Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces
The article deals with the set-valued superposition operator \(H(\varphi)(x) = h(x,\varphi(x))\) generated by the function \(h(\cdot,\cdot): X \times C \to \text{clb}(Z)\), where \(X\) is a set with two metrics \(\rho_1, \rho_2:\;X \times X \to {\mathbb R}\), \(C \subset Y\) is a convex cone, \(Y\) and \(Z\) are real normed spaces and \(\text{clb}(Z)\) is the set of all nonempty, bounded, closed, and convex subsets of \(Z\). It is assumed that the operator \(H\) acts between \(\text{Lip}((X,\rho_1),C)\) and \(\text{Lip}((X,\rho_2),\text{clb}(Z))\) and is uniformly bounded. Under these conditions, it is proved that there exist \(a: X \times C \to \text{clb}(Z)\) and \(b: X \to \text{clb}(Z)\) such that \[ H(\varphi)(x) = a(x,\varphi(x)) \overset{*}{+} b(x), \quad \varphi \in \text{Lip}((X,\rho_1),C), \;x \in X \] (\(A \overset{*}{+} B = \text{co}\, (A + B)\)). Also, the properties of \(a(\cdot,y)\), \(b\) are described in detail. In the case \(X = [0,1]\) and \(\rho_j(x,y) = |x - y|^{\alpha_j}\) or \(\rho_j(x,y) = \alpha_j(|x - y|)\), \(j = 1,2\), these results cover the cases when the superposition operator \(H\) acts between the usual and the generalized Hölder spaces.
generalized Hölder function metric space
scientific article; zbMATH DE number 6039974
scientific article