Nonlinear analysis is a remarkable mixture of topology, analysis and applied mathematics. Mathematicians have good reason to become acquainted with this important, rapidly developing subject. But it is a BIG subject. You can feel it: just hold Eberhard Zeidler's Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems [Z} in your hand. It's heavy, as a 900 page book must be. Yet this is no encyclopedia; the preface accurately describes the " ... very careful selection of material ... " it contains. And what you are holding is only Part I of a five-part work. So how do you get started learning nonlinear analysis? Zeidler's book has a first page, and some people are quite comfortable beginning right there. For an alternative, the bibliography in [Z], which is 42 pages long, contains exposition as well as research results: monographs that explain portions of the subject to a variety of audiences. In particular, [D} covers much of the material of Zeidler's book. What makes this book different? The answer is in three parts: this book is (i) topological (ii) goal-oriented and (iii) a model of its subject
CONTENT
I: Fixed Point Existence Theory -- 1. The Topological Point of View -- 2. Ascoli-Arzela Theory -- 3. Brouwer Fixed Point Theory -- 4. Schauder Fixed Point Theory -- 5. Equilibrium Heat Distribution -- 6. Generalized Bernstein Theory -- II: Degree and Bifurcation -- 7. Some Topological Background -- 8. Brouwer Degree -- 9. Leray-Schauder Degree -- 10. Properties of the Leray-Schauder Degree -- 11. A Separation Theorem -- 12. Compact Linear Operators -- 13. The Degree Calculation -- 14. The Krasnoselskii-Rabinowitz Bifurcation Theorem -- 15. Nonlinear Sturm-Liouville Theory -- 16. Euler Buckling -- Appendices -- A. Singular Homology -- B. Additivity and Product Properties -- References
