Continuity and Fréchet-differentiability of Nemytskij operators in Hölder spaces
The paper is a continuation of \textit{M. Goebel} [Glasg. Math. J. 33, No. 1, 1-5 (1991; Zbl 0724.47041)] and deals with Nemytskij operators (superposition operators), \((Fy)(t)=f(t,y(t))\), which are generated by a function \(f: [a,b]\times\mathbb{R}^ n\to\mathbb{R}\). There are given sufficient conditions for \(f\), such that \(F\) is a bounded, continuous, locally Lipschitz continuous and continuously Fréchet-differentiable, respectively, map between the Hölder spaces \(H^ \alpha([a,b],\mathbb{R}^ n)\) and \(H^ \alpha[a,b]\), \(0<\alpha\leq 1\). The presented criteria require less strong assumptions than those so far known from the literature; all proofs are straightforward.
scientific article; zbMATH DE number 164939
scientific article