Some results on stability of extended derivations

Abstract

In this paper we prove the generalized Hyers-Ulam-Rassias stability of extended derivations on unital Banach algebras associated to a generalized Jensen equation.

A note on envelopes of homotopies

“…Amyari et al [2] showed the generalized Hyers -Ulam -Rassias stability of extended derivations on unital Banach algebras. In the following theorems, A denotes a unital Banach algebra with unit 1 and K denotes an integer greater than 1.…”

“…In detail, we consider the specific homotopies O; I and, what is more, prove that the additivity of homotopies is equivalent to the additivity of the corresponding envelopes of homotopies (Proposition 2.10). In Section 3, we introduce the stabilities of a functional equation and the stabilities of derivations and especially describe the interesting theorems of Gâvruta [18] and Amyari et al [2]. Next, using the Gâvruta's theorem, we investigate the stability for the envelopes of homotopies (Theorem 3.5).…”

K-stable subsets of extended derivations on Banach algebras

“…Let 1 G be a group and 2 G a metric group with the metric ) , ( ⋅ ⋅ d . Given a ε > 0, does there exist δ >0 such that if a function 2 1 :…”

“…Amyari, F. Rahbarniab and Gh. Sadeghi [2] show that the generalized Hyers-Ulam-Rassias stability of extended derivations on unital Banach algebras associated to a generalized Jensen equation as the following form: [2]) Let Α be a Banach algebra with the unit element e and let K be an integer greater than 1. Suppose f : Α Α is a mapping with f (0) = 0 for which there exist real numbers β 0 and …”

“…We say that E is K-stable subset of extended derivations to associated to a generalized Jensen inequality (2) if for every mapping f : Α Α with the inequality (2), there exists a unique extended derivation D :…”