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Approximately intertwining mappings
J. Math. Anal. Appl. 332 (2007) 171–178 www.elsevier.com/locate/jmaa
Approximately intertwining mappings Mohammad Sal Moslehian a,b,1 a Department of Mathematics, Ferdowsi University, PO Box 1159, Mashhad 91775, Iran b Functional Analysis Group, School of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK
Received 28 April 2006 Available online 7 November 2006 Submitted by T. Krisztin
Abstract Let A be a Banach algebra, and let E be a weak Banach A-bimodule. An approximately intertwining mapping corresponding to a functional equation E(f ) = 0 is a mapping f : A → E with f (0) = 0 such that E(f ) ε, and for each a ∈ A the mappings fa (x) = f (ax) − af (x), a f (x) = f (xa) − f (x)a,
are continuous at a point. In this paper, we show that every approximately intertwining mapping corresponding to Cauchy, generalized Jensen or Trif functional equation can be estimated by an intertwining mapping. © 2006 Elsevier Inc. All rights reserved. Keywords: Stability; Intertwining mapping; Banach algebra; Cauchy equation; Generalized Jensen equation; Trif equation
1. Introduction In 1940, S.M. Ulam proposed the following problem during his talk before a Mathematical Colloquium at the University of Wisconsin [20]: E-mail address: [email protected]. 1 The author was supported by Iran National Science Foundation (INSF) (No. 84053).
0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.10.013
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Given a group G1 , a metric group (G2 , d), and a positive number ε, does there exist δ > 0 such that, if a function f : G1 → G2 satisfies the inequality d(f (xy), f (x)f (y)) δ for all x, y ∈ G1 , then there exists a homomorphism T : G1 → G2 such that d(f (x), T (x)) ε for all x ∈ G1 ? In 1941 D.H. Hyers [9] partially solved this problem for linear mappings in the context of Banach spaces, as follows. Suppose that E1 and E2 are Banach spaces, and f : E1 → E2 satisfies the condition that there is ε > 0 such that f (x + y) − f (x) − f (y) ε for all x, y ∈ E1 . Then there is a unique additive mapping T : E1 → E2 such that f (x) − T (x) ε for all x ∈ E1 . In 1951, D.G. Bourgin [5] treated the Ulam problem for additive mappings. In 1978, Th.M. Rassias [16] extended the theorem of Hyers by considering the unbounded Cauchy difference f (x + y) − f (x) − f (y) ε(xp + yp ), where 0 p < 1 and > 0 are fixed. This result has been significant in the development of what we now call Hyers–Ulam–Rassias stability of functional equations. In the last decades, the topic of stability of functional equations was extensively studied in several ways by a number of mathematicians; see [6,10–12,17,18] and references therein. Let A be a Banach algebra, and let E be a weak Banach A-bimodule. By a weak Banach A-module we mean a Banach space E which is an A-bimodule such that for each a ∈ A, the mappings z → az and z → za defined on E are continuous. A linear mapping S : A → E is called intertwining if for each a ∈ A, the mappings x → S(ax) − aS(x) and x → S(xa) − S(x)a from A into E are continuous. These mappings were first introduced by Bade and Curtis [3] and were generalized by K.B. Laursen [13]. Evidently, every A-module homomorphism of A into E and any derivation of A into E are intertwining; cf. [7,8]. By an approximately intertwining mapping corresponding to a functional equation E(f ) = 0, we mean a mapping f : A → E with f (0) = 0 such that E(f ) ε, and for each a ∈ A the mappings fa (x) = f (ax) − af (x),
(1.1)
a f (x) = f (xa) − f (x)a,
(1.2)
are continuous at a point. In this paper, by using the direct method (cf. [9]), the stability of intertwining mappings on Banach algebras via Cauchy, generalized Jensen and Trif equations are established. Throughout the paper, A denotes a Banach algebra, and E is a weak Banach A-bimodule. 2. Stability via the Cauchy equation We start our work with establishing the stability of intertwining mappings via the Cauchy equation. Theorem 2.1. For each approximately intertwining mapping f corresponding to the Cauchy inequality f (λx + λy) − λf (x) − λf (y) ε, (2.1)
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173
where x, y ∈ A and λ ∈ T = {z ∈ C: |z| = 1}, there is a unique intertwining mapping S such that f (x) − S(x) ε (2.2) for all x ∈ A. Proof. Fix a ∈ A. We have fa (λx + λy) − λfa (x) − λfa (y) f (λax + λay) − λf (ax) − λf (ay) + af (λx + λy) − λaf (x) − λaf (y) 1 + a ε
(2.3)
for all x, y ∈ A and λ ∈ T. Putting λ = 1 and y = x in (2.3) we get fa (2x) − 2fa (x) 1 + a ε (x ∈ A).
(2.4)
One can apply (2.4) and use induction to show that n−1 fa (2n x) fa (2m x) − 2− 1 + a ε 2n m 2
(2.5)
=m
n
x) for all integers n > m 0 and all x ∈ A. It follows that the sequence ( fa (2 2n ) is Cauchy in E, and so is convergent, since E is a complete space. Set
fa (2n x) . n→∞ 2n By replacing x, y by 2n x, 2n y, respectively, in (2.3), we get −n n 2 fa 2 (λx + λy) − λ2−n fa 2n x − λ2−n fa 2n y 2−n 1 + a ε. Sa (x) := lim
(2.6)
Taking limits as n → ∞, we obtain Sa (λx + λy) = λSa (x) + λSa (y)
(2.7)
for all x, y ∈ A and all λ ∈ T. Obviously, Sa (0x) = 0 = 0Sa (x). Next, let μ ∈ C (μ = 0), and let M be a natural number greater than |μ|. By an easily geometric argument, one can conclude that there exist two numbers μ = λ1 +λ2 . By the additivity of Sa we get Sa ( 12 x) = 12 Sa (x) for all x ∈ A. λ1 , λ2 ∈ T such that 2 M Therefore M 1 μ μ μ M · 2 · x = MSa · 2 · x = Sa 2 · x Sa (μx) = Sa 2 M 2 M 2 M M M Sa (λ1 x) + Sa (λ2 x) = Sa (λ1 x + λ2 x) = 2 2 μ M M ·2· = μSa (x) = (λ1 + λ2 )Sa (x) = 2 2 M for all x ∈ A, so that Sa is C-linear. To show the continuity of Sa we use the strategy of [9] (see also [14]). Assume that fa is continuous at a point x0 ∈ A and there is a point such that Sa is not continuous at this point. Then there exist a positive number η and a sequence (xn ) in A such that
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limn→∞ xn = 0 and Sa (xn ) > η. Set ε > 3α η . Then Sa (εxn + x0 ) − Sa (x0 ) = Sa (εxn ) > 3α for all n. Since limn→∞ fa (εxn + x0 ) = fa (x0 ), there exists an integer N such that, for all n > N , we have Sa (εxn + x0 ) − Sa (x0 ) Sa (εxn + x0 ) − fa (εxn + x0 ) + fa (εxn + x0 ) − fa (x0 ) + fa (x0 ) − Sa (x0 ) < 3α, which is a contradiction. n Similarly, by (2.1), there exists a linear mapping S : A → E given by S(x) := limn→∞ f (22n x) such that (2.2) holds. It is routine to show that the additive mapping S satisfying f (x) − S(x) ε (x ∈ A) is unique; see, e.g. [2]. Replacing x, y by 2n x, 2n y, respectively, in (1.2), dividing the both sides of the obtained inequality by 2−n and tending n to infinity and using the definition of Sa and S, we get Sa (x) = S(ax) − aS(x). In a similar manner, one can show that for each a ∈ A the linear mapping a S : A → E defined n x) by a S(x) := limn→∞ a f (2 is continuous on A and a S(x) = S(xa) − S(x)a. Thus S is an 2n intertwining mapping. 2 3. Stability via the generalized Jensen equation A generalization of the Jensen equation 2f ( x+y 2 ) = f (x) + f (y) is the equation sx + ty rf = sf (x) + tf (y), r where f is a mapping between linear spaces and r, s, t are given constant values with r > max{s, t}. It is easy to see that a mapping f between linear spaces with f (0) = 0 satisfies the generalized Jensen equation if and only if it is additive; cf. [4,15]. In this section, we prove the stability of intertwining mappings via the generalized Jensen equation. Theorem 3.1. Let f be an approximately intertwining mapping corresponding to the generalized Jensen inequality rf λsx + λty − λsf (x) − λtf (y) ε x, y ∈ A, λ ∈ J = {1, i} . (3.1) r Fix a ∈ A and suppose that, for every fixed b ∈ A, there is a positive number rb such that the real functions t → fa (tb) and t → a f (tb) are bounded on the interval [0, rb ]. Then there is a unique intertwining mapping S such that f (x) − S(x) r ε (x ∈ A). (3.2) r −s Proof. We have rfa λsx + λty − λsfa (x) − λtfa (y) r λsax + λtay rf − λsf (ax) − λtf (ay) r
M.S. Moslehian / J. Math. Anal. Appl. 332 (2007) 171–178
λsx + λty − λsaf (x) − λtaf (y) + raf r 1 + a ε (x, y ∈ A, λ ∈ J).
175
(3.3)
Set λ = 1 and y = 0 in (3.3) to get rfa sx − sfa (x) 1 + a ε, r whence −1 r 1 fa (x) − r 1 + a ε fa x s s r for all x ∈ A. Using induction, we deduce that −n n −m m n−1 −k r 1 r r r r 1 + a ε fa x − fa x s s s s r s
(3.4)
k=m
for all integers n > m 0 and all x ∈ A. Hence the sequence (( rs )−n fa (( rs )n x)) is Cauchy, and so it is convergent in the complete space E. Thus we can define the mapping Sa : A → E by −n n r r Sa (x) := lim fa x . (3.5) n→∞ s s Replace x, y by ( rs )n x, ( rs )n y, respectively, in (3.3) to obtain −n r n −n n −n n r s( s ) x − t ( rs )n y r r r r r −s fa fa x +t fa y s r s s s s −n r 1 + a ε s for all x ∈ A and all n. Letting n → ∞, we deduce that Sa satisfies the generalized Jensen functional equation, and so it is additive. In addition, inequality (3.4) with m = 0 and (3.5) yield fa (x) − Sa (x) r 1 + a ε r −s for all x ∈ A. Using the hypothesis, we have Sa (ix) = iSa (x). To prove the homogeneous property of the additive mapping Sa , fix b ∈ A and F in the dual A∗ of A, and define the additive function Γ : R → R by Γ (t) = F (Sa (tb)). The function Γ is bounded since Γ (t) F Sa (tb) F Sa (tb) − fa (tb) + fa (tb)
F 1 + a ε + sup fa (tb): t ∈ [0, rb ] . It follows from Corollary 2.5 of [1] that Γ (t) = Γ (1)t for all real numbers t. Hence F (Sa (tb)) = F (tSa (b)) for all t ∈ R and F ∈ A∗ . Therefore Sa (tb) = tSa (b). Now, for each complex number λ = u + iv and each b ∈ A, we have Sa (λb) = Sa (ub + ivb) = Sa (ub) + Sa (ivb) = uSa (b) + ivSa (b) = λSa (b).
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Using a similar argument as in the proof of Theorem 2.1, we conclude the existence of an intertwining mapping S satisfying (3.2). Now let S be another linear mapping satisfying r ε (x ∈ A). Then f (x) − S (x) r−s −j j j r r S S(x) − S (x) = r x − S x s s s −j j j r r r S x −f x s s s −j j j r r r f + x − S x s s s −j r r 2 ε (x ∈ A). s r −s The right-hand side tends to zero as j → ∞, and hence S(x) = S (x) for all x ∈ A.
2
Remark 3.2. One may start the proof of Theorem 3.1 by setting λ = 1 and x = 0 in (3.3). Then we get the intertwining mapping S defined by S(x) := limn→∞ ( st )−n f (( st )n x) (x ∈ A). r ε (x ∈ A). Moreover, S satisfies f (x) − S(x) r−t 4. Stability via the Trif equation T. Trif [19] proved the generalized stability for the so-called Trif functional equation d x j1 + · · · + x jl x1 + · · · + xd l−2 l−1 dCd−2 + Cd−2 , f f (xj ) = lf d l j =1
1j1 <···
r! where Crk denotes k!(r−k)! . It is not hard to see that a function T : X → Y between linear spaces with T (0) = 0 satisfies Trif’s equation if and only if it is additive; see [19]. In this section, we deal with stability of intertwining mappings corresponding to the Trif l equation. We use Trif’s approach; cf. [19]. Let q = l(d−1) d−l and r = − d−l for positive integers l, d with 2 l d − 1. For the sake of convenience we use the following notation: d l−2 λx1 + · · · + λxd l−1 dC + C g λg(xj ) Dλ g(x1 , . . . , xd ) = d−2 d−2 d j =1 x j1 + · · · + x jl , − lλg l 1j1 <···
where g : A → E is a mapping, λ is scalar and x1 , . . . , xd ∈ A. Theorem 4.1. For each approximately intertwining mapping f corresponding to the Trif inequality Dλ f (x1 , . . . , xd ) ε, where x1 , . . . , xd ∈ A and λ ∈ T, there is a unique intertwining mapping S such that qε f (x) − S(x) l−1 (1 − q)l · Cd−1 for all x ∈ A.
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Proof. Fix a ∈ A. We have Dλ fa (x1 , . . . , xd ) Kε
(4.1)
for some constant K, all x1 , . . . , xd ∈ A and all λ ∈ T. Let x ∈ A. Set λ = 1 and replace x1 , . . . , xd by qx, rx, . . . , rx in (4.1) to get l−1 C fa (qx) − l · C l−1 fa (x) Kε. d−2 d−1 Using induction one can prove that −n n q fa q x − q −m fa q m x
1 l
n−1
l−1 · Cd−1 j =m
q −j Kε
(4.2)
for all nonnegative integers m < n. Hence the sequence (q −n fa (q n x))n∈N is Cauchy and so convergent. Therefore we can define the mapping Sa : A → E by 1 n fa q x n→∞ q n
Sa (x) := lim
(x ∈ A).
(4.3)
Since D1 Sa (x1 , . . . , xd ) = lim q −n D1 fa q n x1 , . . . , q n xd n→∞
lim q −n Kε n→∞
= 0, we conclude that Sa satisfies the Trif equation and so it is additive (note that (4.3) implies that Sa (0) = 0). It follows from (4.2) with m = 0 and (4.3) that fa (x) − Sa (x)
qKε l−1 (1 − q)l · Cd−1
for all x ∈ A. l−2 l−1 Let λ ∈ T. Put x1 = · · · = xd = x in (4.1) and note that dCd−2 = lCdl−d Cd−2 to obtain l−2 Kε (x ∈ A). dC d−2 fa (λx) − λfa (x) Therefore l−2 n q −n dCd−2 fa λq x − λfa q n x q −n Kε for all x ∈ A. Since the right-hand side tends to zero as n → ∞, we have −n n q fa λq x − λq −n fa q n x → 0 as n → ∞ for all λ ∈ T and x ∈ A. Hence fa (q n λx) λfa (q n x) = lim = λSa (x) n→∞ n→∞ qn qn
Sa (λx) = lim
for all λ ∈ T and x ∈ A. The rest is similar to Theorem 2.1.
2
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Acknowledgment This work was written whilst the author was visiting the University of Leeds during his sabbatical leave in 2006; he would like to sincerely thank Professor H.G. Dales for his warm hospitality and the members of School of Mathematics for their kindness.
References [1] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Encyclopedia Math. Appl., vol. 31, Cambridge Univ. Press, Cambridge, 1989. [2] C. Baak, M.S. Moslehian, Stability of J ∗ -homomorphisms, Nonlinear Anal. 63 (2005) 42–48. [3] W.G. Bade, P.C. Curtis Jr., Prime ideals and automatic continuity problems for Banach algebras, J. Funct. Anal. 29 (1978) 88–103. [4] D.-H. Boo, S.-Q. Oh, C.-G. Park, J.-M. Park, Generalized Jensen’s equations in Banach modules over a C ∗ -algebra and its unitary group, Taiwanese J. Math. 7 (2003) 641–655. [5] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951) 223– 237. [6] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing, River Edge, NJ, 2002. [7] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. (N.S.), vol. 24, Clarendon Press, Oxford University Press, Oxford, 2000. [8] H.G. Dales, A.R. Villena, Continuity of derivations, intertwining maps, and cocycles from Banach algebras, J. London Math. Soc. (2) 63 (2001) 215–225. [9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941) 222–224. [10] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998. [11] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2001. [12] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Panstwowe Wydawnictwo Naukowe, Warsaw, 1985. [13] K.B. Laursen, Automatic continuity of generalized intertwining operators, Dissertationes Math. (Rozprawy Mat.) 189 (1981). [14] M.S. Moslehian, Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl. 318 (2006) 758–771. [15] M.S. Moslehian, L.L. Székelyhidi, Stability of ternary homomorphisms via generalized Jensen equation, Results Math. 49 (2006) 289–300. [16] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [17] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000) 264–284. [18] Th.M. Rassias (Ed.), Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 2003. [19] T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002) 604–616. [20] S.M. Ulam, Problems in Modern Mathematics, science ed., Wiley, New York, 1964.
Approximately intertwining mappings Mohammad Sal Moslehian a,b,1 a Department of Mathematics, Ferdowsi University, PO Box 1159, Mashhad 91775, Iran b Functional Analysis Group, School of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK
Received 28 April 2006 Available online 7 November 2006 Submitted by T. Krisztin
Abstract Let A be a Banach algebra, and let E be a weak Banach A-bimodule. An approximately intertwining mapping corresponding to a functional equation E(f ) = 0 is a mapping f : A → E with f (0) = 0 such that E(f ) ε, and for each a ∈ A the mappings fa (x) = f (ax) − af (x), a f (x) = f (xa) − f (x)a,
are continuous at a point. In this paper, we show that every approximately intertwining mapping corresponding to Cauchy, generalized Jensen or Trif functional equation can be estimated by an intertwining mapping. © 2006 Elsevier Inc. All rights reserved. Keywords: Stability; Intertwining mapping; Banach algebra; Cauchy equation; Generalized Jensen equation; Trif equation
1. Introduction In 1940, S.M. Ulam proposed the following problem during his talk before a Mathematical Colloquium at the University of Wisconsin [20]: E-mail address: [email protected]. 1 The author was supported by Iran National Science Foundation (INSF) (No. 84053).
0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.10.013
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Given a group G1 , a metric group (G2 , d), and a positive number ε, does there exist δ > 0 such that, if a function f : G1 → G2 satisfies the inequality d(f (xy), f (x)f (y)) δ for all x, y ∈ G1 , then there exists a homomorphism T : G1 → G2 such that d(f (x), T (x)) ε for all x ∈ G1 ? In 1941 D.H. Hyers [9] partially solved this problem for linear mappings in the context of Banach spaces, as follows. Suppose that E1 and E2 are Banach spaces, and f : E1 → E2 satisfies the condition that there is ε > 0 such that f (x + y) − f (x) − f (y) ε for all x, y ∈ E1 . Then there is a unique additive mapping T : E1 → E2 such that f (x) − T (x) ε for all x ∈ E1 . In 1951, D.G. Bourgin [5] treated the Ulam problem for additive mappings. In 1978, Th.M. Rassias [16] extended the theorem of Hyers by considering the unbounded Cauchy difference f (x + y) − f (x) − f (y) ε(xp + yp ), where 0 p < 1 and > 0 are fixed. This result has been significant in the development of what we now call Hyers–Ulam–Rassias stability of functional equations. In the last decades, the topic of stability of functional equations was extensively studied in several ways by a number of mathematicians; see [6,10–12,17,18] and references therein. Let A be a Banach algebra, and let E be a weak Banach A-bimodule. By a weak Banach A-module we mean a Banach space E which is an A-bimodule such that for each a ∈ A, the mappings z → az and z → za defined on E are continuous. A linear mapping S : A → E is called intertwining if for each a ∈ A, the mappings x → S(ax) − aS(x) and x → S(xa) − S(x)a from A into E are continuous. These mappings were first introduced by Bade and Curtis [3] and were generalized by K.B. Laursen [13]. Evidently, every A-module homomorphism of A into E and any derivation of A into E are intertwining; cf. [7,8]. By an approximately intertwining mapping corresponding to a functional equation E(f ) = 0, we mean a mapping f : A → E with f (0) = 0 such that E(f ) ε, and for each a ∈ A the mappings fa (x) = f (ax) − af (x),
(1.1)
a f (x) = f (xa) − f (x)a,
(1.2)
are continuous at a point. In this paper, by using the direct method (cf. [9]), the stability of intertwining mappings on Banach algebras via Cauchy, generalized Jensen and Trif equations are established. Throughout the paper, A denotes a Banach algebra, and E is a weak Banach A-bimodule. 2. Stability via the Cauchy equation We start our work with establishing the stability of intertwining mappings via the Cauchy equation. Theorem 2.1. For each approximately intertwining mapping f corresponding to the Cauchy inequality f (λx + λy) − λf (x) − λf (y) ε, (2.1)
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173
where x, y ∈ A and λ ∈ T = {z ∈ C: |z| = 1}, there is a unique intertwining mapping S such that f (x) − S(x) ε (2.2) for all x ∈ A. Proof. Fix a ∈ A. We have fa (λx + λy) − λfa (x) − λfa (y) f (λax + λay) − λf (ax) − λf (ay) + af (λx + λy) − λaf (x) − λaf (y) 1 + a ε
(2.3)
for all x, y ∈ A and λ ∈ T. Putting λ = 1 and y = x in (2.3) we get fa (2x) − 2fa (x) 1 + a ε (x ∈ A).
(2.4)
One can apply (2.4) and use induction to show that n−1 fa (2n x) fa (2m x) − 2− 1 + a ε 2n m 2
(2.5)
=m
n
x) for all integers n > m 0 and all x ∈ A. It follows that the sequence ( fa (2 2n ) is Cauchy in E, and so is convergent, since E is a complete space. Set
fa (2n x) . n→∞ 2n By replacing x, y by 2n x, 2n y, respectively, in (2.3), we get −n n 2 fa 2 (λx + λy) − λ2−n fa 2n x − λ2−n fa 2n y 2−n 1 + a ε. Sa (x) := lim
(2.6)
Taking limits as n → ∞, we obtain Sa (λx + λy) = λSa (x) + λSa (y)
(2.7)
for all x, y ∈ A and all λ ∈ T. Obviously, Sa (0x) = 0 = 0Sa (x). Next, let μ ∈ C (μ = 0), and let M be a natural number greater than |μ|. By an easily geometric argument, one can conclude that there exist two numbers μ = λ1 +λ2 . By the additivity of Sa we get Sa ( 12 x) = 12 Sa (x) for all x ∈ A. λ1 , λ2 ∈ T such that 2 M Therefore M 1 μ μ μ M · 2 · x = MSa · 2 · x = Sa 2 · x Sa (μx) = Sa 2 M 2 M 2 M M M Sa (λ1 x) + Sa (λ2 x) = Sa (λ1 x + λ2 x) = 2 2 μ M M ·2· = μSa (x) = (λ1 + λ2 )Sa (x) = 2 2 M for all x ∈ A, so that Sa is C-linear. To show the continuity of Sa we use the strategy of [9] (see also [14]). Assume that fa is continuous at a point x0 ∈ A and there is a point such that Sa is not continuous at this point. Then there exist a positive number η and a sequence (xn ) in A such that
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limn→∞ xn = 0 and Sa (xn ) > η. Set ε > 3α η . Then Sa (εxn + x0 ) − Sa (x0 ) = Sa (εxn ) > 3α for all n. Since limn→∞ fa (εxn + x0 ) = fa (x0 ), there exists an integer N such that, for all n > N , we have Sa (εxn + x0 ) − Sa (x0 ) Sa (εxn + x0 ) − fa (εxn + x0 ) + fa (εxn + x0 ) − fa (x0 ) + fa (x0 ) − Sa (x0 ) < 3α, which is a contradiction. n Similarly, by (2.1), there exists a linear mapping S : A → E given by S(x) := limn→∞ f (22n x) such that (2.2) holds. It is routine to show that the additive mapping S satisfying f (x) − S(x) ε (x ∈ A) is unique; see, e.g. [2]. Replacing x, y by 2n x, 2n y, respectively, in (1.2), dividing the both sides of the obtained inequality by 2−n and tending n to infinity and using the definition of Sa and S, we get Sa (x) = S(ax) − aS(x). In a similar manner, one can show that for each a ∈ A the linear mapping a S : A → E defined n x) by a S(x) := limn→∞ a f (2 is continuous on A and a S(x) = S(xa) − S(x)a. Thus S is an 2n intertwining mapping. 2 3. Stability via the generalized Jensen equation A generalization of the Jensen equation 2f ( x+y 2 ) = f (x) + f (y) is the equation sx + ty rf = sf (x) + tf (y), r where f is a mapping between linear spaces and r, s, t are given constant values with r > max{s, t}. It is easy to see that a mapping f between linear spaces with f (0) = 0 satisfies the generalized Jensen equation if and only if it is additive; cf. [4,15]. In this section, we prove the stability of intertwining mappings via the generalized Jensen equation. Theorem 3.1. Let f be an approximately intertwining mapping corresponding to the generalized Jensen inequality rf λsx + λty − λsf (x) − λtf (y) ε x, y ∈ A, λ ∈ J = {1, i} . (3.1) r Fix a ∈ A and suppose that, for every fixed b ∈ A, there is a positive number rb such that the real functions t → fa (tb) and t → a f (tb) are bounded on the interval [0, rb ]. Then there is a unique intertwining mapping S such that f (x) − S(x) r ε (x ∈ A). (3.2) r −s Proof. We have rfa λsx + λty − λsfa (x) − λtfa (y) r λsax + λtay rf − λsf (ax) − λtf (ay) r
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λsx + λty − λsaf (x) − λtaf (y) + raf r 1 + a ε (x, y ∈ A, λ ∈ J).
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(3.3)
Set λ = 1 and y = 0 in (3.3) to get rfa sx − sfa (x) 1 + a ε, r whence −1 r 1 fa (x) − r 1 + a ε fa x s s r for all x ∈ A. Using induction, we deduce that −n n −m m n−1 −k r 1 r r r r 1 + a ε fa x − fa x s s s s r s
(3.4)
k=m
for all integers n > m 0 and all x ∈ A. Hence the sequence (( rs )−n fa (( rs )n x)) is Cauchy, and so it is convergent in the complete space E. Thus we can define the mapping Sa : A → E by −n n r r Sa (x) := lim fa x . (3.5) n→∞ s s Replace x, y by ( rs )n x, ( rs )n y, respectively, in (3.3) to obtain −n r n −n n −n n r s( s ) x − t ( rs )n y r r r r r −s fa fa x +t fa y s r s s s s −n r 1 + a ε s for all x ∈ A and all n. Letting n → ∞, we deduce that Sa satisfies the generalized Jensen functional equation, and so it is additive. In addition, inequality (3.4) with m = 0 and (3.5) yield fa (x) − Sa (x) r 1 + a ε r −s for all x ∈ A. Using the hypothesis, we have Sa (ix) = iSa (x). To prove the homogeneous property of the additive mapping Sa , fix b ∈ A and F in the dual A∗ of A, and define the additive function Γ : R → R by Γ (t) = F (Sa (tb)). The function Γ is bounded since Γ (t) F Sa (tb) F Sa (tb) − fa (tb) + fa (tb)
F 1 + a ε + sup fa (tb): t ∈ [0, rb ] . It follows from Corollary 2.5 of [1] that Γ (t) = Γ (1)t for all real numbers t. Hence F (Sa (tb)) = F (tSa (b)) for all t ∈ R and F ∈ A∗ . Therefore Sa (tb) = tSa (b). Now, for each complex number λ = u + iv and each b ∈ A, we have Sa (λb) = Sa (ub + ivb) = Sa (ub) + Sa (ivb) = uSa (b) + ivSa (b) = λSa (b).
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Using a similar argument as in the proof of Theorem 2.1, we conclude the existence of an intertwining mapping S satisfying (3.2). Now let S be another linear mapping satisfying r ε (x ∈ A). Then f (x) − S (x) r−s −j j j r r S S(x) − S (x) = r x − S x s s s −j j j r r r S x −f x s s s −j j j r r r f + x − S x s s s −j r r 2 ε (x ∈ A). s r −s The right-hand side tends to zero as j → ∞, and hence S(x) = S (x) for all x ∈ A.
2
Remark 3.2. One may start the proof of Theorem 3.1 by setting λ = 1 and x = 0 in (3.3). Then we get the intertwining mapping S defined by S(x) := limn→∞ ( st )−n f (( st )n x) (x ∈ A). r ε (x ∈ A). Moreover, S satisfies f (x) − S(x) r−t 4. Stability via the Trif equation T. Trif [19] proved the generalized stability for the so-called Trif functional equation d x j1 + · · · + x jl x1 + · · · + xd l−2 l−1 dCd−2 + Cd−2 , f f (xj ) = lf d l j =1
1j1 <···
r! where Crk denotes k!(r−k)! . It is not hard to see that a function T : X → Y between linear spaces with T (0) = 0 satisfies Trif’s equation if and only if it is additive; see [19]. In this section, we deal with stability of intertwining mappings corresponding to the Trif l equation. We use Trif’s approach; cf. [19]. Let q = l(d−1) d−l and r = − d−l for positive integers l, d with 2 l d − 1. For the sake of convenience we use the following notation: d l−2 λx1 + · · · + λxd l−1 dC + C g λg(xj ) Dλ g(x1 , . . . , xd ) = d−2 d−2 d j =1 x j1 + · · · + x jl , − lλg l 1j1 <···
where g : A → E is a mapping, λ is scalar and x1 , . . . , xd ∈ A. Theorem 4.1. For each approximately intertwining mapping f corresponding to the Trif inequality Dλ f (x1 , . . . , xd ) ε, where x1 , . . . , xd ∈ A and λ ∈ T, there is a unique intertwining mapping S such that qε f (x) − S(x) l−1 (1 − q)l · Cd−1 for all x ∈ A.
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Proof. Fix a ∈ A. We have Dλ fa (x1 , . . . , xd ) Kε
(4.1)
for some constant K, all x1 , . . . , xd ∈ A and all λ ∈ T. Let x ∈ A. Set λ = 1 and replace x1 , . . . , xd by qx, rx, . . . , rx in (4.1) to get l−1 C fa (qx) − l · C l−1 fa (x) Kε. d−2 d−1 Using induction one can prove that −n n q fa q x − q −m fa q m x
1 l
n−1
l−1 · Cd−1 j =m
q −j Kε
(4.2)
for all nonnegative integers m < n. Hence the sequence (q −n fa (q n x))n∈N is Cauchy and so convergent. Therefore we can define the mapping Sa : A → E by 1 n fa q x n→∞ q n
Sa (x) := lim
(x ∈ A).
(4.3)
Since D1 Sa (x1 , . . . , xd ) = lim q −n D1 fa q n x1 , . . . , q n xd n→∞
lim q −n Kε n→∞
= 0, we conclude that Sa satisfies the Trif equation and so it is additive (note that (4.3) implies that Sa (0) = 0). It follows from (4.2) with m = 0 and (4.3) that fa (x) − Sa (x)
qKε l−1 (1 − q)l · Cd−1
for all x ∈ A. l−2 l−1 Let λ ∈ T. Put x1 = · · · = xd = x in (4.1) and note that dCd−2 = lCdl−d Cd−2 to obtain l−2 Kε (x ∈ A). dC d−2 fa (λx) − λfa (x) Therefore l−2 n q −n dCd−2 fa λq x − λfa q n x q −n Kε for all x ∈ A. Since the right-hand side tends to zero as n → ∞, we have −n n q fa λq x − λq −n fa q n x → 0 as n → ∞ for all λ ∈ T and x ∈ A. Hence fa (q n λx) λfa (q n x) = lim = λSa (x) n→∞ n→∞ qn qn
Sa (λx) = lim
for all λ ∈ T and x ∈ A. The rest is similar to Theorem 2.1.
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Acknowledgment This work was written whilst the author was visiting the University of Leeds during his sabbatical leave in 2006; he would like to sincerely thank Professor H.G. Dales for his warm hospitality and the members of School of Mathematics for their kindness.
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