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On the Hyers-Ulam stability of the functional equations that have the quadratic property. (English) Zbl 0928.39013

Let X be a real normed linear space and Y be a real Banach space. The Hyers-Ulam stability of the quadratic functional equation


for f:XY on the restricted domain x+y>d for a d>0 is investigated. Furthermore, the Hyers-Ulam stability of another quadratic functional equation


with condition f(x)+f(-x)γ for a γ>0 and xX or f(x)-f(-x)γ for a γ>0 and xX is treated, first on the whole domain and then on the restricted domain x+y+z>d. The results are applied to the study of the asymptotic behaviour of equation (1) and (2).

39B72Systems of functional equations and inequalities
39B52Functional equations for functions with more general domains and/or ranges