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Continuity and differentiability properties of the Nemitskii operator in Hölder spaces. (English) Zbl 0637.47035
Given a bounded domain \(\Omega\) in \({\mathbb{R}}^ n\), the author gives (sufficient) conditions for a real function f on \({\bar \Omega}\times {\mathbb{R}}\) under which the nonlinear superposition operator \(Fu(x)=f(x,u(x))\) acts in the Hölder space \(C^{\alpha}({\bar \Omega},{\mathbb{R}})\) and is continuous, locally Lipschitz, or continuously differentiable. In the last section, these results are generalized to vector valued functions, including an application to nonlinear elliptic boundary value problems.
Reviewer: J.Appell

MSC:
47J05 Equations involving nonlinear operators (general)
46G05 Derivatives of functions in infinite-dimensional spaces
47H99 Nonlinear operators and their properties
46E40 Spaces of vector- and operator-valued functions
35J65 Nonlinear boundary value problems for linear elliptic equations
26A16 Lipschitz (Hölder) classes
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References:
[1] Elworthy, Proc. Symp. Pure Mathematics 18 (1970)
[2] Valent, Rend. Sent. Mat. Univ. Padova 74 pp 63– (1985)
[3] Berger, Nonlinearity and functional analysis (1977) · Zbl 0368.47001
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