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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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