Journal of Linear and Topological Algebra Vol. 06, No. 04, 07, 69-76 A fixed point method for proving the stability of ring (α, β, γ)-derivations in -Banach algebras M. Eshaghi Gordji a, S. Abbaszadeh b, a Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan 3595-363, Iran. b Department of Mathematics, Payame Noor University, P.O. BOX 9395-4697, Tehran, Iran. Received 3 October 07; Revised 4 November 07; Accepted 0 December 07. Communicated by Ghasem Soleimani Rad Abstract. In this paper, we first present the new concept of -normed algebra. We investigate the structure of this algebra and give some examples. Then we apply a fixed point theorem to prove the stability and hyperstability of (α, β, γ)-derivations in -Banach algebras. c 07 IAUCTB. All rights reserved. Keywords: Fixed point theorem, -normed algebras, (α, β, γ)-derivations, hyperstability. 00 AMS Subject Classification: 39B5, 39B7, 39B8, 47H0, 3N5.. Introduction and preliminaries One of the essential questions in the theory of functional equations giving the notion of stability is when is it true that the solution of an equation differing slightly from a given one, must be close to the solution of the given equation? Such a problem was formulated by Ulam [] in 940 and solved in the next year for the Cauchy functional equation by Hyers []. Since Hyers, many authors have studied the stability theory (now named the Hyers-Ulam stability) for functional equations (see e.g. [3 7]). S. Gähler [8, 9] in 963-964, introduced the concept of linear -normed spaces: Corresponding author. E-mail address: meshaghi@semnan.ac.ir (M. Eshaghi Gordji); abbaszadeh@pnu.ac.ir (S. Abbaszadeh). Print ISSN: 5-00 c 07 IAUCTB. All rights reserved. Online ISSN: 345-5934 http://jlta.iauctb.ac.ir
70 M. Eshaghi Gordji and S. Abbaszadeh / J. Linear. Topological. Algebra. 06(04) (07) 69-76. Definition. Let X be a real linear space with dim X > and let, : X R be a mapping. Then (X,, ) is called a linear -normed space if () x, z = 0, if and only if x, z are linearly dependent, () x, z = z, x, (3) αx, z = α x, z, for any α R, (4) x + y, z x, z + y, z, for all x, y, z X. Sometimes the condition (4) is called the triangle inequality. In the recent years, -normed spaces have been considered by several authors (cf., e.g., [0, ]). The concept of -normed algebra was introduced by Srivastava et al. [, 3]. They also gave some examples satisfying their definition and showed that there exist - normed algebras (with or without unity) which are not normable and a -Banach algebra need not be a -Banach space. Definition. (Srivastava et al. []) Let E be subalgebra of an algebra B with dim E >,, be a -norm in B and a, a B be linearly independent, non-invertible and be such that for all x, y E, xy, a i x, a i y, a i, i =,. Then E is called a -normed algebra with respect to a, a. The concept of -normed spaces was developed by Park [4]. Now, we are going to develop the concept of -normed algebra with the help of Park s definitions. Lemma.3 Let (X,, ) be a linear -normed algebra. If x X and x, y = 0 for all y X, then x = 0. For a linear -normed algebra (X,, ), the functions x x, y are continuous functions of X into R for each fixed y X as follows. Remark Let (X,, ) be a linear -normed algebra. Then, by the conditions () and (4), we have x, z y, z x y, z () for all x, y, z X. Hence the functions x x, y are continuous functions of X into R for each fixed y X. Lemma.4 For a convergent sequence {x n } in a linear -normed algebra X, for all y X. lim x n, y = lim x n, y Definition.5 A sequence {x n } in a linear -normed algebra X is called a Cauchy sequence if there are two points y, z X such that y and z are linearly independent and lim x n x m, y = 0, m, lim x n x m, z = 0. m, Definition.6 Let (X,, ) be a linear -normed algebra. A sequence {x n } is said to be convergent in X, if there exists x X such that lim x n x, y = 0
M. Eshaghi Gordji and S. Abbaszadeh / J. Linear. Topological. Algebra. 06(04) (07) 69-76. 7 for all y X. In this case, x is called the limit of the sequence {x n } and we denote it by lim x n = x. Every convergent sequence in -normed algebra is Cauchy. If each Cauchy sequence be convergent, then the -normed algebra is called -Banach algebra. Example.7 (Srivastava et al. []). Let X be a finite dimensional (dimx = n ) algebra over the field K, where K is the field of real or complex numbers and let {e,..., e n } be a basis for X. Let a, b be two symbols (not in X) and define B := {x + αa + βb : x X, α, β K}, with the agreement that x + αa + βb = 0, if and only if x = 0, α = β = 0, for x X, α, β K. For y i = x i + α i a + β i b B, i =,, α K, define y + y = (x + x ) + (α + α )a + (β + β )b, αy = αx + (αα )a + (αβ )b, y y = x x + α α a + β β b. Then B is an algebra over K and if X has unit e, then ẽ = e + a + b is the unit of B. For x = n i= α ie i + s a + s b, y = n i= β ie i + t a + t b B, define x, y by ( n ) ( n x, y = α i + s + s β i + t + t ) n α i βi + s t + s t. i= i= i= Then, defines a -norm in B. On X, define by x = x, a for x X. Note that is an norm on X. Let be an norm on X so that (X, ) is an normed algebra. Then X becomes finite dimensional, both norms and on X become equivalent and hence there exist k, k > 0 such that for every x X, x k x and x k x. For x, y X, we have xy, a = x.y k x.y k x y k k x k y = k k x, a y, a and so for a = k k a, we have xy, a x, a y, a for all x, y X. Similarly, for suitably chosen k 3 > 0, we have for a = k 3 b, xy, a x, a y, a for all x, y X, and (X,, ) becomes a -normed algebra over K with respect to a, a. Now, we show that (X,, ) is a -Banach algebra with respect to a, a. Let {x n } be a sequence in X so that lim n,m x n x m, a i = 0, i =,. Then lim n,m x n x m = 0. By finite dimensionality of X, (X, ) is a Banach space and hence there is an x X such that lim x n x = 0 or equivalently lim x n x, a = 0. For x X, if we define x = x, a, then (X, ) also becomes an normed space. Therefore, the norms and become equivalent on X and lim x n x = 0 or equivalently lim x n x, a = 0. Hence, (X,, ) is a -Banach algebra with respect to a, a In 99, J. Baker [5] used the Banach fixed point theorem for prove the Hyers-Ulam stability. The method was generalized in [6]. We recall this fundamental result as follows. Theorem.8 (Banach contraction principle) Let (X, m) be a complete generalized metric space and consider a mapping T : X X as a strictly contractive mapping, that is m(t x, T y) Lm(x, y)
7 M. Eshaghi Gordji and S. Abbaszadeh / J. Linear. Topological. Algebra. 06(04) (07) 69-76. for all x, y X and for some (Lipschitz constant) 0 < L <. Then T has one and only one fixed point x = T (x ); x is globally attractive, that is, lim T n x = x for any starting point x X; One has the following estimation inequalities for all x X and n 0 m(t n x, x ) L n m(x, x ), m(t n x, x ) L Ln m(t n x, T n+ x), m(x, x ) Lm(x, T x). Theorem.9 (The Alternative of Fixed Point [7]) Suppose that we are given a complete generalized metric space (X, m) and a strictly contractive mapping T : X X with Lipschitz constant L. Then, for each given element x X, either m(t n x, T n+ x) = + for all nonnegative integers n or there exists a positive integer n 0 such that m(t n x, T n+ x) < + for all n n 0. If the second alternative holds, then The sequence (T n x) is convergent to a fixed point y of T ; y is the unique fixed point of T in the set Y = {y X, m(t n 0x, y) < + }; m(y, y ) Lm(y, T y), y Y. Fixed point theorems have already been applied in the theory of the stability of functional equations by several authors (see for instance [8, 8 30]). Let A be a normed algebra and B be an A-bimodule normed algebra. An additive mapping f : A B is called a ring (α, β, γ)-derivation if there exist α, β, γ C such that for all x, y A. αf(xy) = βf(x)y + γxf(y) (). The main results Hereafter, let A be a Banach algebra and B be an A-bimodule unital -Banach algebra with unit and with dim(b) >. Let j {, }, α, β, γ C and let φ, ϕ : A 3 [0, ) be mappings such that for all x, y, z A. φ( nj x, nj y, z) lim nj = 0, (3) ϕ( nj x, y, z) lim nj = 0 (4) Theorem. Suppose that f : A B is a mapping satisfying f(x + y) f(x) f(y), z φ(x, y, z), (5) αf(xy) βf(x)y γxf(y), z ϕ(x, y, z) (6) for all x, y, z A. If there exists 0 < L = L(j) < such that ( x φ(x, y, z) L j φ j, y ) j, z (7)
M. Eshaghi Gordji and S. Abbaszadeh / J. Linear. Topological. Algebra. 06(04) (07) 69-76. 73 for all x, y, z A, then there exists a unique ring (α, β, γ)-derivation d : A B such that for all x, z A. f(x) d(x), z L j φ(x, x, z) (8) ( L) Proof. Let E = {e : A B e(0) = 0}. For x, z A, define m : E E [0, ] by { } m(e, e ) = inf K R + : e (x) e (x), z Kφ(x, x, z). It is easy to see that (E, m) is a complete generalized metric space. Let us consider the linear mapping T : E E, T e(x) = j e( j x) for all x A. T is a strictly contractive mapping with the Lipschitz constant L. Indeed, for given e and e in E such that m(e, e ) < and any K > 0 satisfying m(e, e ) < K, we have e (x) e (x), z Kφ(x, x, z) j e ( j x) j e ( j x), z j Kφ(j x, j x, z) j e ( j x) j e ( j x), z LKφ(x, x, z) m(t e, T e ) LK. Put K = m(e, e ) + n for positive integers n. Then m(t e, T e ) L(m(e, e ) + n ). Letting n gives m(t e, T e ) Lm(e, e ) for all e, e E. Putting y := x in (5), we get f(x) f(x), z φ(x, x, z) (9) and so f(x) f(x), z φ(x, x, z) for all x, z A. Moreover, replacing x in (9) by x implies the appropriate inequality for j = ( x ) ( x f(x) f, z φ, x ), z L φ(x, x, z) for all x, z A. Therefore m(f, T f) L j. By Theorem.9, there exists a mapping d : A B which is the fixed point of T and satisfies f( nj x) d(x) = lim nj, (0)
74 M. Eshaghi Gordji and S. Abbaszadeh / J. Linear. Topological. Algebra. 06(04) (07) 69-76. since lim m(t n f, d) = 0. The mapping d is the unique fixed point of T in the set M = {e E : m(f, e) < }. Using Theorem.9 we get which yields m(f, d) m(f, T f) L f(x) d(x), z L j φ(x, x, z) ( L) for all x, z A. By Lemma.4 and inequality (5), we also have d(x + y) d(x) d(y), z = lim = lim nj f( nj (x + y) ) nj f(nj x) nj f(nj y), z f( nj nj x + nj y) f( nj x) f( nj y), z lim nj φ(nj x, nj y, z) = 0 for all x, y, z A. By Lemma.3, we conclude that d is additive. Let r(x, y) = αf(xy) βf(x)y γxf(y) for all x, y A. Using inequality (6), we have nj r(nj x, y), z = nj αf ( ( nj x)y ) βf( nj x)y γ nj xf(y), z nj ϕ(nj x, y, z) for all x, y, z A and n N. By Lemma.3, Lemma.4 and (4) we obtain lim for all x, y A. Applying (0), we get for all x, y A. Indeed αd(xy) = lim nj r(nj x, y) = 0 αd(xy) = βd(x)y + γxf(y) () nj αf( nj (xy) ) = lim nj αf( ( nj x)y ) βf( nj x)y + nj γxf(y) + r( nj x, y) = lim nj ( βf( nj ) x)y = lim nj + γxf(y) + r(nj x, y) nj = βd(x)y + γxf(y)
M. Eshaghi Gordji and S. Abbaszadeh / J. Linear. Topological. Algebra. 06(04) (07) 69-76. 75 for all x, y A. Using () and the additivity of d, we have for all x, y A and n N. Hence, nj βd(x)y + γxf( nj y) = βd(x) nj y + γxf( nj y) = αd ( x( nj y) ) = αd ( ( nj x)y ) = βd( nj x)y + nj γxf(y) = nj βd(x)y + nj γxf(y) for all x, y A and n N. Tending n to infinity, we see that xf(y) = x f(nj y) nj () xf(y) = xd(y) (3) for all x, y A. Combining () and (3), we conclude that d satisfies (). Next, we are going to establish the hyperstability of ring (α, β, γ)-derivations. Theorem. Assume that f : A B is a mapping satisfying f(x + y) f(x) f(y), z φ(x, y, z) αf(xy) βf(x)y γxf(y), z ϕ(x, y, z) for all x, y, z A. Let B be an A-bimodule unital -Banach algebra without order, i.e. if b B, then Ab = 0 or ba = 0 implies that b = 0. If there exists 0 < L = L(j) < such that the mapping φ has the property then f is a ring (α, β, γ)-derivation. ( x φ(x, y, z) L j φ j, y ) j, z, Proof. According to Theorem., there exists a unique ring (α, β, γ)-derivation d : A B such that f( nj x) d(x) = lim nj (4) for all x A. By applying (4) in (), we conclude that x ( f(y) d(y) ) = 0 for all x, y A. Therefore f = d. Acknowledgements The authors would like to thank from the anonymous reviewers for carefully reading of the manuscript and giving useful comments, which will help to improve the paper. The
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