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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).

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
26A16 Lipschitz (Hölder) classes
Full Text: DOI
[1] Prö\(\beta\)dorf, Einige Klassen singulärer Gleichungen (1974) · doi:10.1007/978-3-0348-5827-4
[2] Nugari, Glasgow Math. J. 30 pp 59– (1988)
[3] Drábek, Comment. Math. Univ. Carotin 16 pp 37– (1975)
[4] Appell, The superposition operator in function spaces – a survey (1987)
[5] Appell, Z. Anal. Anwendungen 6 pp 193– (1987)
[6] Appell, Nonlinear superposition operators (1989)
[7] Bondarenko, Dokl. Akad. Nauk SSR 222 pp 1265– (1975)
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