Continuity properties of the superposition operator

Abstract

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(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 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. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 47 H 15, 46 E 30, 26 B 40, 28 A 20. © 1989, Australian Mathematical Society. All rights reserved.