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Continuity and differentiability of multivalued superposition operators with atoms and parameters. I. (English) Zbl 1237.47065
The article deals with a single- or multivalued nonlinear superposition operator $$S_f(\lambda,x):\;{\mathcal M}(S,U) \to {\mathcal M}(S,V)$$, where $$U$$ and $$V$$ are quasi-pseudonormed spaces, $${\mathcal M}(S,U)$$ and $${\mathcal M}(S,V)$$ are spaces of measurable functions defined on $$S$$ and taking values in $$U$$ and $$V$$, correspondingly, $$f:\Lambda \times S \times {\mathcal M}(S,U) \to {\mathcal M}(S,V)$$ is a given single- or multivalued function, $S_f(\lambda,x) = \{\phi(s) \in f(\lambda,s,x(s)), \;\phi(\cdot) \in {\mathcal M}(S,V), \phi(\cdot) \text{ are constant on } S_i, \;i \in I\},$ and, at last, $$(S,\Sigma,\mu)$$ is a complete $$\sigma$$-finite measure space and $$S_i \in \Sigma$$ ($$i \in I$$) a fixed family of pairwise disjoint sets of positive measure (atoms). The first main result of this article gives natural conditions for the upper and lower semicontinuity of the operator $$S_f(\lambda,x)$$ at a point $$(\lambda_0,x_0)$$. The other results are concerned with the differentiability properties of the operator $$S_f(\lambda,x)$$. More precisely, the author describes conditions under which the following equations hold: $\lim_{x \to 0} \sup_{\lambda \in \Lambda} \frac{\sup \{y: y \in S_f(\lambda,x)\}}{\|x\|} = 0$ or $\lim_{x \to 0,\, \lambda \to \lambda_0} \frac{\sup \{y: y \in S_f(\lambda,x)\}}{\|x\|} = 0.$ Apart from the main results which concern quasi-pseudonormed and ideal spaces, the author also considers the corresponding results in Orlicz and Lebesgue spaces.

MSC:
 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 47H04 Set-valued operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E40 Spaces of vector- and operator-valued functions
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