On Lipschitzian operators of substitution generated by set-valued functions

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.