Apart from the space S = S(Ω), which was studied in detail in Chapter 1, the space C = C(Ω) of all continuous functions on a complete metric space Ω plays an important role in both linear and nonlinear analysis. If Ω is a compact subset of Euclidean space without isolated points, most results on the superposition operator in C(Ω) are of course well-known “folklore”. For instance, in this case F maps C into itself if and only if f is continuous on Ω × ℝ, and F is always bounded and continuous. A somewhat more careful analysis is required, however, if Ω, has isolated points.

Before studying the superposition operator in the space C, we discuss a certain continuity property “up to small sets” of functions f = f(s, u) which is usually called the Scorza–Dragoni property. It turns out that the functions having this property are precisely the Carathéodory functions.

The main sections of this chapter are devoted to the study of the superposition operator from C into C, from C into S, and from S into C. Since C is a thick set in S, there is no essential difference between the cases F(S) ⊆ S and F(C) ⊆ S. On the other hand, the requirement that F(S) ⊆ C leads to a strong degeneracy, as will be shown at the end of Section 6.4.

Introduction

This chapter is intended to give a brief review of the basic notions in the theory of iteration which (with one exception) will be needed in what follows. It is one of our leading ideas, expressed also in the title of the book, to emphasize the fruitful and intriguing interplay between iteration and the theory of functional equations in a single variable. We feel that any attempt to divorce iteration from functional equation investigations would be an extremely, indeed totally, fruitless task. In the whole book we shall endeavour to make the reader believe this.

Some readers may find some serious omissions in this chapter but we have attempted to minimize the contents of weighty terminology and, on the other hand, we are very far from an aspiration to any kind of completeness. Basic to this chapter are the following questions: iteration sequences (splinters) and orbits, cycles, attractive fixed points and the domain of attraction. Fixed points play a distinguished role both in functional equations and in iteration theory. Some fixed-point theorems which will be useful in the sequel are also included in the present chapter. One may consider them as various generalizations of the Banach contraction principle in a complete metric space.

In this chapter we study the superposition operator in various spaces of functions which are characterized by certain smoothness properties. We begin with a necessary and sufficient acting and continuity condition for F in the space Ck of k-times continuously differentiate functions. Surprisingly, without the continuity requirement for F the generating function f need not even be continuous. Afterwards, we show that a (global) Lipschitz condition for F is “never” satisfied, while a (local) Darbo condition holds “always”. This is in sharp contrast to the situation in spaces of measurable functions dealt with in Chapters 2 – 5, and also in the space C.

In the second part we try to develop a parallel theory in the spaces of all functions from Ck whose k-th. derivatives belong to the Hölder space Hφ. In particular, we give a sufficient acting and boundedness condition.

The last part is concerned with the superposition operator in various classes of smooth (i.e. C∞) functions, including Roumieu spaces, Beurling spaces, Gevrey spaces, and their projective and inductive limits. It turns out that an acting condition for the operator F in such classes, together with suitable additional growth conditions on the derivatives of the function f, guarantees not only the boundedness and continuity, but also the compactness of F.

Symmetric spaces are ideal spaces whose norm may be defined by means of the decreasing rearrangement of measurable functions. Thus, all general results discussed in Chapter 2 carry over to such spaces, but some results may be sharpened. For instance, the main statements on the boundedness, Lipschitz continuity, or differentiability of the superposition operator between symmetric spaces can be formulated more explicitly in terms of the so-called fundamental function.

The most important examples of symmetric spaces, apart from those discussed in Chapters 3 and 4, are the Lorentz space ∧φ and the Marcinkiewicz space Mφ. These spaces play a fundamental role, for example, in interpolation theory of linear operators.

After recalling the notions and properties of symmetric spaces, in general, and Lorentz or Marcinkiewicz spaces, in particular, we formulate some elementary results on the superposition operator between such spaces. Unfortunately, the theory is here much less advanced than in, say, Lebesgue and Orlicz spaces. The results presented here are mainly combinations of special properties of symmetric spaces and general results obtained in Chapter 2.

Symmetric spaces

Let Ω be an arbitrary set, M some σ-algebra of subsets of Ω, and µ a σ-finite and count ably additive measure on M; as before, by λ we denote some equivalent normalized measure on M.

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.

First, we give conditions for the local and global boundedness of the superposition operator F between ideal spaces X and Y which are typically ensured by special properties of the “source space” X. Second, special properties, such as absolute boundedness and compactness, are treated. Afterwards, we give conditions for the continuity and uniform continuity of F which are now typically ensured by special properties of the “target space” Y. For example, F is “always” continuous if Y is regular, and “never” continuous if Y is completely irregular (see the definitions below).

Weak continuity of F between ideal spaces is also considered; here we mention the surprising fact that, loosely speaking, only linear superposition operators are weakly continuous.

Next, we give necessary and sufficient conditions under which F satisfies a Lipschitz or Darbo condition. It turns out that in many spaces these two conditions are in fact equivalent.

Finally, the last part of this chapter is concerned with differentiability conditions for the superposition operator between ideal spaces.

In this chapter we study the superposition operator Fx(s) = f(s,x(s)) in the complete metric space S of measurable functions over some measure space Ω. First, we consider some classes of functions f which generate a superposition operator F from S into S; a classical example is the class of Carathéodory functions, a more general class that of Shragin functions.

As a matter of fact, there exist functions f, called “monsters”, which generate the zero operator Fx ≡ θ, but are not measurable on Ω × ℝ, and hence are not Carathéodory functions; this disproves the old-standing Nemytskij conjecture. On the other hand, we show that a function which generates a continuous superposition operator (in measure) is “almost” a Carathéodory function.

We give a necessary and sufficient condition for the function f to generate a bounded superposition operator F in the space S. In particular, this conditions holds always if f is a Carathéodory function. On the other hand, we show that the superposition operator F is “never” compact in the space S, except for the trivial case when F is constant.

Finally, we consider superposition operators which are generated by functions f with special properties (e.g. monotonicity), and characterize the points of discontinuity of such operators.

The space S

Let Ω be an arbitrary set, M some σ-algebra of subsets of Ω (which will be called measurable in what follows), and µ a countably additive and σ-finite measure on M.