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Zeitschrift für Analysis und ihre Anwendungen


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Volume 31, Issue 1, 2012, pp. 93–124
DOI: 10.4171/ZAA/1450

Published online: 2011-12-27

Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters I

Martin Väth[1]

(1) Czech Academy of Sciences, Prague, Czech Republic

For a given single- or multivalued function $f$ and "atoms'' $S_i$, let $S_f(\lambda,x)$ be the set of all measurable selections of the function $s\mapsto f(\lambda,s,x(s))$ which are constant on each $S_i$. Continuity and differentiability of such operators are studied in spaces of measurable functions containing ideal, Orlicz and $L_p$ spaces with new results for the parameter-dependent case even for single-valued superposition operators without atoms. A motivation is to apply the results for variant of such maps $S_f$ in %%% here alteration:Sobolev spaces in the second part of this article [Z. Anal. Anwend. 31 (2011) (to appear)].

Keywords: Superposition operator, Nemytskij operator, multivalued map, atom, parameter dependence, continuity, uniform differentiability, generalized ideal space, Orlicz space, Lebesgue-Bochner space

Väth Martin: Continuity and Differentiability of Multivalued Superposition Operators with Atoms and Parameters I. Z. Anal. Anwend. 31 (2012), 93-124. doi: 10.4171/ZAA/1450