On the Autonomous Nemytskij Operator in Hölder Spaces

Abstract

The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces $H^{k+\alpha }[a,b],(k,\alpha )\in {\mathbb{Z}}_{+}\times [0,1].Westudyacting,continuity,Lipschitzcontinuity,andFr\acute{e}chetdifferentiabilityconditions.For$ k = 0$,$ \alpha \in [0, 1] $and$ k \in \mathbb N, \alpha = 1 $therespectiveconditionsarebothnecessaryandsufficient.For$ 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.