(2 m n 1 x) 2 m 2 l( 1 + a ) ε (2.5) l=m fo all intege n>m 0 and all x A. It follow that the equence ( f a(2 n x) 2 n ) i Cauchy in E, and o i convegent, ince E i a complete pace. Set f a (2 n x) S a (x) := lim n 2 n. (2.6) By eplacing x,y by 2 n x,2 n y, epectively, in (2.3), we get 2 n ( f a 2 n (λx + λy) ) λ2 n ( f a 2 n x ) λ2 n ( f a 2 n y ) 2 n( 1 + a ) ε. Taking limit a n, we obtain S a (λx + λy) = λs a (x) + λs a (y) (2.7) fo all x,y A and all λ T. Obviouly, S a (0x) = 0 = 0S a (x). Next, let μ C (μ 0), and let M be a natual numbe geate than μ. By an eaily geometic agument, one can conclude that thee exit two numbe λ 1,λ 2 T uch that 2 μ M = λ 1 +λ 2. By the additivity of S a we get S a ( 1 2 x) = 1 2 S a(x) fo all x A. Theefoe ( M S a (μx) = S a 2 2 μ ) ( 1 M x = MS a 2 2 μ ) M x = M ( 2 S a 2 μ ) M x = M 2 S a(λ 1 x + λ 2 x) = M ( Sa (λ 1 x) + S a (λ 2 x) ) 2 = M 2 (λ 1 + λ 2 )S a (x) = M 2 2 μ M = μs a(x) fo all x A, o that S a i C-linea. To how the continuity of S a we ue the tategy of [9] (ee alo [14]). Aume that f a i continuou at a point x 0 A and thee i a point uch that S a i not continuou at thi point. Then thee exit a poitive numbe η and a equence (x n ) in A uch that

4 174 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) lim n x n = 0 and S a (x n ) >η. Set ε> 3α η. Then S a(εx n +x 0 ) S a (x 0 ) = S a (εx n ) > 3α fo all n. Since lim n f a (εx n +x 0 ) = f a (x 0 ), thee exit an intege N uch that, fo all n>n, we have Sa (εx n + x 0 ) S a (x 0 ) Sa (εx n + x 0 ) f a (εx n + x 0 ) + f a (εx n + x 0 ) f a (x 0 ) + f a (x 0 ) S a (x 0 ) < 3α, which i a contadiction. f(2 Similaly, by (2.1), thee exit a linea mapping S : A E given by S(x) := lim n x) n 2 n uch that (2.2) hold. It i outine to how that the additive mapping S atifying f(x) S(x) ε(x A) i unique; ee, e.g. [2]. Replacing x,y by 2 n x,2 n y, epectively, in (1.2), dividing the both ide of the obtained inequality by 2 n and tending n to infinity and uing the definition of S a and S, we get S a (x) = S(ax) as(x). In a imila manne, one can how that fo each a A the linea mapping a S : A E defined by a S(x) := lim af(2 n x) n 2 n i continuou on A and a S(x) = S(xa) S(x)a. Thu S i an intetwining mapping. 3. Stability via the genealized Jenen equation A genealization of the Jenen equation 2f( x+y 2 ) = f(x)+ f(y)i the equation x + ty f = f (x) + tf (y), whee f i a mapping between linea pace and,, t ae given contant value with > max{,t}. It i eay to ee that a mapping f between linea pace with f(0) = 0 atifie the genealized Jenen equation if and only if it i additive; cf. [4,15]. In thi ection, we pove the tability of intetwining mapping via the genealized Jenen equation. Theoem 3.1. Let f be an appoximately intetwining mapping coeponding to the genealized Jenen inequality λx + λty f λf (x) λtf (y) ε x,y A, λ J ={1, i}. (3.1) Fix a A and uppoe that, fo evey fixed b A, thee i a poitive numbe b uch that the eal function t f a (tb) and t a f(tb) ae bounded on the inteval [0, b ]. Then thee i a unique intetwining mapping S uch that f(x) S(x) ε (x A). (3.2) Poof. We have λx + λty f a λf a (x) λtf a (y) λax + λtay f λf (ax) λtf (ay)

5 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) λx + λty af λaf (x) λtaf (y) ( 1 + a ) ε (x,y A, λ J). (3.3) Set λ = 1 and y = 0 in (3.3) to get x f a f a (x) ( 1 + a ) ε, whence 1 f a(x) f a x a ε fo all x A. Uing induction, we deduce that ( ) n ( n f a x) ( ) m ( m ) f a x 1 n 1 k 1 + a ε (3.4) fo all intege n>m 0 and all x A. Hence the equence (( ) n f a (( )n x)) i Cauchy, and o it i convegent in the complete pace E. Thu we can define the mapping S a : A E by n ( n S a (x) := lim f a x). (3.5) n Replace x,y by ( )n x,( )n y, epectively, in (3.3) to obtain n ( ( f ) n x t( )n y a n 1 + a ε ) ( k=m ) n ( n f a x) + t ( ) n ( n ) f a y fo all x A and all n. Letting n, we deduce that S a atifie the genealized Jenen functional equation, and o it i additive. In addition, inequality (3.4) with m = 0 and (3.5) yield f a (x) S a (x) 1 + a ε fo all x A. Uing the hypothei, we have S a (ix) = is a (x). To pove the homogeneou popety of the additive mapping S a,fixb A and F in the dual A of A, and define the additive function Γ : R R by Γ(t)= F(S a (tb)). The function Γ i bounded ince Γ(t) F Sa (tb) F ( S a (tb) f a (tb) + f a (tb) ) F (( 1 + a ) ε + up { f a (tb) : t [0, b ] }). It follow fom Coollay 2.5 of [1] that Γ(t)= Γ(1)t fo all eal numbe t. Hence F(S a (tb)) = F(tS a (b)) fo all t R and F A. Theefoe S a (tb) = ts a (b). Now, fo each complex numbe λ = u + iv and each b A,wehave S a (λb) = S a (ub + ivb) = S a (ub) + S a (ivb) = us a (b) + ivs a (b) = λs a (b).

6 176 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) Uing a imila agument a in the poof of Theoem 2.1, we conclude the exitence of an intetwining mapping S atifying (3.2). Now let S be anothe linea mapping atifying f(x) S (x) ε(x A). Then S(x) S (x) j ( j ( j = S x) S x) j ( j ( j ) S x) f x j ( j ( j + f x) S x) j 2 ε (x A). The ight-hand ide tend to zeo a j, and hence S(x) = S (x) fo all x A. Remak 3.2. One may tat the poof of Theoem 3.1 by etting λ = 1 and x = 0 in (3.3). Then we get the intetwining mapping S defined by S(x) := lim n ( t ) n f(( t )n x) (x A). Moeove, S atifie f(x) S(x) ε(x A). 4. Stability via the Tif equation t T. Tif [19] poved the genealized tability fo the o-called Tif functional equation dc l 2 d 2 f x1 + +x d d + C l 1 xj1 + +x jl d d 2 f(x j ) = lf, l j=1 1 j 1 < <j l d whee C k denote! k!( k)!. It i not had to ee that a function T : X Y between linea pace with T(0) = 0 atifie Tif equation if and only if it i additive; ee [19]. In thi ection, we deal with tability of intetwining mapping coeponding to the Tif equation. We ue Tif appoach; cf. [19]. Let q = l(d 1) d l and = l d l fo poitive intege l,d with 2 l d 1. Fo the ake of convenience we ue the following notation: D λ g(x 1,...,x d ) = dcl 2 d 2 g λx1 + +λx d d + C l 1 d d 2 λg(x j ) j=1 xj1 + +x jl lλg, l 1 j 1 < <j l d whee g : A E i a mapping, λ i cala and x 1,...,x d A. Theoem 4.1. Fo each appoximately intetwining mapping f coeponding to the Tif inequality D λ f(x 1,...,x d ) ε, whee x 1,...,x d A and λ T, thee i a unique intetwining mapping S uch that f(x) S(x) qε (1 q)l C l 1 d 1 fo all x A.

7 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) Poof. Fix a A.Wehave D λ f a (x 1,...,x d ) Kε fo ome contant K, allx 1,...,x d A and all λ T. Let x A. Set λ = 1 and eplace x 1,...,x d by qx,x,...,x in (4.1) to get C l 1 d 2 f a(qx) l C l 1 d 1 f a(x) Kε. Uing induction one can pove that (4.1) q n f a ( q n x ) q m f a ( q m x ) 1 l C l 1 d 1 n 1 q j Kε (4.2) fo all nonnegative intege m<n. Hence the equence (q n f a (q n x)) n N i Cauchy and o convegent. Theefoe we can define the mapping S a : A E by 1 S a (x) := lim n q n f ( a q n x ) (x A). (4.3) Since ( D 1 S a (x 1,...,x d ) = lim n q n D 1 f a q n x 1,...,q n ) x d lim n q n Kε = 0, we conclude that S a atifie the Tif equation and o it i additive (note that (4.3) implie that S a (0) = 0). It follow fom (4.2) with m = 0 and (4.3) that f a (x) S a (x) qkε (1 q)l C l 1 d 1 j=m fo all x A. Let λ T. Put x 1 = =x d = x in (4.1) and note that dc l 2 dc l 2 d 2( fa (λx) λf a (x) ) Kε (x A). Theefoe d 2 = lcl d d q n dcd 2( l 2 ( fa λq n x ) ( λf a q n x )) q n Kε fo all x A. Since the ight-hand ide tend to zeo a n,wehave q n ( f a λq n x ) λq n ( f a q n x ) 0 a n fo all λ T and x A. Hence f a (q n λx) λf a (q n x) S a (λx) = lim n q n = lim n q n = λs a (x) fo all λ T and x A. The et i imila to Theoem 2.1. C l 1 d 2 to obtain

8 178 M.S. Molehian / J. Math. Anal. Appl. 332 (2007) Acknowledgment Thi wok wa witten whilt the autho wa viiting the Univeity of Leed duing hi abbatical leave in 2006; he would like to inceely thank Pofeo H.G. Dale fo hi wam hopitality and the membe of School of Mathematic fo thei kindne. Refeence [1] J. Aczél, J. Dhombe, Functional Equation in Seveal Vaiable, Encyclopedia Math. Appl., vol. 31, Cambidge Univ. Pe, Cambidge, [2] C. Baak, M.S. Molehian, Stability of J -homomophim, Nonlinea Anal. 63 (2005) [3] W.G. Bade, P.C. Cuti J., Pime ideal and automatic continuity poblem fo Banach algeba, J. Funct. Anal. 29 (1978) [4] D.-H. Boo, S.-Q. Oh, C.-G. Pak, J.-M. Pak, Genealized Jenen equation in Banach module ove a C -algeba and it unitay goup, Taiwanee J. Math. 7 (2003) [5] D.G. Bougin, Clae of tanfomation and bodeing tanfomation, Bull. Ame. Math. Soc. 57 (1951) [6] S. Czewik, Functional Equation and Inequalitie in Seveal Vaiable, Wold Scientific Publihing, Rive Edge, NJ, [7] H.G. Dale, Banach Algeba and Automatic Continuity, London Math. Soc. Monog. (N.S.), vol. 24, Claendon Pe, Oxfod Univeity Pe, Oxfod, [8] H.G. Dale, A.R. Villena, Continuity of deivation, intetwining map, and cocycle fom Banach algeba, J. London Math. Soc. (2) 63 (2001) [9] D.H. Hye, On the tability of the linea functional equation, Poc. Natl. Acad. Sci. USA 27 (1941) [10] D.H. Hye, G. Iac, Th.M. Raia, Stability of Functional Equation in Seveal Vaiable, Bikhäue, Bael, [11] S.-M. Jung, Hye Ulam Raia Stability of Functional Equation in Mathematical Analyi, Hadonic Pe Inc., Palm Habo, Floida, [12] M. Kuczma, An Intoduction to the Theoy of Functional Equation and Inequalitie, Pantwowe Wydawnictwo Naukowe, Waaw, [13] K.B. Lauen, Automatic continuity of genealized intetwining opeato, Dietatione Math. (Rozpawy Mat.) 189 (1981). [14] M.S. Molehian, Appoximately vanihing of topological cohomology goup, J. Math. Anal. Appl. 318 (2006) [15] M.S. Molehian, L.L. Székelyhidi, Stability of tenay homomophim via genealized Jenen equation, Reult Math. 49 (2006) [16] Th.M. Raia, On the tability of the linea mapping in Banach pace, Poc. Ame. Math. Soc. 72 (1978) [17] Th.M. Raia, On the tability of functional equation in Banach pace, J. Math. Anal. Appl. 251 (2000) [18] Th.M. Raia (Ed.), Functional Equation, Inequalitie and Application, Kluwe Academic Publihe, Dodecht, [19] T. Tif, On the tability of a functional equation deiving fom an inequality of Popoviciu fo convex function, J. Math. Anal. Appl. 272 (2002) [20] S.M. Ulam, Poblem in Moden Mathematic, cience ed., Wiley, New Yok, 1964.