An introduction to nonlinear analysis : theory

An Introduction To Nonlinear Analysis: Theory Is An Overview Of Some Basic, Important Aspects Of Nonlinear Analysis, With An Emphasis On Those Not Included In The Classical Treatment Of The Field. Today Nonlinear Analysis Is A Very Prolific Part Of Modern Mathematical Analysis, With Fascinating Theory And Many Different Applications Ranging From Mathematical Physics And Engineering To Social Sciences And Economics. Topics Covered In This Book Include The Necessary Background Material From Topology, Measure Theory And Functional Analysis (banach Space Theory). The Text Also Deals With Multivalued Analysis And Basic Features Of Nonsmooth Analysis, Providing A Solid Background For The More Applications-oriented Material Of The Book An Introduction To Nonlinear Analysis: Applications By The Same Authors. The Book Is Self-contained And Accessible To The Newcomer, Complete With Numerous Examples, Exercises And Solutions. It Is A Valuable Tool, Not Only For Specialists In The Field Interested In Technical Details, But Also For Scientists Entering Nonlinear Analysis In Search Of Promising Directions For Research. List Of Figures -- Preface -- Acknowledgements -- 1. Elements Of Topology -- 2. Elements Of Measure Theory -- 3. Banach Spaces -- 4. Set-valued Analysis -- 5. Nonsmooth Analysis -- References -- Index. By Zdzisław Denkowski, Stanisław Migórski, Nikolas S. Papageorgiou. This book provides a self-contained and systematic introduction to the mathematical foundations of nonlinear analysis. The authors have written a splendid text. The book can be used as a textbook. It can also serve as a reference for researchers. The work is organized as follows: Chapter 1 gives a thorough treatment of topology. Theorems are precisely formulated and all proofs (here and in subsequent chapters as well) are given. Chapter 2, entitled "Elements of measure theory", contains a detailed elaboration of the subject. Chapter 3 is devoted to Banach spaces. Among others, Sobolev spaces and vector-valued functions and the Bochner integral are presented here. The last two chapters, namely Chapter 4, "Set-valued analysis", and Chapter 5, "Nonsmooth analysis", are, in the opinion of the reviewer, major parts of the monograph. Both of these chapters constitute the foundations of modern nonlinear analysis with a view towards applications in many areas. The authors give a deep insight into the theory of multifunctions and into nonsmooth analysis. Each chapter contains illustrative examples perfectly adjusted to the context. At the end of each chapter there are valuable comments, both historical and bibliographical; they provide also, without proofs, additional material. The well-chosen numerous (circa 55) exercises at the end of each chapter with complete solutons make the book especially attractive as a text both for students and teachers. To sum up, the work is an important addition to a rather meager set of text/reference books on modern nonlinear analysis. Because of contents and a lucid and transparent style of presentation it was a pleasure to study and review the book. The second volume of the work is devoted to various applications of modern nonlinear analysis Front Matter....Pages i-xv Elements of Topology....Pages 1-102 Elements of Measure Theory....Pages 103-253 Banach Spaces....Pages 255-403 SET-Valued Analysis....Pages 405-516 Nonsmooth Analysis....Pages 517-664 Back Matter....Pages 665-689 The purely metric space formulation of introductory analysis is sadly inadequate in many cases and so we have to introduce more general concepts. In Definition T.3.7.45 we introduced the notion of compact linear operator and developed some basic properties of them.