Methods In Nonlinear Analysis

This book contains the basic theories and methods with many interesting problems from PDE, ODE, differential geometry and mathematical physics as applications, and covers the necessary preparations to almost all important aspects in contemporary studies. There are five chapters: Linearizations, Fixed Point Theorems, Degree theory and applications, Minimizations, and Topological and Variational Methods. Each chapter is a very nice combination of abstract analysis, classical analysis and applied analysis. Chapter 1 emphasizes on the applications of the Implicit Function Theorem, including the continuation method, bifurcation theory, perturbation technique, gluing method and the transversality. Chapter 2 contains fixed point theorem obtained by compactness and convexity. All theorems are based on Ky Fan's inequality. Besides the basic theory and standard applications of the degree theory, the following topics are studied in Chapter 3: Positive solutions for semilinear elliptic BVP, Multiple solutions problems, Bifurcation at infinity etc. Chapters 4 and 5 consist of an overall view of modern calculus of variations: Direct Method (constraint problem, Legendre transformation, quasi-convexity and Morrey Theorem, Young measure, relaxing method, BV space, Hardy space and compensation compactness, concentration compactness and best constants, and the segmentation in the image processing), Infinite dimensional Morse theory, Minimax Principles and the Conley theory on metric spaces.