## On a modified Hyers-Ulam sequence.(English)Zbl 0746.46038

It is shown that, given a mapping $$f: E\to F$$ between Banach spaces, which is continuous along rays through 0, and for which there is a $$\delta\geq 0$$ and a $$0\leq p<1$$ such that $\| f(x+y)-f(x)- f(y)\|\leq\delta\cdot(\| x\|^ p+\| y\|^ p)\text{ for all } x,y,$ there is a unique linear mapping $$T: E\to F$$ such that for all integers $$k\geq 2$$ one has $\| f(x)-T(x)\|\leq\delta\cdot\| x\|^ p\cdot {{k+\sum_{j=2}^{k-1}j^ k} \over {k-k^ p}} \text{ for all } x.$ In fact $$T(x)=\lim_{n\to\infty} {{f(k^ n x)}\over{k^ n}}$$. The author poses the question whether the minimum is attained for $$k=2$$.
Reviewer: A.Kriegl (Wien)

### MSC:

 46G05 Derivatives of functions in infinite-dimensional spaces 41A35 Approximation by operators (in particular, by integral operators)

### Keywords:

Gâteaux-derivative; approximation by linear operators
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### References:

 [1] Hyers, D. H., On the stability of the linear functional equation, (Proc. Natl. Acad. Sci. U.S.A., 27 (1941)), 222-224 · Zbl 0061.26403 [2] Hyers, D. H., The stability of homomorphisms and related topics, (Rassias, Th. M., Global Analysis—Analysis on Manifolds. Global Analysis—Analysis on Manifolds, Teubner-Texte zur Mathematik, Leipzig, Band 57 (1983)), 140-153 · Zbl 0517.22001 [3] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, (Proc. Amer. Math. Soc., 72 (1978)), 297-300 · Zbl 0398.47040 [4] Ulam, S. M., Problems in Modern Mathematics, ((1960), Wiley: Wiley New York), Chap. VI · Zbl 0137.24201 [5] Ulam, S. M., Sets, Numbers, and Universes. Selected Works (1974), MIT Press: MIT Press Cambridge, MA, Part III
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