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

Găvruţa, P.

Nonlinear Funct. Anal. Appl. 9, No. 3, 415-428 (2004).

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.

Reviewer: Jacek Chmielinski (Kraków)

MSC:

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

Keywords:

quadratic functional equation; functional inequality; quadratic mappings