# zbMATH — the first resource for mathematics

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