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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 $f(x+y)+ f(x-y)=2f(x) +2f(y),\quad x,y\in X\tag{1}$ for $$f:X\to Y$$ on the restricted domain $$\| x\|+\| y\|>d$$ for a $$d>0$$ is investigated. Furthermore, the Hyers-Ulam stability of another quadratic functional equation $f(x+y+z)+ f(x)+f(y)+f(z)= f(x+y)+f(y+z)+ f(z+x),\quad x,y,z\in X,\tag{2}$ with condition $$\| f(x)+f(-x) \|\leq\gamma$$ for a $$\gamma>0$$ and $$x\in X$$ or $$\| f(x)-f(-x) \|\leq\gamma$$ for a $$\gamma>0$$ and $$x\in X$$ 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).
Reviewer: M.C.Zdun (Kraków)

##### MSC:
 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
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##### References:
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