Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces (Q2906301)

Let \(X\), \(Y\), \(Z\) be nonempty sets and \(h:X\times Y\to Z\) a function. The mapping \(H:Y^X\to Z^X\) defined by \(H( \phi)(x):=h(x, \phi(x))\) is called a composition operator. Let \(\rho_1\) and \( \rho_2\) be metrics on \(X\), let \(Y\) and \(Z\) be normed spaces, \(C\subset Y\) a convex cone, and let \(h\) be a mapping from \(X\times C\) into the set \(cc(Z)\) of all nonempty convex compact subsets of \(Z\) which is continuous in the second variable. Assume that the composition operator \(H\) generated by \(h\) maps \(\text{Lip}((X,\rho_1),C)\) into \(\text{Lip}((X,\rho_2),cc(Z))\). In this situation, the authors prove that \(h\) is affine in the second variable provided that \(H\) is uniformly bounded.