An Introduction To Nonlinear Analysis And Fixed Point Theory

This book systematically introduces the theory of nonlinear analysis, providing an overview of topics such as geometry of Banach spaces, differential calculus in Banach spaces, monotone operators, and fixed point theorems. It also discusses degree theory, nonlinear matrix equations, control theory, differential and integral equations, and inclusions. The book presents surjectivity theorems, variational inequalities, stochastic game theory and mathematical biology, along with a large number of applications of these theories in various other disciplines. Nonlinear analysis is characterised by its applications in numerous interdisciplinary fields, ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. Organised into ten chapters, the book shows the elegance of the subject and its deep-rooted concepts and techniques, which provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in diverse applied fields. It is intended for graduate and undergraduate students of mathematics and engineering who are familiar with discrete mathematical structures, differential and integral equations, operator theory, measure theory, Banach and Hilbert spaces, locally convex topological vector spaces, and linear functional analysis.

This self-contained textbook provides the basic, abstract tools used in nonlinear analysis and their applications to semilinear elliptic boundary value problems and displays how various approaches can easily be applied to a range of model cases. Complete with a preliminary chapter, an appendix that includes further results on weak derivatives, and chapter-by-chapter exercises, this book is a practical text for an introductory course or seminar on nonlinear functional analysis.

This third edition is addressed to the mathematician or graduate student of mathematics - or even the well-prepared undergraduate - who would like, with a minimum of background and preparation, to understand some of the beautiful results at the heart of nonlinear analysis. Based on carefully-expounded ideas from several branches of topology, and illustrated by a wealth of figures that attest to the geometric nature of the exposition, the book will be of immense help in providing its readers with an understanding of the mathematics of the nonlinear phenomena that characterize our real world. Included in this new edition are several new chapters that present the fixed point index and its applications. The exposition and mathematical content is improved throughout. This book is ideal for self-study for mathematicians and students interested in such areas of geometric and algebraic topology, functional analysis, differential equations, and applied mathematics. It is a sharply focused and highly readable view of nonlinear analysis by a practicing topologist who has seen a clear path to understanding. "For the topology-minded reader, the book indeed has a lot to offer: written in a very personal, eloquent and instructive style it makes one of the highlights of nonlinear analysis accessible to a wide audience."-Monatshefte fur Mathematik (2006)

This book offers an exposition of the main applications of Nonlinear Analysis, beginning with a chapter on Nonlinear Operators and Fixed Points, a connecting point and bridge from Nonlinear Analysis theory to its applications. The topics covered include applications to ordinary and partial differential equations, optimization, optimal control, calculus of variations and mathematical economics. The presentation is supplemented with the inclusion of many exercises and their solutions.

Contents: Fixed Point Theory and Nonlinear Problems (Th Rassias)Global Linearization Iterative Methods and Nonlinear Partial Differential Equations III (M Altman)On Generalized Power Series and Generalized Operational Calculus and Its Application (M Al-Bassam)Multiple Solutions to Parametrized Nonlinear Differential Systems from Nielsen Fixed Point Theory (R Brown)The topology of Ind-Affine Sets (P Cherenack)Almost Approximately Polynomial Functions (P Cholewa)Cohomology Classes and Foliated Manifolds (M Craioveanu & M Puta)Bifurcation and Nonlinear Instability in Applied Mathematics (L Debnath)The Stability of Weakly Additive Functional (H Drljevic)Index Theory for G-Bundle Pairs with Applications to Borsuk-Ulam Type Theorems for G-Sphere Bundles (E Fadell & S Husseini)Nonlinear Approximation and Moment Problem (J S Hwang & G D Lin)Periods in Equicontinuous Topological Dynamical Systems (A Iwanik et al.)Continuation Theorems for Semi-Linear Equations in Banach Spaces: A Survey (J Mawhin & K Rybakowski)On Contractifiable Self-Mappings (P Meyers)Normal Structures and Nonexpansive Mappings in Banach Spaces (J Nelson et al.): Survey on Uniqueness and Classification Theorems for Minimal Surfaces (Th Rassias)Contractive Definitions (B Rhoades)On KY Fan's Theorem and Its Applications (S Singh)Fixed Points of Amenable Semigroups of Differentiable Operators (P Soardi)Research Problems on Nonlinear Equations (Th Rassias) Readership: Mathematicians and applied scientists. Keywords:Nonlinear Analysis;Nonlinear Partial Differential Equations III;Polynomial Functions;Cohomology Classes;Foliated Manifolds;Topological Dynamical Systems;Minimal Surfaces;Differentiable Operators;Nonlinear Equations

The topological methods based on fixed-point theory and on local topological degree which have been developed by Leray, Schauder, Nirenberg, Cesari and others for the study of nonlinear differential equations are here described in detail, beginning with elementary considerations. The reader is not assumed to have any knowledge of topology beyond the theory of point sets in Euclidean n-space which ordinarily forms part of a course in advanced calculus. The methods are first developed for Euclidean n-space and applied to the study of existence and stability of periodic and almost-periodic solutions of systems of ordinary differential equations, both quasi-linear and with ``large'' nonlinearities. Then, after being extended to infinite-dimensional ``function-spaces'', these methods are applied to integral equations, partial differential equations and further problems concerning periodic solutions of ordinary differential equations.

Progress in the theory of economic equilibria and in game theory has proceeded hand in hand with that of the mathematical tools used in the field, namely nonlinear analysis and, in particular, convex analysis. Jean-Pierre Aubin, one of the leading specialists in nonlinear analysis and its application to economics, has written a rigorous and concise - yet still elementary and self-contained - textbook providing the mathematical tools needed to study optima and equilibria, as solutions to problems, arising in economics, management sciences, operations research, cooperative and non-cooperative games, fuzzy games etc. It begins with the foundations of optimization theory, and mathematical programming, and in particular convex and nonsmooth analysis. Nonlinear analysis is then presented, first game-theoretically, then in the framework of set valued analysis. These results are then applied to the main classes of economic equilibria. The book contains numerous exercises and problems: the latter allow the reader to venture into areas of nonlinear analysis that lie beyond the scope of the book and of most graduate courses.

This is the second of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self -contained and accessible to the nonspecialist. Part II concerns the theory of monotone operators. It is divided into two subvolumes, II/A and II/B, which form a unit. The present Part II/A is devoted to linear monotone operators. It serves as an elementary introduction to the modern functional analytic treatment of variational problems, integral equations, and partial differential equations of elliptic, parabolic and hyperbolic type. This book also represents an introduction to numerical functional analysis with applications to the Ritz method along with the method of finite elements, the Galerkin methods, and the difference method. Many exercises complement the text. The theory of monotone operators is closely related to Hilbert's rigorous justification of the Dirichlet principle, and to the 19th and 20th problems of Hilbert which he formulated in his famous Paris lecture in 1900, and which strongly influenced the development of analysis in the twentieth century.

The theory of Fixed Points is one of the most powerful tools of modern mathematics. This book contains a clear, detailed and well-organized presentation of the major results, together with an entertaining set of historical notes and an extensive bibliography describing further developments and applications. From the reviews: "I recommend this excellent volume on fixed point theory to anyone interested in this core subject of nonlinear analysis." --MATHEMATICAL REVIEWS

Metric Fixed Point Theory has proved a flourishing area of research for many mathematicians. This book aims to offer the mathematical community an accessible, self-contained account which can be used as an introduction to the subject and its development. It will be understandable to a wide audience, including non-specialists, and provide a source of examples, references and new approaches for those currently working in the subject.