# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.