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 Hk+α[a,b],(k,α)Z+×[0,1].Westudyacting,continuity,Lipschitzcontinuity,andFreˊchetdifferentiabilityconditions.ForH^{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] andand k \in \mathbb N, \alpha = 1 therespectiveconditionsarebothnecessaryandsufficient.Forthe 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