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.


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
Full Text: DOI


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