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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)].

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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