×

On characterization of Lipschitzian operators of substitution in the class of Hölder functions. (English) Zbl 0599.46032

Let \(h: <a,b>\times R\to R\), let \(Lip^{\alpha}<a,b>\), \((\alpha \in (0,1>)\), be the Banach space of all Hölder functions \(\phi\) : \(<a,b>\to R\) and let \(N(\phi)(x)=h(x,\phi (x))\) for \(\phi \in Lip^{\alpha}<a,b>\). Suppose that N: Lip\({}^{\alpha}<a,b>\to Lip^{\beta}<a,b>\) \((\beta \in (0,1>)\). It is shown, that
1) in the case \(\alpha\geq \beta\) N is a Lipschitz map if and only if there are \(G,H\in Lip^{\beta}<a,b>\) such that \[ h(x,y)=G(x)y+H(x),\quad x\in <a,b>,\quad y\in R, \] 2) in the case \(\alpha <\beta\) N is a Lipschitz map if and only if N is constant, i.e. there is a \(H\in Lip^{\beta}<a,b>\) such that \[ h(x,y)=H(x),\quad x\in <a,b>,\quad y\in R. \]

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
47H99 Nonlinear operators and their properties
47B38 Linear operators on function spaces (general)
PDFBibTeX XMLCite