Inclusion and uniform continuity properties of the Nemytskij-operator (Q1084656)

Let (T,\(\mu)\) be a measure space, X a topological space, Z a Banach space, and \(f: T\times X\to Z\) and \(g_ 1,g_ 2:T\times X\to [0,\infty]\) three given functions such that, whenever \(x: T\to X\) is measurable and \(g_ 1(\cdot,x(\cdot))\in L^ 1(\mu)\), \(g_ 2(\cdot,x(\cdot))\in L^{\infty}(\mu)\), then \(f(\cdot,x(\cdot))\in L^ q(\mu)\) (1\(\leq q\leq \infty)\). Under these assumptions, the author shows that the function f satisfies certain growth estimates which generalize the classical Krasnosel'skij condition [see e.g. \textit{J. Appell} and \textit{P. P. Zabrejko}, Nonlin. Anal., Theory Methods Appl. 7, 695-706 (1983; Zbl 0522.47056)]. Moreover, continuity properties of the superposition operator generated by the function f are studied in detail.