On approximation of approximate solutions of Dhombres' equation

Let f:S→Xf:S→X map an abelian semigroup (S,+)(S,+) into a Banach space (X‖⋅‖)(X‖⋅‖). We deal with stability of Dhombres' equationf(x)+f(y)≠0⟹f(x+y)=f(x)+f(y). We assume that‖f(x)+f(y)‖>Φ1(x,y)⟹‖f(x+y)−f(x)−f(y)‖⩽Φ2(x,y) for all x,y∈Sx,y∈S, where Φ1,Φ2:S→R+Φ1,Φ2:S→R+ are given functions and prove that, under some additional assumptions on Φ1Φ1, Φ2Φ2, there exists a solution of Dhombres' equation a:S→Xa:S→X such that‖f(x)−a(x)‖⩽Ψ(x)for x∈S, where Ψ:S→R+Ψ:S→R+ is a function which can be explicitly computed starting from Φ1Φ1 and Φ2Φ2.