Ebook: Methods of Nonlinear Analysis : Applications to Differential Equations

Of Methods Presented in This Book.In this book, fundamental methods of nonlinear analysis are introduced, discussed and illustrated in straightforward examples. Each method considered is motivated and explained in its general form, but presented in an abstract framework as comprehensively as possible. A large number of methods are applied to boundary value problems for both ordinary and partial differential equations. In this edition we have made minor revisions, added new material and organized the content slightly differently. In particular, we included evolutionary equations and differential equations on manifolds. The appl. Read more... Methods of Nonlinear Analysis; Applications to Differential Equations; Contents; Preface; Chapter 1 Preliminaries; 1.1 Elements of Linear Algebra; 1.2 Normed Linear Spaces; Chapter 2 Properties of Linear and Nonlinear Operators; 2.1 Linear Operators; 2.2 Compact Linear Operators; 2.3 Contraction Principle; Chapter 3 Abstract Integral and Differential Calculus; 3.1 Integration of Vector Functions; 3.2 Differential Calculus in Normed Linear Spaces; 3.2A Newton Method; Chapter 4 Local Properties of Differentiable Mappings; 4.1 Inverse Function Theorem; 4.2 Implicit Function Theorem. 4.3 Local Structure of Differentiable Maps, Bifurcations4.3A Differentiable Manifolds, Tangent Spaces and Vector Fields; 4.3B Differential Forms; 4.3C Integration on Manifolds; Chapter 5 Topological Methods; 5.1 Brouwer Fixed Point Theorem; 5.1A Contractible Sets; 5.2 Schauder Fixed Point Theorem; 5.2A Fixed Point Theorems for Noncompact Operators; 5.3 Classical Solutions of PDEs, Functional Setting; 5.4 Classical Solutions, Applications of Fixed Point Theorems; 5.5 Weak Solutions of PDEs, Functional Setting; 5.6 Weak Solutions of PDEs, Applications of Fixed Point Theorems. 5.7 Brouwer Topological Degree5.7A Brouwer Topological Degree on Manifolds; 5.8 Leray-Schauder Topological Degree; 5.8A Global Bifurcation Theorem; 5.8B Topological Degree for Generalized Monotone Operators; 5.9 Weak Solutions of PDEs, Applications of Degree Theory; 5.9A Weak Solutions of PDEs, Application of the Degree of Generalized Monotone Mappings; Chapter 6 Monotonicity Methods; 6.1 Theory of Monotone Operators; 6.1A Browder and Leray-Lions Theorem; 6.2 Weak Solutions of PDEs, Application of Monotone Operators; 6.2A Weak Solutions of PDEs, Application of Leray-Lions Theorem. 6.3 Supersolutions, Subsolutions, Monotone Iterations6.3A Minorant Principle and Krein-Rutman Theorem; 6.3B Supersolutions, Subsolutions and Topological Degree; 6.4 Maximum Principle for ODEs; 6.5 Maximum Principle for PDEs; Chapter 7 Variational Methods; 7.1 Local Extrema; 7.2 Global Extrema; 7.2A Supersolutions, Subsolutions and Global Extrema; 7.2B Ritz Method; 7.3 Weak Solutions of PDEs, Applications of Global Minimizers; 7.4 Mountain Pass Theorem; 7.4A Pseudogradient Vector Fields in Banach Spaces; 7.5 Weak Solutions of PDEs, Applications of Mountain Pass Theorem. 7.6 Saddle Point Theorem7.6A Linking Theorem; 7.7 Weak Solutions of PDEs, Applications of Saddle Point Theorem; 7.7A Weak Solutions of PDEs, Applications of General Saddle Point Theorem; 7.8 Relative Extrema and Lagrange Multipliers; 7.8A Lusternik-Schnirelmann Method; 7.8B Krasnoselski Potential Bifurcation Theorem; Chapter 8 Some Applications to Partial Differential Equations; 8.1 Linear Evolution Equations and Semigroups; 8.2 Semilinear Evolution Equations; 8.3 Linearization of Quasilinear PDEs and Fixed Point Theorems; 8.4 Equations on Riemann Manifolds