On the autonomous Nemytskij operator in Hölder spaces. (English) Zbl 0941.47053
Building on previous work by the first author [Glasg. Math. J. 33, 1-5 (1991; Zbl 0724.47041) and Monatsh. Math. 113, 107-119 (1992; Zbl 0765.47022)] and both authors [Nonlinear Anal., Theory Methods Appl. 30, No. 1, 513-519 (1997; Zbl 0894.47051)], the authors give acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions for the Nemytskij operator \(u\mapsto f\circ u\) in the Hölder space \(H^{k,\alpha}[a,b]\) in terms of the generating function \(f\). All these conditions are both necessary and sufficient in the spaces \(H^{0,\alpha}[a,b]\) \((0<\alpha\leq 1)\) and \(H^{k,1}[a,b]\) \((k\in\mathbb{N})\). For general \(k\in\mathbb{N}\) and \(0<\alpha<1\), only the acting condition is necessary and sufficient, while the other conditions are either necessary or sufficient.
Reviewer: Jürgen Appell (Würzburg)
MSC:
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 26A16 | Lipschitz (Hölder) classes |
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