Let be a unital Banach algebra with norm , and let , be left Banach -modules. A quadratic mapping is called -quadratic if for all , . Let be a function such that one of the series and converges for every . Denote by the sum of the convergent series. The following theorem is proved:
Theorem. Let be a mapping such that and
for all and all . If is continuous in for each fixed , then there exists a unique -quadratic mapping such that for all .
The similar results are obtained for the other functional equations:
and for the classical quadratic functional equation.