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Solution of the Ulam stability problem for quartic mappings. (English) Zbl 0951.39008
The author studies the Hyers-Ulam stability of the functional equation \[ f(x+2y) + f(x-2y) + 6 f(x) = 4 \left [ f(x+y) + f(x-y) + 6 f(y) \right ]\tag{FE} \] using the so called direct method of Hyers. A function \(F\) is called a quartic mapping if it satisfies the above functional equation (FE). The author proves the following result: Let \(X\) be a normed linear space and \(Y\) be a real complete normed linear space. If \(f: X \to Y\) satisfies the inequality \[ \|f(x+2y) + f(x-2y) + 6 f(x) - 4 \left [ f(x+y) + f(x-y) + 6 f(y) \right ] \|\leq \varepsilon \tag{FI} \] for all \(x, y \in X\) with a constant \(\varepsilon \geq 0\) (independent of \(x\) and \(y\)), then there exists a unique quartic function \(F : X \to Y\) such that \(\|F(x) - f(x) \|\leq {{17} \over {180}} \varepsilon \). This result is obtained through six lemmas.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
39B52 Functional equations for functions with more general domains and/or ranges
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