On uniformly continuous Nemytskij operators generated by set-valued functions

Abstract

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H α (I, C) of all Hölder functions \({\varphi : I \to C}\) into the set H β (I, clb(Z)) of all Hölder set-valued functions \({\phi : I \to clb(Z)}\) and is uniformly continuous, then$$F(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C$$for some set-valued functions A, B such that \({A(\cdot,y),B \in H_{\beta}(I, clb(Z)), y \in C}\) and A(x, ·), \({x \in I}\) are *additive and continuous on C into clb(Z). A converse result is also investigated.