Ulam's stability of a generalization of the Fréchet functional equation

We prove some general stability and hyperstability results for a generalization of the well known Fréchet equation, stemming from the characterization of the inner product spaces due to Jordan and von Neumann. The main result yields several stability outcomes for various other well known functional equations. Thus we obtain in particular some new inequalities characterizing the inner product spaces. As the main tool we use a fixed point theorem for some function spaces.