Title: 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]. We study acting, continuity, Lipschitz...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.Read More