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Perspectives in Nonlinearity

其他書名	An Introduction to Nonlinear Analysis
出版	W. A. Benjamin, 1968
ISBN	08053066849780805306682
URL	http://books.google.com.hk/books?id=gq6mAAAAIAAJ&hl=&source=gbs_api

註釋These notes are based on lectures given at the University of Minnesota and the Courant Institute of Mathematical Sciences, New York University, in 1966 and 1967. Our aim has been to present some qualitative aspects of nonlinear analysis, which we think are important, in as simple and direct a manner as possible. Thus we have neither striven for results of the utmost generality nor complicated the text by introducing an excess number of new concepts. In this way we hope to make the ideas presented accessible to persons who enjoy mathematics and its applications but are not specialists in nonlinear analysis. To accomplish this goal in a small book we have had to sketch the ideas of a few proofs and to specialize the general theory of nonlinear analysis on finite and infinite dimensional differentiable manifolds. The interested reader will find this theory discussed in the monographs, Lectures on Nonlinear Functional Analysis by J. T. Schwartz, and Foundations of Global Nonlinear Analysis by R. S. Palais. Furthermore, our choice of material was necessarily selective, for example, iterative results such as Newton's method and Nash's implicit function theorem have been omitted. Nonetheless we believe that the material discussed here has sufficient beauty to induce the reader to further excursions into nonlinear analysis. Our text is divided into four chapters and two appendices. Chapter 1 is intended to be a partial answer to the question: What are some of the problems of nonlinear analysis and how have they been studied in previous generations? Chapter 2 introduces the concepts of the degree of a continuous mapping, and the theory of critical points of real-valued functions in finite dimensional Euclidean spaces Rn . In Chapter 3 we show how the ideas of Chapter 2 can be carried over to infinite dimensional spaces. Appendices 1 and 2 at the end of the book include some preliminary material necessary to the understanding of Chapters 2 and 1. In Chapter 4 we select a few specific nonlinear problems and indicate just how the methods of the previous chapters can be used to study these problems. The first-mentioned problem in Chapter 4, global univalence, is of great interest outside of mathematics (for example, to mathematical economists in the study of international trade and to applied mathematicians studying elastic deformations). Similarly the topics of differential equations (ordinary and partial) discussed in Chapter 4 are basic to the understanding of physical processes of nature.