We characterize derivations and 2-local derivations from Mn(A) into Mn(M), n ≥ 2, where A is a unital algebra over C and M is a unital A-bimodule. We show that every derivation D : Mn(A) → Mn(M), n ≥ 2, is the sum of an inner derivation and a derivation induced by a derivation from A to M. We say that A commutes with M if am = ma for every a ∈ A and m ∈ M. If A commutes with M we prove that eve...

In this paper, we establish the generalized Hyers-Ulam stability of Jordan homomorphisms and Jordan derivations between ternary algebras via the generalized Jensen equation rf( sx+ty r ) = sf(x) + tf(y).

One of the interesting questions in the theory of functional equations concerning the problem of the stability of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation? The first stability problem was raised by Ulam during his talk at the University of Wisconsin in 194...

Let N be a near ring. An additive mapping f : N → N is said to be a right generalized (resp., left generalized) derivation with associated derivation d onN if f(xy) = f(x)y + xd(y) (resp., f(xy) = d(x)y + xf(y)) for all x, y ∈ N. A mapping f : N → N is said to be a generalized derivation with associated derivation d onN iff is both a right generalized and a left generalized derivation with asso...

and Applied Analysis 3 Generalized derivations first appeared in the context of operator algebras 7 . Later, these were introduced in the framework of pure algebra 8, 9 . Definition 1.1. LetA be an algebra and let X be anA-bimodule. A linear mapping d : A → X is called i derivation if d ab d a b ad b , for all a, b ∈ A; ii generalized derivation if there exists a derivation in the usual sense δ...

In this paper, we establish the stability and superstability of J∗−derivations in J∗−algebras for the generalized Jensen–type functional equation rf( x+ y r ) + rf( x− y r ) = 2f(x). Finally, we investigate the stability of J∗−derivations by using the fixed point alternative.

Amenability is a cohomological property of Banach algebras which was introduced by Johnson in [14]. Let A be a Banach algebra, and suppose that X is a Banach A−bimodule such that the following statements hold ∥a · x∥ ≤ ∥a∥∥x∥ and ∥x · a∥ ≤ ∥a∥∥x∥ for each a ∈ A and x ∈ X. We can define the right and left actions of A on dual space X∗ of X via ⟨x, λ · a⟩ = ⟨a · x, λ⟩ ⟨x, a · λ⟩ = ⟨x · a, λ⟩, for...