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.


39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
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[1] Aczél, J.; Dhombres, J., Functional Equations in Several Variables (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0685.39006
[2] Chung, J. K.; Sahoo, P. K., On the general solution of a quartic functional equation, Bull. Korean Math. Soc., 40, 565-576 (2003) · Zbl 1048.39017
[3] Găvruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043
[4] Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27, 222-224 (1941) · Zbl 0061.26403
[5] Rassias, J. M., Solution of the Ulam stability problem for quartic mappings, Glasnik Matematički, 34, 243-252 (1999) · Zbl 0951.39008
[6] Rassias, Th. M., On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
[7] Ulam, S. M., A Collection of Mathematical Problems (1968), Interscience Publishers: Interscience Publishers New York, p. 63 · Zbl 0086.24101
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