Topological Analysis : From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions

This monograph aims to give a self-contained introduction into the whole field of topological analysis: Requiring essentially only basic knowledge of elementary calculus and linear algebra, it provides all required background from topology, analysis, linear and nonlinear functional analysis, and multivalued maps, containing even basic topics like separation axioms, inverse and implicit function theorems, the Hahn-Banach theorem, Banach manifolds, or the most important concepts of continuity of multivalued maps. Thus, it can be used as additional material in basic courses on such topics. The main intention, however, is to provide also additional information on some fine points which are usually not discussed in such introductory courses.

The selection of the topics is mainly motivated by the requirements for degree theory which is presented in various variants, starting from the elementary Brouwer degree (in Euclidean spaces and on manifolds) with several of its famous classical consequences, up to a general degree theory for function triples which applies for a large class of problems in a natural manner. Although it has been known to specialists that, in principle, such a general degree theory must exist, this is the first monograph in which the corresponding theory is developed in detail.

"The book invites the reader to go on a journey from the basics of topology, through the fields of multivalued maps and Fredholm operators, up to constructions of topological degrees (both those which have been known for decades as well as more recent ones) which are presented in great detail. […] The book gives a very detailed study of different topological concepts that play important roles in a modern approach to nonlinear analysis problems, with special emphasis put on the study of problems involving set-valued maps and operators. It can surely be recommended to researchers and students interested in homotopy invariants and their applications in the theory of differential equations/inclusions." Zentralblatt für Mathematik