Nikolas S. Papageorgiou

SET-Valued Analysis

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

Elements of Measure Theory

One of the most important tools which one combines with nonlinear analysis in the context of applied problems is “Measure Theory”. The subject started at the end of the nineteenth century with the works of Jordan, Borel, W.H. Young and Lebesgue. By that time it was clear to mathematicians that the Riemann integral had to be replaced by a new type of integral which will be more general (i.e. more functions will be integrable) and more flexible (in particular produce better convergence results). The construction of Lebesgue turned out to be the most fruitful and launched “Measure Theory” as a separate discipline in mathematical analysis. In contrast to the Riemann integral, the Lebesgue approach starts by partitioning the range of the function into small pieces, determining regions in the domain on which the function is approximately constant (these regions can be quite complicated) measuring the size of these regions, summing and passing to the limit as the size of the pieces in the range goes to zero. A prerequisite for this method to work, is the ability to measure the size of very general and complicated sets in the domain. This was the starting point of “Measure Theory”, which developed rigorously during the twentieth century. The aim of this chapter is to survey some parts of this theory which are needed in the understanding of certain aspects of nonlinear analysis. Of course our treatment is incomplete. Afterall this is impossible within a chapter of a book. We only present those items that are necessary for the discussion of future topics and special emphasis is placed on the interplay between Measure Theory and Topology.

Elements of Topology

In this chapter, we review the basic facts of general topology that will be used in this book. A detailed study of set-theoretic topology would be out of place here. Nevertheless, topology, continuity and convergence are integral parts in any study of nonlinear analysis having a serious claim to completeness. Topological notions play a central role in constructing the objects studied in Nonlinear Analysis and in carrying out proofs. Moreover, there are intimate connections between many basic fields of Nonlinear Analysis and topology. Measure theory, integration and differentiation, Banach space theory, degree theory, nonsmooth analysis, fixed point theory and critical point theory, to mention only a few, depend extensively on topological concepts and results. In asking the reader to go through the basic theory of point-set topology, with its high level of abstraction, we ask for a considerable preliminary effort. The reward will be a much more thorough presentation of contemporary Nonlinear Analysis.

Evolution Inclusions of the Subdifferential Type

In this section we study a second class of evolution inclusions, which are driven by time-dependent subdifferential operators. The t-dependence of the convex function φ(t, ·) is an important feature of our inclusions and general enough to incorporate cases where domφ(t, ·) ⋂ domφ(s, ·) = O for t ≠ s. So the abstract framework of this chapter is suitable for the analysis of problems with time-varying obstacles. The structure of this chapter basically parallels that of chapter I.

Evolution Inclusions Involving Monotone Coercive Operators

In this chapter we study the first major class of evolution inclusions. They are problems defined in the framework of an evolution triple, which involve coercive nonlinear operators of monotone type. We investigate the existence of solutions for both “convex” and “nonconvex” problems and conduct a detailed analysis of the properties of the solution set. Note that the solution of the problem is not unique, due to the presence of multivalued terms in the equation.

Special Topics in Mathematical Economics and Optimization

In this chapter we have gathered certain specialized applications of multivalued analysis. So in section 1, we deal with the resource allocation problem from mathematical economics. This is a dynamic constrained optimization problem with infinite planning horizon. We present a relaxation method based on the so-called generalized plans. In contrast to the control-theoretic relaxation that we studied in chapter IV, in this model the utility has to be considered as part of the plan before the completion is performed. Otherwise, the utility of a generalized plan is not well-defined. We prove the existence of optimal generalized plans and characterize them. In section 2, we study the central question of including information as a parameter in the dynamic economic models and investigating how the various economic variables depend on this parameter. This requires the introduction of appropriate topologies in the space of information. Since information is modeled by sequences of sub-σ-fields of the original σ-field and the σ-fields can be identified with the corresponding conditional expectation operators, it is natural to look for convenient topologies in the realm of the operator topologies. Two such topologies are introduced and studied in detail. In particular, we examine how they can be used to obtain qualitative information in the context of economic models.

Special Topics in Differential and Evolution Inclusions

In chapters I and II we examined in detail two large classes of evolution inclusions. In this chapter we close the subject of multivalued differential equations, by presenting certain topics that were not yet covered and can not be found in the existing books. So, in section 1 we consider differential inclusions in R N . First we present a unified treatment of the convex and nonconvex problems, which so far have been approached using distinct methods. Using the notions of directionally continuous selectors (see section A-I.5) and the well-known multivalued Filippov regularization of a discontinuous vector field, we are able to transform a nonconvex problem to an equivalent convex one which is solved by standard methods. In the second half of section 1, we go beyond the confines of the existence theory and of the analysis of the topological structure of the solution set of a differential inclusion in R N , by introducing a “metric likelihood” map which is entirely independent of probability theory. This notion allows to characterize the extremal trajectories of a differential inclusion and to give a precise mathematical meaning to a class of estimation, prediction and filtering problems, in a context which is independent of any probability theory (deterministic problems). In section 2 we continue the study of differential inclusions in R N and investigate boundary value problems for second order differential inclusions in R N . The boundary conditions that we consider are nonlinear and include as special cases the Dirichlet (Picard), the Neumann and the periodic problems. In section 3 we investigate evolution inclusions driven by m-accretive operators. We conduct a detailed study analogous to the ones in chapters I and II. Moreover, in the process of our investigation we derive some additional results about semigroups of nonlinear contractions which complement the ones presented in section A-III.8. In section 4 we repeat the same for Volterra integral inclusions in Banach spaces. The Hausdorff measure of noncompactness plays a prominent role in this analysis. In section 5 we examine the reachable set of evolution inclusions and the solution set of differential inclusions in R N with maximal monotone operators. For the reachable set of semilinear evolution inclusions, we determine the infinitesimal generators of the reachable map. For the maximal monotone differential inclusions in R N we consider the solution set as a multifunction of the initial state and determine its differentiability properties and the differential inclusion that the derivative satisfies (variational inclusion). Finally, in section 6 we deal with differential equations with discontinuities. Since such problems need not have solutions, in order to have a reasonable existence theory we pass to a multivalued version of the problem by, roughly speaking, filling in the gaps at the discontinuity points. We consider such problems for ordinary differential equations, elliptic equations and parabolic equations. The ode’s and elliptic equations are solved using a variational approach based on the nonsmooth critical point theory for locally Lipschitz energy functionals, which is outlined in the beginning of section 5. The parabolic problems are analyzed using the method of upper and lower solutions and truncation and penalization techniques. The analytical formalism of evolution triples (see section 1.1) and the theory of operators of monotone-type (see section A-III.6) play a crucial role.