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)


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