zbMATH — the first resource for mathematics

On stability of additive mappings. (English) Zbl 0739.39013
Let \(E_ 1\), \(E_ 2\) be real normed spaces with \(E_ 2\) complete, and let \(p\), \(\varepsilon\) be real numbers with \(\varepsilon\geq 0\). When \(f: E_ 1\to E_ 2\) satisfies the inequality \(\| f(x+y)-f(x)- f(y)\|\leq\varepsilon(\| x\|^ p+\| y\|^ p)\) for all \(x,y\in E\), it was shown by T. M. Rassias [Proc. Amer. Math. Soc. 72, 299-300 (1978; Zbl 0398.47040)] that there exists a unique additive mapping \(T: E_ 1\to E_ 2\) such that \(\| f(x)- T(x)\|\leq\delta\| x\|^ p\) for all \(x\in E_ 1\), providing that \(p<1\), where \(\delta=2\varepsilon/(2-2^ p)\).
The relationship between \(f\) and \(T\) was given by the formula \(T(x)=\lim_{n\to\infty}2^{-n}f(2^ nx)\). Rassias also proved that if the mapping from \(\mathbb{R}\) to \(E_ 2\) given by \(t\to f(tx)\) is continuous for each fixed \(x\in E\), then \(T\) is linear.
In the present paper the author extends these results to the case \(p>1\), but now the additive mapping \(T\) is given by \(T(x)=\lim_{n\to\infty}2^ nf(2^{-n}x)\), and the corresponding value of \(\delta\) is \(\delta=2\varepsilon/(2^ p-2)\). The author also gives a counterexample to show that the theorem is false for the case \(p=1\), and any choice of \(\delta>0\) when \(\varepsilon>0\).
Reviewer: Hyers, Donald H.

39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI EuDML