Stability of an alternative functional equation

AbstractLet f:S→X map an abelian semigroup (S,+) into a Banach space (X‖⋅‖). We deal with stability of the following alternative functional equationf(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)forx∈S, where Ψ:S→R+ is a function which can be explicitly computed starting from Φ1 and Φ2