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Zbl 0951.39008
Rassias, John Michael
Solution of the Ulam stability problem for quartic mappings.
(English)
[J] Glas. Mat., III. Ser. 34, No.2, 243-252 (1999). ISSN 0017-095X

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.
[Prasanna Sahoo (Louisville)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B62 Systems of functional equations
39B52 Functional equations for functions with more general domains

Keywords: functional equation; quartic map; functional inequality; Hyers-Ulam stability; direct method; normed linear space

Cited in: Zbl 1237.39026

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