By Jürgen Appell, Petr P. Zabrejko

This publication is a self-contained account of data of the idea of nonlinear superposition operators: a generalization of the thought of features. the speculation constructed here's acceptable to operators in a wide selection of functionality areas, and it's right here that the fashionable idea diverges from classical nonlinear research. the aim of this booklet is to gather the fundamental evidence in regards to the superposition operator, to offer the most principles that are valuable in learning its homes and to supply a comparability of its behaviour in numerous functionality areas. a few purposes also are thought of, for instance to regulate concept and optimization. a lot of the paintings right here has simply seemed earlier than in examine literature, which itself is catalogued intimately right here.

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To see this, consider in the case Ω = [0, 1] the sequences xn = π2 (4n + 1)χDn , yn = π2 (4n − 1)χDn (n = 1, 2, . . ), where Dn are subsets 1 , and in the case Ω = N the sequences xn = 2n+1 χ{n} , yn = of [0, 1] with λ(Dn ) = 2πn 2n 2n−1 χ{n} (n = 1, 2, . . ). Obviously, these sequences are bounded and satisfy the condition 2n lim xn − yn = 0; on the other hand, the sequence F xn − F yn is bounded away from n→∞ zero, and hence F cannot be uniformly continuous on bounded sets. 7. 9. Let X and Y be two ideal spaces, and let f be a Caratheodory function.

Vajnberg [78], [164], [165], [238-241], [341346]; in the last mentioned paper, the superpo- sition operator is called "Nemytskij operator"for the ﬁrst time. P. Zabrejko. 9) characterizes the superposition operator, or if there are other operators which are locally determined. 6) is the only locally determined operator on the space S (over a separable measure): in fact, since the space S has the cardinality of the continuum, its elements can be indexed by the ordinals from 1 to ω, the ﬁrst uncountable ordinal.

24 Chapter 2 The superposition operator in ideal spaces In this chapter we are concerned with the basic properties of the superposition operator in so-called ideal spaces which are, roughly speaking, Banach spaces of measurable functions with monotone norm. To formulate our results in a suﬃciently general framework, we must introduce a large number of auxiliary notions which will be justiﬁed by the results in concrete function spaces given in subsequent chapters; we request the reader’s indulgence until then.