TY  - JOUR
ID  - Ludew2007
LB  - Ludew2007
AU  - Ludew, Jakub Jan
TI  - On Lipschitzian operators of substitution generated by set-valued functions
ST  - On Lipschitzian operators of substitution generated by set-valued functions
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2007
VL  - 27
IS  - 1
SP  - 13
EP  - 24
DO  - http://dx.doi.org/10.7494/OpMath.2007.27.1.13
UR  - http://dx.doi.org/10.7494/OpMath.2007.27.1.13
AB  - 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.
KW  - Nemytskii operator
KW  - Hölder functions
KW  - set-valued functions
KW  - Jensen equation
SN  - 1232-9274
ER  -