## On the Hyers-Ulam-Rassias stability of a quadratic functional equation.(English)Zbl 0926.39013

The author examines the Hyers-Ulam-Rassias stability [see D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhäuser, Boston (1998; Zbl 0907.39025)] of the quadratic functional equation $f(x-y-z)+f(x)+f(y)+f(z) = f(x-y)+f(y+z)+f(z-x)$ and proves that if a mapping $$f$$ from a normed space $$X$$ into a Banach space $$Y$$ satisfies the inequality $|f(x-y-z)+f(x)+f(y)+f(z) - f(x-y)-f(y+z)-f(z-x)| \leq \epsilon$ for all $$x,y,z \in X$$ with $$\|x\|+\|y\|+\|z\|\geq d$$, then there exists a unique quadratic function $$Q : X \to Y$$ such that $\|f(x)-Q(x)\|\leq 39 \epsilon$ for all $$x \in X$$. Here $$d$$ and $$\epsilon$$ are positive real numbers.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges

Zbl 0907.39025
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### References:

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