Approximation of Generalized Left Derivations

Let A be an algebra over the real or complex field F. An additive mapping d : A → A is said to be a left derivation resp., derivation if the functional equation d xy xd y yd x resp., d xy xd y d x y holds for all x, y ∈ A. Furthermore, if the functional equation d λx λd x is valid for all λ ∈ F and all x ∈ A, then d is a linear left derivation resp., linear derivation . An additive mapping G : A → A is called a generalized left derivation resp., generalized derivation if there exists a left derivation resp., derivation δ : A → A such that the functional equation G xy xG y yδ x resp., G xy xG y δ x y is fulfilled for all x ∈ A. In addition, if the functional equations G λx λG x and δ λx λδ x hold for all λ ∈ F and all x ∈ A, then G is a linear generalized left derivation resp., linear generalized derivation . It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation. The study of stability problems had been formulated by Ulam 1 during a talk in 1940: “Under what condition does there exists a homomorphism near an approximate homomorphism?” In the following year 1941, Hyers 2 was answered affirmatively the question of Ulam for Banach spaces, which states that if ε > 0 and f : X → Y is a map with X, a normed space, Y, a Banach space, such that