J Austral Math Soc Ser A 47 pp186--210, 1989.

Continuity Properties of the SuperPosition Operator

Jürgen Appell and Pjotr P. Zabrejko

(Received 4 March 1987; revised 18 November 1987)

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 function¾f 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 AMS Subject Classification (1985 Revision): 47H15, 46E30, 26B40, 28A20

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Authors

Jürgen Appell: Institut für Mathematik, Universität Würzburgm, Am Hubland, D-9700 Würzburg, West Germany.
Pjotr P. Zabrejko: Mech.-Mat. Fakultet, Belgosuniversitet Minsk, P1. Lenina 1, SU-220080 Minsk, USSR.

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