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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.

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
Full Text: DOI
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