
The author examines the Hyers-Ulam-Rassias stability [see \textit{D. H. Hyers, G. Isac} and \textit{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.
Banach space, Applied Mathematics, Stability, separation, extension, and related topics for functional equations, Functional equations for functions with more general domains and/or ranges, normed space, Hyers-Ulam-Rassias stability, Hyers–Ulam–Rassias stability, quadratic functional equation, Analysis
Banach space, Applied Mathematics, Stability, separation, extension, and related topics for functional equations, Functional equations for functions with more general domains and/or ranges, normed space, Hyers-Ulam-Rassias stability, Hyers–Ulam–Rassias stability, quadratic functional equation, Analysis
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