This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations.
Author: Melvyn S. Berger
Publisher: Academic Press
ISBN: 0080570445
Category: Mathematics
Page: 417
View: 624
Nonlinearity and Functional Analysis is a collection of lectures that aim to present a systematic description of fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis. For decades, great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences. This volume comprises six chapters and begins by presenting some background material, such as differential-geometric sources, sources in mathematical physics, and sources from the calculus of variations, before delving into the subject of nonlinear operators. The following chapters then discuss local analysis of a single mapping and parameter dependent perturbation phenomena before going into analysis in the large. The final chapters conclude the collection with a discussion of global theories for general nonlinear operators and critical point theory for gradient mappings. This book will be of interest to practitioners in the fields of mathematics and physics, and to those with interest in conventional linear functional analysis and ordinary and partial differential equations.
Nonlinearity and Functional Analysis , Academic Press , New York - San Francisco - London . Berger , M.S. and Church , P.T. ( 1979 ) . Complete integrability and perturbation of a nonlinear Dirichlet problem I , Indiana Univ . Math .
Author: P. S. Milojevic
Publisher: CRC Press
ISBN: 0824782550
Category: Mathematics
Page: 284
View: 575
This book is based on the lectures presented at the Special Session on Nonlinear Functional Analysis of the American Mathematical Society Regional Meeting, held at New Jersey Institute of Technology. It explores global invertibility and finite solvability of nonlinear differential equations.
Y. Benyamini , Constants of simultaneous extension of continuous functions , Israel J. Math . 16 ( 1973 ) , 258-262 . ( cited on p 34 ) 48 . ... M. S. Berger , Nonlinearity and Functional Analysis . Lectures on Nonlinear Problems in ...
Author: Yoav Benyamini
Publisher: American Mathematical Soc.
ISBN: 0821869639
Category: Mathematics
Page: 512
View: 495
This book presents a systematic and unified study of geometric nonlinear functional analysis. This area has its classical roots in the beginning of the twentieth century and is now a very active research area, having close connections to geometric measure theory, probability, classical analysis, combinatorics, and Banach space theory. The main theme of the book is the study of uniformly continuous and Lipschitz functions between Banach spaces (e.g., differentiability, stability, approximation, existence of extensions, fixed points, etc.). This study leads naturally also to the classification of Banach spaces and of their important subsets (mainly spheres) in the uniform and Lipschitz categories. Many recent rather deep theorems and delicate examples are included with complete and detailed proofs. Challenging open problems are described and explained, and promising new research directions are indicated.
A class of operator equations in nonlinear mechanics. Applicable Anal. ... Benilan, P., Crandall, M., and Pazy, A. (1989): Nonlinear Evolution Governed by Accretive Operators ... Berger, M. (1977): Nonlinearity and Functional Analysis.
Author: E. Zeidler
Publisher: Springer Science & Business Media
ISBN: 9781461209812
Category: Mathematics
Page: 741
View: 217
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.
J.P. Aubin [1993]: Optima and Equilibria—An Introduction to Nonlinear Analysis, Springer, Berlin. J.P. Aubin [2000]: Applied Functional Analysis, Second Edition, Wiley-Interscience, New York (First Edition: 1979).
Author: Philippe G. Ciarlet
Publisher: SIAM
ISBN: 9781611972597
Category: Mathematics
Page: 832
View: 731
This single-volume textbook covers the fundamentals of linear and nonlinear functional analysis, illustrating most of the basic theorems with numerous applications to linear and nonlinear partial differential equations and to selected topics from numerical analysis and optimization theory. This book has pedagogical appeal because it features self-contained and complete proofs of most of the theorems, some of which are not always easy to locate in the literature or are difficult to reconstitute. It also offers 401 problems and 52 figures, plus historical notes and many original references that provide an idea of the genesis of the important results, and it covers most of the core topics from functional analysis.
In that paper we considered both C” and C* small perturbations of a fixed nonlinear function f(u). For C* perturbations, because of the openness ... M. S. Berger, Nonlinearity and functional analysis, Academic Press, New York, 1977. 2.
M. S. Berger, Nonlinearity and functional analysis, Academic Press, New York, 1977. 5. H. Brezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. (N.S.) 8 (1983), 409–426. 6.
7, 1–28 (1981) Amann, H., Bazley, N., Kirchgässner, K. (eds): [1] Applications of nonlinear analysis in the physical sciences. ... 14, 349–381 (1973) Amerio, L., Prouse, G.: [1] Almost periodic functions and functional equations.
Author: Klaus Deimling
Publisher: Springer Science & Business Media
ISBN: 9783662005477
Category: Mathematics
Page: 450
View: 300
topics. However, only a modest preliminary knowledge is needed. In the first chapter, where we introduce an important topological concept, the so-called topological degree for continuous maps from subsets ofRn into Rn, you need not know anything about functional analysis. Starting with Chapter 2, where infinite dimensions first appear, one should be familiar with the essential step of consider ing a sequence or a function of some sort as a point in the corresponding vector space of all such sequences or functions, whenever this abstraction is worthwhile. One should also work out the things which are proved in § 7 and accept certain basic principles of linear functional analysis quoted there for easier references, until they are applied in later chapters. In other words, even the 'completely linear' sections which we have included for your convenience serve only as a vehicle for progress in nonlinearity. Another point that makes the text introductory is the use of an essentially uniform mathematical language and way of thinking, one which is no doubt familiar from elementary lectures in analysis that did not worry much about its connections with algebra and topology. Of course we shall use some elementary topological concepts, which may be new, but in fact only a few remarks here and there pertain to algebraic or differential topological concepts and methods.
Equations of evolution, 282 Evolution equation, 145, 245 With a nonlinearity of Riccati type, 173 Free boundary value problem, 36 Initial value problems, 105 “Irrationality” hypothesis, 17 Leray-Schauder existence theorem, 92 Lipschitz ...
Another point that makes the text introductory is the use of an essentially uniform mathematical language and way of thinking, one which is no doubt familiar from elementary lectures in analysis that did not worry much about its connections ...
Author: Klaus Deimling
Publisher: Springer
ISBN: 3662005492
Category: Mathematics
Page: 450
View: 256
topics. However, only a modest preliminary knowledge is needed. In the first chapter, where we introduce an important topological concept, the so-called topological degree for continuous maps from subsets ofRn into Rn, you need not know anything about functional analysis. Starting with Chapter 2, where infinite dimensions first appear, one should be familiar with the essential step of consider ing a sequence or a function of some sort as a point in the corresponding vector space of all such sequences or functions, whenever this abstraction is worthwhile. One should also work out the things which are proved in § 7 and accept certain basic principles of linear functional analysis quoted there for easier references, until they are applied in later chapters. In other words, even the 'completely linear' sections which we have included for your convenience serve only as a vehicle for progress in nonlinearity. Another point that makes the text introductory is the use of an essentially uniform mathematical language and way of thinking, one which is no doubt familiar from elementary lectures in analysis that did not worry much about its connections with algebra and topology. Of course we shall use some elementary topological concepts, which may be new, but in fact only a few remarks here and there pertain to algebraic or differential topological concepts and methods.
Release on 2014-05-10 | by Eduardo H. Zarantonello
Preface Linearity is such a deep-seated notion among mathematicians that any outside venture, especially in functional analysis, is immediately qualified as “nonlinear,” as if linearity were the normal way of life in mathematics.
Author: Eduardo H. Zarantonello
Publisher: Academic Press
ISBN: 9781483266626
Category: Mathematics
Page: 686
View: 919
Contributions to Nonlinear Functional Analysis contains the proceedings of a Symposium on Nonlinear Functional Analysis, held in Madison, Wisconsin, on April 12-14, 1971, under the sponsorship of the University of Wisconsin's Mathematics Research Center. The symposium provided a forum for discussing various topics related to nonlinear functional analysis, from transversality in nonlinear eigenvalue problems to monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. Comprised of 15 chapters, this book begins by presenting an extension of Leray-Schauder degree and an application to a nonlinear elliptic boundary value problem. The discussion then turns to the use of degree theory to prove the existence of global continua of solutions of nonlinear eigenvalue problems; transversality in nonlinear eigenvalue problems; and how variational structure can be used to study some local questions in bifurcation theory. Subsequent chapters deal with the notion of monotone operators and monotonicity theory; a nonlinear version of the Hille-Yosida theorem; a version of the penalty method for the Navier-Stokes equations; and various types of weak solutions for minimizing problems in the spirit of duality theory for convex functionals. This monograph will be of interest to students and practitioners in the field of mathematics who want to learn more about nonlinear functional analysis.
We have in hand several other examples including a quite nonlinear 3 X 3 case. ... The Relationship of the Majorant Theory to Local and Global Analysis The theory given here yields theorems which are local in nature but they differ from ...
Author: Louis B. Rall
Publisher: Elsevier
ISBN: 9781483272443
Category: Mathematics
Page: 594
View: 969
Nonlinear Functional Analysis and Applications provides information pertinent to the fundamental aspects of nonlinear functional analysis and its application. This book provides an introduction to the basic concepts and techniques of this field. Organized into nine chapters, this book begins with an overview of the possibilities for applying ideas from functional analysis to problems in analysis. This text then provides a systematic exposition of several aspects of differential calculus in norms and topological linear spaces. Other chapters consider the various settings in nonlinear functional analysis in which differentials play a significant role. This book discusses as well the generalized inverse for a bounded linear operator, whose range is not necessarily closed. The final chapter deals with the equations of hydrodynamics, which are usually highly nonlinear and difficult to solve. This book is a valuable resource for mathematicians. Readers who are interested in nonlinear functional analysis will also find this book useful.
[ 3 ] Bifurcation Theory and Nonlinear Eigenvalue Problems . Edited by Joseph B. Keller and Stuart Antman . W. A. Benjamin , New York - Amsterdam , 1969 . [ 4 ] Contributions to Nonlinear Functional Analysis . Proceedings of a Symposium ...
Author: L. Nirenberg
Publisher: American Mathematical Soc.
ISBN: 0821883461
Category: Mathematics
Page: 162
View: 675
Since its first appearance as a set of lecture notes published by the Courant Institute in 1974, this book served as an introduction to various subjects in nonlinear functional analysis. The current edition is a reprint of these notes, with added bibliographic references. Topological and analytic methods are developed for treating nonlinear ordinary and partial differential equations. The first two chapters of the book introduce the notion of topological degree and develop its basic properties. These properties are used in later chapters in the discussion of bifurcation theory (the possible branching of solutions as parameters vary), including the proof of Rabinowitz global bifurcation theorem. Stability of the branches is also studied. The book concludes with a presentation of some generalized implicit function theorems of Nash-Moser type with applications to Kolmogorov-Arnold-Moser theory and to conjugacy problems. For more than 20 years, this book continues to be an excellent graduate level textbook and a useful supplementary course text. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.
The nonlinear function g may be unbounded and "touching" of the eigenvalue (m + 1)2 (respectively (m - 1)*) on a subset of [0, 2T ] of positive measure is allowed. ... Nonlinear Functional Analysis and Its Applications, 277–289.
Author: S.P. Singh
Publisher: Springer Science & Business Media
ISBN: 9789400946323
Category: Mathematics
Page: 430
View: 725
A NATO Advanced Study Institute on Nonlinear Functional Analysis and Its Applications was held in Hotel Villa del Mare, Maratea, It.a1y during April 22 - May 3, 1985. This volume consists of the Proceedings of the Institute. These Proceedings include the invited lectures and contributed papers given during the Institute. The papers have been refereed. The aim of these lectures was to bring together recent and up-to-date development of the subject, and to give directions for future research. The main topics covered include: degree and generalized degree theory, results related to Hamiltonian Systems, Fixed Point theory, linear and nonlinear Differential and Partial Differential Equations, Theory of Nielsen Numbers, and applications to Dynamical Systems, Bifurcation Theory, Hamiltonian Systems, Minimax Theory, Heat Equations, Pendulum Equation, Nonlinear Boundary Value Problems, and Dirichlet and Neumann problems for elliptic equations and the periodic Dirichlet problem for semilinear beam equations. I express my sincere thanks to Professors F. E. Browder, R. Conti, A. Do1d, D. E. Edmunds and J. Mawhin members of the Advisory Committee.
This book addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the problem of analyzing the structure of the set of zeroes of a general class of nonlinear operators defined between ...
Author: Julian Lopez-Gomez
Publisher: CRC Press
ISBN: 9781420035506
Category: Mathematics
Page: 280
View: 146
This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure of the set of zeroes of a general class of nonlinear operators. It features the construction of an optimal algebraic/analytic invariant for calculating the Leray-Schauder degree, new methods for solving nonlinear equations in Banach spaces, and general properties of components of solutions sets presented with minimal use of topological tools. The author also gives several applications of the abstract theory to reaction diffusion equations and systems. The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers. Appealing to a broad audience, Spectral Theory and Nonlinear Functional Analysis contains many important contributions to linear algebra, linear and nonlinear functional analysis, and topology and opens the door for further advances.
Release on 2021-12-28 | by Maxim Olegovich Korpusov
[18] R. E. Edwards, Functional Analysis. Theory and Applications. ... [20] S. Fuc ́ık and A. Kufner, Nonlinear Differential Equations, Elsevier, Amsterdam–Oxford–New York (1980). [21] H. Fujita, On the blowing up solutions of the Cauchy ...
Author: Maxim Olegovich Korpusov
Publisher: World Scientific
ISBN: 9789811248948
Category: Mathematics
Page: 376
View: 505
This book is a systematic presentation of basic notions, facts, and ideas of nonlinear functional analysis and their applications to nonlinear partial differential equations. It begins from a brief introduction to linear functional analysis, including various types of convergence and functional spaces. The main part of the book is devoted to the theory of nonlinear operators. Various methods of the study of nonlinear differential equations based on the facts of nonlinear analysis are presented in detail. This book may serve as an introductory textbook for students and undergraduates specializing in modern mathematical physics.
Li, F.Y.; Long, L.; Huang, Y.Y.; Zhang, Z.P. Ground state for Choquard equation with doubly critical growth nonlinearity. Electron. J. Qual. Theory Differ. Equ. 2019, 33, 1–15. [CrossRef] 27. Moroz, V.; Van Schaftingen, ...
Author: Radu Precup
Publisher: MDPI
ISBN: 9783036502403
Category: Mathematics
Page: 146
View: 461
This book consists of nine papers covering a number of basic ideas, concepts, and methods of nonlinear analysis, as well as some current research problems. Thus, the reader is introduced to the fascinating theory around Brouwer's fixed point theorem, to Granas' theory of topological transversality, and to some advanced techniques of critical point theory and fixed point theory. Other topics include discontinuous differential equations, new results of metric fixed point theory, robust tracker design problems for various classes of nonlinear systems, and periodic solutions in computer virus propagation models.
Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Brezis,HandNirenberg,L., Positive solutions of nonlinearelliptic equations involving critical Sobolev exponents, Comm. on Pure ...
Author: Antonio Ambrosetti
Publisher: Springer Science & Business Media
ISBN: 9780817681142
Category: Mathematics
Page: 199
View: 701
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.
... because of their nonlinearity, the mathematical study of these equations is difficult and requires the full power of modern functional analysis. Even now, despite all the important work done on these equations, our understanding of ...
Author: Roger Temam
Publisher: SIAM
ISBN: 1611970059
Category: Technology & Engineering
Page: 155
View: 975
This second edition, like the first, attempts to arrive as simply as possible at some central problems in the Navier-Stokes equations in the following areas: existence, uniqueness, and regularity of solutions in space dimensions two and three; large time behavior of solutions and attractors; and numerical analysis of the Navier-Stokes equations. Since publication of the first edition of these lectures in 1983, there has been extensive research in the area of inertial manifolds for Navier-Stokes equations. These developments are addressed in a new section devoted entirely to inertial manifolds. Inertial manifolds were first introduced under this name in 1985 and, since then, have been systematically studied for partial differential equations of the Navier-Stokes type. Inertial manifolds are a global version of central manifolds. When they exist they encompass the complete dynamics of a system, reducing the dynamics of an infinite system to that of a smooth, finite-dimensional one called the inertial system. Although the theory of inertial manifolds for Navier-Stokes equations is not complete at this time, there is already a very interesting and significant set of results which deserves to be known, in the hope that it will stimulate further research in this area. These results are reported in this edition.
Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications Aref Jeribi, ... are the nonlinearity of the boundary condition K and the nonlinear dependence of the function r(, , , ) on b.
Author: Aref Jeribi
Publisher: CRC Press
ISBN: 9781498733892
Category: Mathematics
Page: 371
View: 510
Uncover the Useful Interactions of Fixed Point Theory with Topological Structures Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and Banach algebras. The book provides researchers and graduate students with a unified survey of the fundamental principles of fixed point theory in Banach spaces and algebras. The authors present several extensions of Schauder’s and Krasnosel’skii’s fixed point theorems to the class of weakly compact operators acting on Banach spaces and algebras, particularly on spaces satisfying the Dunford–Pettis property. They also address under which conditions a 2×2 block operator matrix with single- and multi-valued nonlinear entries will have a fixed point. In addition, the book describes applications of fixed point theory to a wide range of diverse equations, including transport equations arising in the kinetic theory of gas, stationary nonlinear biological models, two-dimensional boundary-value problems arising in growing cell populations, and functional systems of integral equations. The book focuses on fixed point results under the weak topology since these problems involve the loss of compactness of mappings and/or the missing geometric and topological structure of their underlying domain.