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.

The present book contains an outline of the modern theory of iterative functional equations. The expression functional equations is here understood in a narrow sense (equations of finite kind; see Kuczma [20]). It does not include equations in which infinitesimal operations are performed on the unknown functions. So, e.g., differential equations with transformed argument do not fall under this notion.

Nowadays, mainly owing to various activities of Professor J. Aczél, functional equations have grown to be a large, independent branch of mathematics, with its own methods, rich in results and abounding in applications. In such a large area further subdivisions are indispensable. The main line of division runs between equations in several variables in which at least one unknown function depends on fewer variables than the number of independent variables actually occurring in the equation (see Aczél [2], Kuczma [12]) and equations in a single variable, which can be written using one independent variable only.

Functional equations containing several variables are dealt with in another Encyclopedia volume written by J. Aczél and J. Dhombres [1]. The reader interested in the history of functional equations can consult Dhombres [4], [2] and also Aczél [3], Aczél–Dhombres [1].

‘Iterative functional equations’ is just another name for functional equations in a single variable (such equations are also referred to as equations of rank 1). Thus the subject matter of this book is approximately the same as that of Kuczma [26].

Whenever one has to deal with problems involving rapidly increasing nonlinearities (e.g. of exponential type), Orlicz spaces are more appropriate than Lebesgue spaces. Since Orlicz spaces are ideal spaces, many statements of this section are just reformulations of the general results of Chapter 2, and therefore are cited mostly without proofs. However, in contrast to Lebesgue spaces, several new features occur in Orlicz spaces. For instance, the superposition operator may act from one Orlicz space into another and be bounded but not continuous, or continuous but not bounded.

It turns out again that F is weakly continuous between Orlicz spaces if and only if f is affine in u. Further, the Lipschitz and Darbo conditions for F coincide, as in Lebesgue spaces, and one gets the same “degeneracy” phenomenon if the second space is “essentially smaller” than the first one.

As in Lebesgue spaces, one can give conditions for differentiability and asymptotic linearity which are both necessary and sufficient. Again, f reduces to an affine function if F is differentiate from a “large” into a “small” space. On the other hand, the class of analytic superposition operators in Orlicz spaces is reasonably large; in contrast to Lebesgue spaces, a degeneration to polynomials occurs only under an additional restriction on the first space.

Orlicz spaces

As before, let Ω be an arbitrary set, M some σ-algebra of subsets of Ω, and µ, a σ-finite and countably additive measure on M; together with µ we shall sometimes consider an equivalent normalized countably additive measure λ (see Section 1.1).