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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.

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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