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Multi-valued superpositions. (English) Zbl 0855.47037

Let \(\Omega\) be an arbitrary set, \(X\) a space of functions on \(\Omega\) with values in \(\mathbb{R}^m\), \(Y\) a space of functions on \(\Omega\) with values in \(\mathbb{R}^n\), and \(F\) a multi-valued function which associates to each pair \((t, u)\in \Omega\times \mathbb{R}^m\) a nonempty set \(F(t, u)\subseteq \mathbb{R}^n\). Associating then to each (single-valued) function \(t\mapsto x(t)\) in \(X\) the set of all (single-valued) selections \(t\mapsto y(t)\) of the (multi-valued) function \(t\mapsto F(t, x(t))\) in \(Y\) defines the superposition operator \(N_F\) generated by \(F\) between the spaces \(X\) and \(Y\). The purpose of this paper is to give a systematic description of this operator in terms of the generating multi-valued function \(F\) and the underlying spaces \(X\) and \(Y\).
The survey paper consists of 6 chapters. In Chapter 1 the authors collect the necessary notions and facts on multi-valued functions and their selections. Except for Michael’s selection principle, all results in this chapter are elementary, and therefore the authors state them without proofs.
As the authors are mainly interested in multi-valued functions on the Cartesian product \(\Omega\times \mathbb{R}^m\), they study multi-valued functions of two variables in Chapter 2. Particular emphasis is laid here on functions which satisfy a Carathéodory condition or Scorza Dragoni condition: the first means that, loosely speaking, \(F\) is measurable in the first and continuous in the second argument; the latter means that \(F\) is continuous “up to small sets”. The importance of such functions for differential inclusions is the same as in the single-valued case for differential equations. However, if one replaces the continuity in the second variable by a weaker semicontinuity assumption, many new features occur which are“hidden” in the single-valued theory.
In Chapter 3 the authors give a systematic account of continuity and boundedness properties of the superposition operator in the metric space \(S\) of measurable functions, in the normed space \(C\) of continuous functions, and between ideal spaces (i.e. \(L_\infty\)-Banach modules) of measurable functions. As in the classical single-valued theory, the Carathéodory condition on \(F\) guarantees both the boundedness and continuity of \(N_F\) in the space \(S\), and the continuity of \(F\) guarantees both the boundedness and continuity of \(N_F\) in the space \(C\). On the other hand, it turns out that the boundedness and continuity of \(N_F\) between two ideal spaces \(X\) and \(Y\) relies very much on properties of these spaces, rather than on properties of the generating multi-valued function \(F\).
As a matter of fact, important multi-valued functions arising in applications do not have the necessary properties for applying the results described in the third chapter. This emphasizes the need of passing from a given function \(F\) to some extension \(G\) which either takes values in a “nicer” class of sets, or generates a superposition operator \(N_G\) with “nicer” analytical properties. The most important and useful extensions, in this connection, are the strong closure \(G= \overline F\), the weak closure \(G= \vec F\), and the convexification \(G= F^\square\). These extensions, as well as the superposition operators generated by them, are studied in Chapter 4. In particular, the authors are interested in conditions under which the operations of “taking extensions” and “taking operators” commute, i.e. \(\overline N_F= N_{\overline F}\), \(\vec N_F= N_{\vec F}\), or \(N^\square_F= N_{F^\square}\).
Chapter 5 is concerned with fixed point theorems and integral inclusions. First, the authors recall some fixed point principles for multi-valued operators. Here the basic results are the fixed point theorems of Nadler, Kakutani, and Bohnenblust-Karlin, which may be considered as multi-valued analogues of the classical fixed point theorems of Banach, Brouwer, and Schauder, respectively. These fixed point principles are then applied to operators of the form \(KN_F\), where \(K\) is a linear (single-valued) integral operator, and \(N_F\) is the nonlinear (multi-valued) superposition operator described above. In this way, the authors obtain existence (and sometimes also uniqueness) theorems for nonlinear integral inclusions of Hammerstein type. In the last section they describe a general method which allows them to reduce the study of (multi-valued) integral inclusions for vector functions to the study of (single-valued) integral equations for scalar functions.
The last Chapter 6 is devoted to selected applications. First, an application to an elliptic system with multi-valued right-hand side is sketched; in the “variational formulation” this also gives existence of critical points for nonsmooth energy functionals. Afterwards, the authors describe forced periodic oscillations in nonlinear control systems with “noise”. Finally, we discuss a mathematical model for the theory of heat regulation by thermostats which leads to Hammerstein integral inclusions for two-dimensional vector functions.

MSC:

47H04 Set-valued operators
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
45P05 Integral operators
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
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