@Article{Ludew2007,
author = "Ludew, Jakub Jan",
title = "On Lipschitzian operators of substitution generated by set-valued functions",
journal = "Opuscula Math.",
year = "2007",
volume = "27",
number = "1",
pages = "13--24",
doi = "http://dx.doi.org/10.7494/OpMath.2007.27.1.13",
url = "http://dx.doi.org/10.7494/OpMath.2007.27.1.13",
abstract = "We consider the Nemytskii operator, i.e., the operator of substitution, defined by \((N \phi)(x):=G(x,\phi(x))\), where \(G\) is a given multifunction. It is shown that if \(N\) maps a Hölder space \(H_{\alpha}\) into \(H_{\beta}\) and \(N\) fulfils the Lipschitz condition then \[G(x,y)=A(x,y)+B(x),\tag{1}\] where \(A(x,\cdot)\) is linear and \(A(\cdot ,y),\, B \in H_{\beta}\). Moreover, some conditions are given under which the Nemytskii operator generated by \((1)\) maps \(H_{\alpha}\) into \(H_{\beta}\) and is Lipschitzian.",
keywords = "Nemytskii operator, Hölder functions, set-valued functions, Jensen equation",
issn = "1232-9274"
}