Functional Analysis and Applications

The core of this book is variational methods and their applications in geometry, physics, mechanics engineering control and economics. The author set out to solve the classical and famous problems including Isoperimetric Problem, Brachistochrone Curve Problem, N-Body Problems, Geodesic Curve Problem, Minimal Surface Problem, Dirichlet Principle, Minimax Problems, Rabinowitz Minimal Period Conjecture, etc. The book contains many interesting historic backgrounds and important examples, explains profound theories in simple language, which can help readers to follow in order and advance step by step. The proofs for very difficult theorems are also clearly expressed, and all chapters and appendixes are very well-written. The book has 8 organized appendixes that are important and appropriate supplements to the main texts. Appendices 1 to 7 are related with some famous classical theorems while Appendix 8 is related with the famous Rabinowitz's minimum period conjecture.

The level of this book is between the textbook for graduate students and monograph. The prerequisites on Calculus, Classical Mechanics, Ordinary Differential Equations and Real and Functional Analysis are required. It is very useful for graduate students in mathematics, physics, mechanics and related engineering majors who want to improve their knowledge in Nonlinear Sciences.

Several Classical Examples of Variational Methods.- Introduction to Banach Spaces and Hilbert Spaces.- Generalized Function and Sobolev Spaces.- First-order and Second-order Conditions of Functional Extremum.- Ekeland’s Variational Principles and Applications.- Pontryagin Maximum Principles and Applications.- Conjugate Convex Functional theorems and Application.- Periodic Solutions of N-body problems.- Several Famous Fixed Point Theorems and Applications.- Appendix 1 Zorn’s Lemma.- Appendix 2 Some Important Theorems for Lebesgues’ Measurable Functions and Integrations.- Appendix 3 Eberlein-Shmulyan’s Theorem.- Appendix 4 Ascoli-Arzelà Theorem and Kolmogorov-Riesz-Fréchet Theorem.- Appendix 5 Eigenvalues and Eigenfunctions of Laplace’s Operator.- Appendix 6 Regularities for Weak Solutions.- Appendix 7 The Proof of the Mountain Pass Lemma using Ekeland’s Variational Principle.- Appendix 8 Minimal Period Solutions for Super-Quadratic Convex Hamiltonian Systems.

Professor Shiqing Zhang, Professor of Mathematics at Sichuan University, was born in 1966 and obtained his Ph.D in Chern Institute of Nankai University in 1991. His research interests are Nonlinear Functional Analysis, Celestial Mechanics, Differential Equations and Mathematical Physics.