A systematic study of geometric nonlinear functional analysis. The main theme is the study of uniformly continuous and Lipschitz functions between Banach spaces. This study leads to the classification of Banach spaces and of their important subsets in the uniform and Lipschitz categories.

This book provides a comprehensive presentation of recent approaches to and results about properties of various classes of functional spaces, such as Banach spaces, uniformly convex spaces, function spaces, and Banach algebras. Each of the 12 articles in this book gives a broad overview of current subjects and presents open problems. Each article includes an extensive bibliography. This book is dedicated to Professor Per. H. Enflo, who made significant contributions to functional analysis and operator theory.

This is an collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry. The main purpose of this work is to help in convincing young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study. Some of the problems are longstanding open problems, some are recent, some are more important and some are only local problems. Some would require new ideas, some may be resolved with only a subtle combination of known facts. Regardless of their origin or longevity, each of these problems documents the need for further research in this area.

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 contributed volume discusses aspects of nonlinear analysis in which optimization plays an important role, as well as topics which are applied to the study of optimization problems. Topics include set-valued analysis, mixed concave-convex sub-superlinear Schroedinger equation, Schroedinger equations in nonlinear optics, exponentially convex functions, optimal lot size under the occurrence of imperfect quality items, generalized equilibrium problems, artificial topologies on a relativistic spacetime, equilibrium points in the restricted three-body problem, optimization models for networks of organ transplants, network curvature measures, error analysis through energy minimization and stability problems, Ekeland variational principles in 2-local Branciari metric spaces, frictional dynamic problems, norm estimates for composite operators, operator factorization and solution of second-order nonlinear difference equations, degenerate Kirchhoff-type inclusion problems, and more.

Banach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization and other branches of mathematics. This book introduces the reader to linear functional analysis and to related parts of infinite-dimensional Banach space theory. Key Features: - Develops classical theory, including weak topologies, locally convex space, Schauder bases and compact operator theory - Covers Radon-Nikodým property, finite-dimensional spaces and local theory on tensor products - Contains sections on uniform homeomorphisms and non-linear theory, Rosenthal's L1 theorem, fixed points, and more - Includes information about further topics and directions of research and some open problems at the end of each chapter - Provides numerous exercises for practice The text is suitable for graduate courses or for independent study. Prerequisites include basic courses in calculus and linear. Researchers in functional analysis will also benefit for this book as it can serve as a reference book.

Abstract topological tools from generalized metric spaces are applied in this volume to the construction of locally uniformly rotund norms on Banach spaces. The book offers new techniques for renorming problems, all of them based on a network analysis for the topologies involved inside the problem. Maps from a normed space X to a metric space Y, which provide locally uniformly rotund renormings on X, are studied and a new frame for the theory is obtained, with interplay between functional analysis, optimization and topology using subdifferentials of Lipschitz functions and covering methods of metrization theory. Any one-to-one operator T from a reflexive space X into c0 (T) satisfies the authors' conditions, transferring the norm to X. Nevertheless the authors' maps can be far from linear, for instance the duality map from X to X* gives a non-linear example when the norm in X is Fréchet differentiable. This volume will be interesting for the broad spectrum of specialists working in Banach space theory, and for researchers in infinite dimensional functional analysis.

This unique mathematical volume brings together geometers, analysts, differential equations specialists and graph-theorists to provide a glimpse on recent mathematical trends whose commonalities have hitherto remained, for the most part, unnoticed. The applied mathematician will be pleasantly surprised with the interpretation of a voting system in terms of the fixed points of a mapping given in the book, as much as the classical analyst will be enthusiastic to find detailed discussions on the generalization of the notion of metric space, in which the metric takes values on an abstract monoid. Classical themes on fixed point theory are adapted to the diverse setting of graph theory, thus uncovering a set of tools whose power and versatility will be appreciated by mathematicians working on either area. The volume also includes recent results on variable exponent spaces which reveal much-needed connections with partial differential equations, while the incipient field of variational inequalities on manifolds, also explored here, will be of interest to researchers from a variety of fields.