Stability of an alternative functional equation

Let f : S→X map an abelian semigroup (S,+) into a Banach space (X · ).We deal with stability of the following alternative functional equation f (x +y) +f (x)+ f (y) = 0 ⇒ f (x + y) = f (x) +f (y). We assume that f (x + y)+ f (x) +f (y) >Φ1(x, y) ⇒ f (x +y) −f (x) −f (y) Φ2(x, y) for all x,y ∈ S, where Φ1,Φ2 : S →R+ are given functions and prove that, under some additional assumptions on Φ1,Φ2, there exists a unique additive mapping a : S→X such that f (x)− a(x) Ψ(x) for x ∈ S, where Ψ : S→R+ is a function which can be explicitly computed starting from Φ1 and Φ2. © 2007 Elsevier Inc. All rights reserved