This book emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a nice balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis.

This book offers a systematic presentation of up-to-date material scattered throughout the literature from the methodology point of view. It reviews 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 from mechanics, physics, engineering and geometry.

Geometrical (in particular, topological) methods in nonlinear analysis were originally invented by Banach, Birkhoff, Kellogg, Schauder, Leray, and others in existence proofs. Since about the fifties, these methods turned out to be essentially the sole approach to a variety of new problems: the investigation of iteration processes and other procedures in numerical analysis, in bifur cation problems and branching of solutions, estimates on the number of solutions and criteria for the existence of nonzero solutions, the analysis of the structure of the solution set, etc. These methods have been widely applied to the theory of forced vibrations and auto-oscillations, to various problems in the theory of elasticity and fluid. mechanics, to control theory, theoretical physics, and various parts of mathematics. At present, nonlinear analysis along with its geometrical, topological, analytical, variational, and other methods is developing tremendously thanks to research work in many countries. Totally new ideas have been advanced, difficult problems have been solved, and new applications have been indicated. To enumerate the publications of the last few years one would need dozens of pages. On the other hand, many problems of non linear analysis are still far from a solution (problems arising from the internal development of mathematics and, in particular, problems arising in the process of interpreting new problems in the natural sciences). We hope that the English edition of our book will contribute to the further propagation of the ideas of nonlinear analysis.

This book emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a nice balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis.

A comprehensive book focusing on the Force Analogy Method, a novel method for nonlinear dynamic analysis and simulation This book focusses on the Force Analogy Method, a novel method for nonlinear dynamic analysis and simulation. A review of the current nonlinear analysis method for earthquake engineering will be summarized and explained. Additionally, how the force analogy method can be used in nonlinear static analysis will be discussed through several nonlinear static examples. The emphasis of this book is to extend and develop the force analogy method to performing dynamic analysis on structures under earthquake excitations, where the force analogy method is incorporated in the flexural element, axial element, shearing element and so on will be exhibited. Moreover, the geometric nonlinearity into nonlinear dynamic analysis algorithm based on the force analogy method is included. The application of the force analogy method in seismic design for buildings and structural control area is discussed and combined with practical engineering.

The techniques that can be used to solve non-linear problems are far different than those that are used to solve linear problems. Many courses in analysis and applied mathematics attack linear cases simply because they are easier to solve and do not require a large theoretical background in order to approach them. Professor Schechter's 2005 book is devoted to non-linear methods using the least background material possible and the simplest linear techniques. An understanding of the tools for solving non-linear problems is developed whilst demonstrating their application to problems in one dimension and then leading to higher dimensions. The reader is guided using simple exposition and proof, assuming a minimal set of pre-requisites. For completion, a set of appendices covering essential basics in functional analysis and metric spaces is included, making this ideal as an accompanying text on an upper-undergraduate or graduate course, or even for self-study.

Nonlinear analysis, formerly a subsidiary of linear analysis, has advanced as an individual discipline, with its own methods and applications. Moreover, students can now approach this highly active field without the preliminaries of linear analysis. As this text demonstrates, the concepts of nonlinear analysis are simple, their proofs direct, and their applications clear. No prerequisites are necessary beyond the elementary theory of Hilbert spaces; indeed, many of the most interesting results lie in Euclidean spaces. In order to remain at an introductory level, this volume refrains from delving into technical difficulties and sophisticated results not in current use. Applications are explained as soon as possible, and theoretical aspects are geared toward practical use. Topics range from very smooth functions to nonsmooth ones, from convex variational problems to nonconvex ones, and from economics to mechanics. Background notes, comments, bibliography, and indexes supplement the text.