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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
Keywords:
Nemytskij operator; Hölder space
Full Text:
References:
 [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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