Cubic derivations on Banach algebras
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CUBIC DERIVATIONS ON BANACH ALGEBRAS Abasalt Bodaghi
Received: 3 April 2012 / Revised: 20 December 2012 / Accepted: 10 January 2013 / Published online: 12 October 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract Let A be a Banach algebra and X be a Banach A-bimodule. A mapping D: A −→ X is a cubic derivation if D is a cubic homogeneous mapping, that is, D is cubic and D(λa) = λ3 D(a) for any complex number λ and all a ∈ A, and D(ab) = D(a) · b3 + a 3 · D(b) for all a, b ∈ A. In this paper, we prove the stability of a cubic derivation with direct method. We also employ a fixed point method to establish the stability and the superstability of cubic derivations. Keywords Banach algebra · Cubic derivation · Stability · Superstability Mathematics Subject Classification (2000) 39B52 · 47B47 · 39B72 · 46H25
1 Introduction In 1940, Ulam [19] posed the following question concerning the stability of group homomorphisms: Under what condition does there exist an additive mapping near an approximately additive mapping between a group and a metric group? One year later, Hyers [9] answered the problem of Ulam under the assumption that the groups are Banach spaces. This problem for linear mappings on Banach spaces was solved by J.M. Rassias in [15]. A generalized version of the theorem of Hyers for approximately linear mappings was given by Th.M. Rassias [16]. Subsequently, the stability problems of various functional equations have been extensively investigated by a number of authors (for example, [2, 12] and [14]). In particular, one of the functional equations which has been studied frequently is the cubic functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x).
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A. Bodaghi ( ) Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran e-mail: [email protected]
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The cubic function f (x) = ax 3 is a solution of this functional equation. The stability of the functional equation (1) has been considered on different spaces by a number researchers (for instance, [11] and [17]). In 2003, C˘adariu and Radu applied a fixed point method to the investigation of the Jensen functional equation. They presented a short and a simple proof for the Cauchy functional equation and the quadratic functional equation in [4] and [3], respectively. After that, the method has been applied by many authors to establish miscellaneous functional equations (see [1, 7] and [13]). In [8], Eshaghi Gordji et al. introduced the concept of a cubic derivation which is different from that one of the current paper. In fact, they did not consider the homogeneous property of such derivations. In that paper, the authors studied the stability of cubic derivations on commutative Banach algebras. The stability and the superstability of cubic double centralizers and cubic multipliers on Banach algebras has been earlier proved in [10]. In this paper, we prove the stability of cubic derivations on Banac
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