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Local properties and approximation of operators. (Italian) Zbl 0599.39007
This paper is connected with the theory of functional equations in the meaning of J. Aczél and more exactly with their Hyers stability [cf. D. H. Hyers, Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)]. We state some results and problems concerning the local uniform approximation and the extension of an operator $f:{D}_{f}\subset ℝ\to X$ (X being a Banach space) for which the condition $\parallel f\left(x+y\right)-f\left(x\right)-f\left(y\right)\parallel <\delta$ holds only in a given subset of ${ℝ}^{2}$ for some $\delta >0$. Similar problems are posed in relation to the condition $\parallel f\left(x+y\right)+f\left(x-y\right)-2f\left(x\right)-2f\left(y\right)\parallel <\delta$.

##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges
##### Keywords:
Hyers stability; local uniform approximation; Banach space