On the Hyers-Ulam-Rassias stability of the quadratic mappings. (English) Zbl 1066.39029

Some new aspects of the stability of the quadratic mappings are considered. One of the results refers to the stability of Hyers-Ulam-Rassias type, where the functional inequality of the form \[ \| f(x+y)+f(x-y)-2f(x)-2f(y)\| \leq \varepsilon(\| x\| ^p+\| y\| ^p) \] is dealt with.
In the main result it is shown that, for mappings taking values in the interval \([0,\infty)\), if the quotient \[ \frac{f(x+y)+f(x-y)}{2f(x)+2f(y)} \] is close to 1, then so is the quotient \(\frac{f}{Q}\), for some quadratic mapping \(Q\). It is also shown that if a mapping between two vector spaces satisfies the quadratic equation for all nonzero vectors, then it must be a quadratic one.


39B82 Stability, separation, extension, and related topics for functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.