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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

Citations:

Zbl 0907.39025
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References:

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