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Hyers-Ulam-Rassias stability of a quadratic type functional equation. (English) Zbl 1049.39030
The authors prove that if a function $$f$$, from a real vector space $$X$$ into a real vector space $$Y$$, satisfies the function equation $\begin{split} Df(x,y,z):= a^2 f\Biggl({x+ y+ z\over a}\Biggr)+ a^2 f\Biggl({x- y+z\over a}\Biggr)+ a^2 f\Biggl({x+ y-z\over a}\Biggr)\\ +a^2 f\Biggl({-x+ y+z\over a}\Biggr)- 4f(x)- 4f(y)- 4f(z)= 0\end{split}\tag{1}$ for all $$x,y,z\in X$$ and for a real nonzero constant $$a$$, then there exist a quadratic function $$Q: X\to Y$$ and an additive function $$A: X\to Y$$ such that $f(x)= Q(x)+ A(x)+ f(0)$ for all $$x\in X$$.
Moreover, they also prove a Hyers-Ulam-Rassias stability of the equation (1) as follows: Let $$X$$ and $$Y$$ be a real normed space and a real Banach space, respectively. Assume that $$H: [0,\infty)^3\to [0,\infty)$$ is homogeneous of degree $$p,p\in (0,\infty)- \{1,2\}$$. If a function $$f: X\to Y$$ satisfies $\| Df(x,y,z)\|\leq \delta+ H(\| x\|,\| y\|,\| z\|)$ for all $$x,y,z\in X$$, then there exists a unique quadratic function $$Q: X\to Y$$ and a unique additive function $$A: X\to Y$$ such that $\| f(x)- Q(x)- A(x)- f(0)\|\leq{3\over 4}\delta+ \|(a^2- 3)f(0)\|+ {1\over| 4- 2^p|} h_1(x)+ {1\over| 2- 2^p|} h_2(x)$ for all $$x\in X$$, where we set $$\delta= 0$$ for $$p> 1$$, and we set $$\|(a^2- 3)f(0)\|= 0$$ for $$p> 2$$, and where \begin{aligned} h_1(x) &= \tfrac 12 H(\| x\|,\| x\|,0)+ \textstyle{{1\over 4}}H(\| 2x\|, 0,0),\\ h_2(x) &= \textstyle{{1\over 4}} H(\| x\|,\| x\|, 0)+\textstyle{{1\over 4}} H(\| 2x\|, 0,0).\end{aligned}

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