EMIS ELibM Electronic Journals Zeitschrift für Analysis und ihre Anwendungen
Vol. 18, No. 2, pp. 205 - 229 (1999)

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On the Autonomous Nemytskij Operator in Hölder Spaces

M. Goebel and F. Sachweh

M. Goebel: Martin-Luther-Universität Halle-Wittenberg, FB Math. und Inform., D-06099 Halle (Saale); goebel@mathematik.uni-halle.de
F. Sachweh: Bispinckpl. 1-3, D-48683 Ahaus; f_sachweh@lauder.de

Abstract: The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces $H^{k+\alpha}[a,b]$, $(k,\alpha) \in \Bbb{Z}_+ \times (0,1]$. We study acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions. For $k = 0$, $\alpha \in (0,1]$ and $k \in \Bbb{N}$, $\alpha = 1$ the respective conditions are both necessary and sufficient. For $k \in \Bbb{N}$, $\alpha \in (0,1)$ only the acting condition is both necessary and sufficient; the other investigated properties are characterized by necessary and sufficient conditions different from each other.

Keywords: Hölder spaces, Lipschitz spaces, Nemytskij operator, superposition operator, acting conditions, boundedness, continuity and Lipschitz continuity, Fréchet differentiability

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Electronic fulltext finalized on: 31 Jul 2001. This page was last modified: 9 Nov 2001.

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