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On a bi-quadratic functional equation and its stability. (English) Zbl 1076.39027

The authors consider the system of functional equations on vector spaces: \[ \begin{aligned} f(x+y,z)+f(x-y,z)&= 2f(x,z)+2f(y,z),\\ f(x,y+z)+f(x,y-z)&= 2f(x,y)+2f(x,z) \end{aligned}\tag{1} \] and the functional equations \[ \begin{split} f(x+y,z+w)+f(x+y,z-w)+f(x-y,z+w)+f(x-y,z-w)\\ =4[f(x,z)+f(x,w)+f(y,z)+f(y,w)]\end{split} \tag{2} \] and \[ g(x+2y)+g(x-2y)+6g(x)=4[g(x+y)+g(x-y)+6g(y)]. \tag{3} \] The equivalence of (1) and (2) and their general solution is obtained; some relations between (2) and (3) are shown and the generalized Hyers-Ulam stability of (1) and (2) is proved.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
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References:

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