On hyperstability of the biadditive functional equation

We present results on approximate solutions to the biadditive equation f(x+y,z-w)+f(x-y,z+w)=2f(x,z)-2f(y,w)on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces. Our outcomes are connected with the well known issues of Ulam stability and hyperstability.