Methods in Nonlinear Analysis

Nonlinear analysis has developed rapidly in the last three decades. Theories, techniques and results in many different branches of mathematics have been combined in solving nonlinear problems. This book collects and reorganizes up-to-date materials scattered throughout the literature from the methodology point of view, and presents them in a systematic way. It contains the basic theories and methods with many interesting problems in partial and ordinary differential equations, differential geometry and mathematical physics as applications, and provides the necessary preparation for almost all important aspects in contemporary studies.

There are five chapters that cover linearization, fixed-point theorems based on compactness and convexity, topological degree theory, minimization and topological variational methods. Each chapter combines abstract, classical and applied analysis. Particular topics included are bifurcation, perturbation, gluing technique, transversality, Nash–Moser technique, Ky Fan's inequality and equilibrium in game theory, set­valued mappings and differential equations with discontinuous nonlinear terms, multiple solutions in partial differential equations, direct method, quasi­convexity and relaxation, Young measure, compensation compactness method and Hardy space, concentration compactness and best constants, Ekeland variational principle, infinite-dimensional Morse theory, minimax method, index theory with group action, and Conley index theory.

All methods are illustrated by carefully chosen examples from mechanics, physics, engineering and geometry. The book aims to find a balance between theory and applications and will contribute to filling the gap between texts that either only study the abstract theory, or focus on some special equations.

• Convexity
• Hardy space
• Morse Theory
• Nonlinear Partial Differential Equations
• Nonlinear analysis
• Nonlinear functional analysis
• calculus
• compactness
• differential equation
• fixed-point theorem
• measure
• minimum
• ordinary differential equation
• partial
• partial differential equations

From the reviews:

"This book is based on the lecture notes of a course on nonlinear analysis offered by the author to graduate students at various universities during the past two decades. … This book contains very rich theoretical material, together with the presentation of interesting problems and examples from various branches of mathematics. It will be useful to both students and researchers." (Adriana Buica, Zentralblatt MATH, Vol. 1081, 2006)

"Nonlinear analysis has developed rapidly in the last three decades. … This book collects and reorganizes up-to-date materials scattered throughout the literature from the methodology point of view, and presents them in a systematic way. It contains the basic theories and methods with many interesting problems in partial and ordinary differential equations, differential geometry and mathematical physics as applications, and provides the necessary preparation for almost all important aspects in contemporary studies. … All methods are illustrated by carefully chosen examples … ." (L’Enseignement Mathematique, Vol. 51 (3-4), 2005)

"Nonlinear analysis is a quite young area in mathematical sciences, and it has grown tremendously in the last thirty years. … In addition, all methods discussed in this book are illustrated by carefully chosen examples from applied mathematics, physics, engineering and geometry. … Overall, the book presents a unified approach, and is an excellent contribution to nonlinear analysis." (Claudio H. Morales, Mathematical Reviews, Issue 2007 b)

“This well-written monograph gives an excellent introduction to the fundamental methods of nonlinear analysis and its application to ordinary and partial differential equations. … Each chapter applies the abstract techniques to basic problems for nonlinear differential equations coming from geometry, mechanics or physics, and the book ends with bibliographical notes and a substantial list of references. The book is strongly recommended both as an introductory and a reference book in this very active domain of analysis.” (Jean Mawhin, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)

• School of Mathematical Sciences Peking University, Beijing, People's Republic of China

  Kung-Ching Chang

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