Let A 1 be a subalgebra of a Banach algebra A and let f : A 1 → A satisfies f (x + y) − f (x) − f (y) δ and f (x • y) − x • f (y) − f (x) • y ε, for all x, y ∈ A 1 and for some constants δ, ε 0. Then we prove that there exists a unique derivation d : A 1 → A such that f (x) − d(x) δ, x ∈ A 1 and x • (f (y) − d(y)) = 0, x, y ∈ A 1. Moreover, we also prove the Rassias type stability result for derivations.
Abstract. We present some extension of the concept of an invariant mean to a space of vector-valued mappings defined on a semigroup. Next, we apply it to the study of the stability of some functional equation.
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