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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

f(x+y)+f(x-y)=2f(x)+2f(y),x,yX(1)

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

f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(z+x),x,y,zX,(2)

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).


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