Nonlinear Functional Analysis (Second Edition)

Author: Guo DaJun
Publisher:
Publish Date: 2001-08-01
Features: This book is divided into five chapters: Chapter 1 discusses the general properties of nonlinear operators, including continuity, boundedness, total continuity, differentiability, etc., and presents the implicit function theorem and the inverse function theorem. Chapter 2 establishes the theory of topological degree, not only establishing the important Brouwer degree in finite-dimensional spaces and the Leray-Schauder degree for total continuous fields in Banach spaces, but also discussing the topological degree of (coercive fields) and the generalized topological degree of A-proper mappings. Chapter 3 combines semi-order and topological degree (fixed-point index) to study the positive solutions of nonlinear operator equations, discussing the positive solutions and multiple solutions of commonly used concave and convex operators. Chapter 4 primarily proves the surjectivity of coercive semi-continuous monotone mappings and the surjectivity of coercive multivalued maximal monotone mappings. Chapter 5 discusses variational methods in nonlinear problems, including both classical extremal theory and principles such as the Minimax principle and the Mountain Pass lemma from large-scale variational theory. The book includes applications to nonlinear integral equations, ordinary differential equations, and second-order semilinear elliptic partial differential equations. This book can serve as a textbook for graduate students in mathematics departments of comprehensive universities and normal universities, as well as a selective course textbook for senior undergraduate students. It can also be referenced by university teachers and scientific and technological workers engaged in nonlinear problem research.