Page 43 - Textos de Matemática Vol. 44

ON APPROXIMATE GENERALIZED DERIVATIONS
AJDA FOSˇNER AND MOHAMMAD SAL MOSLEHIAN
Dedicated to Professor Nata´lia Bebiano on the occasion of her 60th birthday
Abstract. We study approximate generalized derivations on a Banach alge- bra A to a Banach A-bimodule. In particular, we use the notion of module extension to reduce the problem of stability of generalized derivations to the problem of stability of ring homomorphisms.
1. Introduction
We say that a functional equation E is stable if any function f approximately satisfying the equation E is near to an exact solution of E. A classical question in the theory of functional equations is: Under what conditions a functional equation E is stabel? This problem was formulated by Ulam in 1940 for group homomorphisms (see [25]). One year later, Ulam’s problem was affirmatively solved by Hyers [13] for the Cauchy functional equation f (a + b) = f (a) + f (b). This gave rise to the stability theory of functional equations. Later, Aoki [2] and Th.M. Rassias [23] considered mappings f from a normed space into a Banach space such that the norm of the Cauchy difference is bounded by the expression ε(∥a∥p + ∥b∥p) for all a, b, some ε ≥ 0 and 0 ≤ p < 1. The terminology Hyers– Ulam–Rassias stability indeed originated from Th.M. Rassias’ paper [23].
In the last few decades, various approaches to the problem have been in- troduced by several authors. Furthermore, it is surprising that in some cases an approximate mapping is actually a true mapping. In such cases we call the equation E superstable. For the history and various aspects of this theory we refer the reader to monographs [8, 14, 17].
Approximate ring homomorphisms were first investigated by Bourgin [6]. Later, the stability of ring homomorphisms was studied by a number of math- ematicians (see, for example, [3, 5, 15, 16, 22, 24] and references therein).
As we are aware, the stability of derivations was first investigated by Jun and Park [18]. During the past few years, approximate derivations were studied by a
2010 Mathematics Subject Classification. 39B82, 39B52, 46H25
Key words and phrases. Stability of functional equation, Banach algebra, Banach bimodule, derivation, generalized derivation, homomorphism.
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