An Introduction to Topological Methods in Nonlinear Analysis

P. AMSTER

SPRINGER

Lugar: La Plata; Año: 2012 p. 240

The present book constitutes an elementary introduction to the application of topological techniques in Nonlinear Analysis. For the reader´s convenience, only boundary value problems for ordinary differential equations shall be considered, although most of the ideas exposed in the text may be generalized for partial differential equations and some other fields. The first chapter is devoted to one of the most basic topological techiques in nonlinear analyisis: the shooting method. For the scalar Dirichlet problem, it is shown that only Bolzano-like theorems are required; however, the study of more difficult problems such as systems of equations or periodic problems serve as a natural introduction to a n-dimensional generalization of these theorems, given by Brouwer´s theorem and related results. We start with a very elementary approach, based on the Green´s Theorem, which allows an interesting and easy-to-understand presentation of the topic. All the results in this chapter can be understood in the context of basic calculus, although some starred sections refer to more general topological results that shall be developed later in the book. The second chapter concerns one of the best known fixed point theorems in functional analysis, which has remarkable applications to many different fields in Mathematics: the Banach contraction principle. Some applications of this theorem to the model equation and related problems are given. In the third chapter, we introduce Shauder´s theorem, that can be regarded as a ´natural´ extension of Brouwer´s theorem to the infinite-dimensional case. Applications to the study of boundary value problems are shown; in particular, an overview of the method of upper and lower solutions is presented. In the fourth chapter we define the topological degree, firstly the finite-dimensional case (Brouwer´s degree) and then the infinite-dimensional case (Leray-Schauder´s degree). The presentation follows the analytic approach and it is elementary, so no particular knowledge of general topology is required. Although there are alternative possible presentations using for example the homology machinery or the language of differential forms, that allow a more concise definition of the degree, the analytic approach was preferred in this book since it permits a more intuitive and self-contained exposition. Finally, in the last chapter we apply some of the techniques introduced in the previous chapters to the study of different boundary problems. In particular, we consider some resonant problems, which have been the subject of investigation of many authors in the last decades.