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Hyers-Ulam-Rassias stability of a quadratic functional equation. (English) Zbl 1014.39019
The stability in the sense of Hyers, Ulam and Rassias of the following functional equation $f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(z+x)\tag{1}$ is considered. This equation may be regarded as related to the classical quadratic functional equation (notice that a function $$f(x)=cx^2$$ fulfils it). The results are proved under approximately even and approximately odd conditions.
\smallskip Given three functions: $$H:({\mathbb R^+})^3\to{\mathbb R^+}$$ such that $$H(tu,tv,tw)\leq t^pH(u,v,w)$$ for all $$t,u,v,w\geq 0$$ and for some $$p\in{\mathbb R}$$, and $$E,O:{\mathbb R^+}\to{\mathbb R^+}$$ such that $$E(tx)\leq t^qE(x)$$, $$O(tx)\leq t^qO(x)$$ for all $$t,x\geq 0$$ and for some $$q\in{\mathbb R}$$. Under some additional assumptions on $$p$$ and $$q$$ it is proved that if a function $$f:X\to Y$$ (where $$X,Y$$ are real normed space and a Banach space, respectively) satisfies for all $$x,y,z\in X\setminus\{0\}$$ two inequalities \begin{aligned} \bigl\|f(x+y+z)+f(x)+f(y)+f(z)-f(x+y)-f(y+z)-f(z+x)\bigr\|&\leq H\bigl(\|x\|,\|x\|,\|x\|\bigr),\\ \bigl\|f(x)-f(-x)\bigr\|&\leq E(\|x\|),\tag{2} \end{aligned} then close to $$f$$ there exists a unique function satisfying (1).
\smallskip If the inequality (2) is replaced by $$\bigl\|f(x)+f(-x)\bigr\|\leq O(\|x\|)$$, then close to $$f$$ there exists a unique additive function.
\smallskip The obtained results generalize earlier results of S.-M. Jung concerning the equation (1) [J. Math. Anal. Appl. 222, No. 1, 126-137 (1998; Zbl 0928.39013)].

MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges