Nonlinear Functional Analysis: A First Course

The main aim of these lectures is to give an introduction to the theory of the topological degree and to some variational methods used in the solution of nonlinear equations in Banach spaces. While the treatment and choice of topics have been kept sufficiently general so as to interest all students of higher mathematics, the material presented will be especially useful to students aspiring to work in applications of mathematics. The first chapter gives a brisk introduction to calculus in normed linear spaces and proves classical results like the implicit function theorem and Sard’s theorem. The second chapter develops the theory of the topological degree in finite dimensional Euclidean spaces, while the third chapter extends this study to cover the theory of the Leray – Schauder degree for maps, which are compact perturbations of the identity. Fixed point theorems and their applications are presented. The fourth chapter gives an introduction to abstract bifurcation theory. The last chapter studies some methods to find critical points of functionals defined on Banach spaces with emphasis on min-max methods. The text is punctuated throughout by several exercises which prove additional results and also indicate applications, especially to nonlinear partial differential equations.