Continuity properties of the superposition operator. (English) Zbl 0683.47045

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator \(Fx(s)=f(s,x(s))\) between spaces of measurable functions are established in terms of the generating function \(f=f(s,u)\). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function \(\tilde f\) which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that F is weakly continuous if and only if f is affine in u. Finally, some continuity results for the integral functional associated with the function f are proved.
Reviewer: J.Appell


47H99 Nonlinear operators and their properties
47J05 Equations involving nonlinear operators (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26B40 Representation and superposition of functions
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence