# On the Autonomous Nemytskij Operator in Hölder Spaces

### Manfred Goebel

Universität Halle-Wittenberg, Germany### F. Sachweh

Ahaus, Germany

## Abstract

The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces $H^{k+\alpha}[a,b], (k, \alpha) \in \mathbb Z_+ \times [0, 1]. We study acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions. For$ k = 0$,$ \alpha \in [0, 1] $and$ k \in \mathbb N, \alpha = 1 $the respective conditions are both necessary and sufficient. For$ k \in \mathbb 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.

## Cite this article

Manfred Goebel, F. Sachweh, On the Autonomous Nemytskij Operator in Hölder Spaces. Z. Anal. Anwend. 18 (1999), no. 2, pp. 205–229

DOI 10.4171/ZAA/878