An Introduction to Nonlinear Analysis: Theory

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.

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.

Banach space theory became a recognized part of mathematical analysis with the appearance of the book of (1932). Since then the theory had a very quick development and found many significant applications, because it is primarily concerned with infinite dimensional function spaces, which arise rather naturally in applied problems. Even the nonlinear theories developed later, exploit particular structures in Banach space theory, like Hilbert spaces or more generally reflexive and/or separable Banach spaces. So any venture into the realm of modern nonlinear analysis, demands knowledge of at least the basic aspects of Banach space theory. The purpose of this chapter is to survey those parts of the theory that will equip the reader with all the necessary tools, to deal with the nonlinear problems that follow. The treatment is by no means exhaustive. Only the very basic things are presented and the knowledgeable reader will undoubtely spot important omissions. The books mentioned in the references of this chapter, which are devoted exclusively on the theory of Banach spaces, provide a more detailed treatment of the subject and the interested reader can find there more details and additional results.

“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.

Nonsmooth Analysis is closely related to Set-Valued Analysis. It is no coincidence that the real explosion in set-valued analysis occurred when nonsmooth analysis appeared. Since then the two fields move in synchronization and provide each other with new tools, ideas and results. This symbiotic relationship gives to both their remarkable vitality and power of applications. So this chapter is very natural continuation of the previous one.