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On Fréchet-differentiability of Nemytskij operators acting in Hölder spaces. (English) Zbl 0724.47041
The author considers the Nemytskij operator \(Fy(t)=f(y(t))\), generated by some continuous real function f, in the Hölder space \(H^{\nu}[a,b]\) \((0<\nu \leq 1)\). He proves that, if f is of class \(C^ 1\) [resp. \(C^ 2]\), the operator F is continuous [resp. continuously Fréchet differentiable] on \(H^{\nu}[a,b]\). Related work is due, among others, to R. Nugari [Glasgow J. Math. 30, 59-65 (1988; Zbl 0637.47035)] and T. Valent [Springer Tracts Nat. Philos. 31 (1987; Zbl 0648.73019)]. A parallel study for the non-autonomous case \(f=f(t,y)\) will be published by the same author in Monatsh. Math. (to appear).

MSC:
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
26A16 Lipschitz (Hölder) classes
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