Generalized Ordinary Differential Equations in Abstract Spaces and Applications
Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Author: Everaldo M. Bonotto

Publisher: John Wiley & Sons

Published: 2021-09-15

Total Pages: 514

ISBN-13: 1119654939

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GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES AND APPLICATIONS Explore a unified view of differential equations through the use of the generalized ODE from leading academics in mathematics Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations, including Measure Functional Differential Equations (measure FDEs). It presents a uniform collection of qualitative results of Generalized ODEs and offers readers an introduction to several theories, including ordinary differential equations, impulsive differential equations, functional differential equations, dynamical equations on time scales, and more. Throughout the book, the focus is on qualitative theory and on corresponding results for other types of differential equations, as well as the connection between Generalized Ordinary Differential Equations and impulsive differential equations, functional differential equations, measure differential equations and dynamic equations on time scales. The book’s descriptions will be of use in many mathematical contexts, as well as in the social and natural sciences. Readers will also benefit from the inclusion of: A thorough introduction to regulated functions, including their basic properties, equiregulated sets, uniform convergence, and relatively compact sets An exploration of the Kurzweil integral, including its definitions and basic properties A discussion of measure functional differential equations, including impulsive measure FDEs The interrelationship between generalized ODEs and measure FDEs A treatment of the basic properties of generalized ODEs, including the existence and uniqueness of solutions, and prolongation and maximal solutions Perfect for researchers and graduate students in Differential Equations and Dynamical Systems, Generalized Ordinary Differential Equations in Abstract Spaces and App­lications will also earn a place in the libraries of advanced undergraduate students taking courses in the subject and hoping to move onto graduate studies.

Existence Theory for Nonlinear Ordinary Differential Equations
Existence Theory for Nonlinear Ordinary Differential Equations

Author: Donal O'Regan

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 207

ISBN-13: 9401715173

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We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to Y'. = I(t,y) (1. 1 ) { yeO) = r n where I: I X R n ---+ R and I = [0, b]. We shall seek solutions that are de fined either locally or globally on I, according to the assumptions imposed on I. Notice that (1. 1) is a system of first order equations because I takes its values in Rn. In section 3. 2 we will first establish some basic existence theorems which guarantee that a solution to (1. 1) exists for t > 0 and near zero. Familiar examples show that the interval of existence can be arbi trarily short, depending on the initial value r and the nonlinear behaviour of I. As a result we will also examine in section 3. 2 the dependence of the interval of existence on I and r. We mention in passing that, in the results which follow, the interval I can be replaced by any bounded interval and the initial value can be specified at any point in I. The reasoning needed to cover this slightly more general situation requires minor modifications on the arguments given here.

Differential Equations in Banach Spaces
Differential Equations in Banach Spaces

Author: Giovanni Dore

Publisher: CRC Press

Published: 1993-08-05

Total Pages: 290

ISBN-13: 9780824790677

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This reference - based on the Conference on Differential Equations, held in Bologna - provides information on current research in parabolic and hyperbolic differential equations. Presenting methods and results in semigroup theory and their applications to evolution equations, this book focuses on topics including: abstract parabolic and hyperbolic linear differential equations; nonlinear abstract parabolic equations; holomorphic semigroups; and Volterra operator integral equations.;With contributions from international experts, Differential Equations in Banach Spaces is intended for research mathematicians in functional analysis, partial differential equations, operator theory and control theory; and students in these disciplines.

Functional Analysis, Sobolev Spaces and Partial Differential Equations
Functional Analysis, Sobolev Spaces and Partial Differential Equations

Author: Haim Brezis

Publisher: Springer Science & Business Media

Published: 2010-11-02

Total Pages: 600

ISBN-13: 0387709142

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This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces
Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

Author: Toka Diagana

Publisher: Springer Science & Business Media

Published: 2013-08-13

Total Pages: 312

ISBN-13: 3319008498

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This book presents a comprehensive introduction to the concepts of almost periodicity, asymptotic almost periodicity, almost automorphy, asymptotic almost automorphy, pseudo-almost periodicity, and pseudo-almost automorphy as well as their recent generalizations. Some of the results presented are either new or else cannot be easily found in the mathematical literature. Despite the noticeable and rapid progress made on these important topics, the only standard references that currently exist on those new classes of functions and their applications are still scattered research articles. One of the main objectives of this book is to close that gap. The prerequisites for the book is the basic introductory course in real analysis. Depending on the background of the student, the book may be suitable for a beginning graduate and/or advanced undergraduate student. Moreover, it will be of a great interest to researchers in mathematics as well as in engineering, in physics, and related areas. Further, some parts of the book may be used for various graduate and undergraduate courses.

Degenerate Differential Equations in Banach Spaces
Degenerate Differential Equations in Banach Spaces

Author: Angelo Favini

Publisher: CRC Press

Published: 1998-09-10

Total Pages: 338

ISBN-13: 9780824716776

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This work presents a detailed study of linear abstract degenerate differential equations, using both the semigroups generated by multivalued (linear) operators and extensions of the operational method from Da Prato and Grisvard. The authors describe the recent and original results on PDEs and algebraic-differential equations, and establishes the analyzability of the semigroup generated by some degenerate parabolic operators in spaces of continuous functions.

Nonlinear Differential Equations in Abstract Spaces
Nonlinear Differential Equations in Abstract Spaces

Author: V. Lakshmikantham

Publisher: Pergamon

Published: 1981

Total Pages: 276

ISBN-13:

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Many problems in partial differential equations which arise from physical models can be considered as ordinary differential equations in appropriate infinite dimensional spaces, for which elegant theories and powerful techniques have recently been developed. This book gives a detailed account of the current state of the theory of nonlinear differential equations in a Banach space, and discusses existence theory for differential equations with continuous and discontinuous right-hand sides. Of special importance is the first systematic presentation of the very important and complex theory of multivalued discontinuous differential equations

Differential Equations on Measures and Functional Spaces
Differential Equations on Measures and Functional Spaces

Author: Vassili Kolokoltsov

Publisher: Springer

Published: 2019-06-20

Total Pages: 536

ISBN-13: 3030033775

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This advanced book focuses on ordinary differential equations (ODEs) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It briefly discusses the fundamentals before moving on to the cutting edge research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality chosen is suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is intended mainly for upper undergraduate and graduate students, but is also of use to researchers in differential equations and their applications. It particularly highlights the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, and also clarifies the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

Functional Spaces for the Theory of Elliptic Partial Differential Equations
Functional Spaces for the Theory of Elliptic Partial Differential Equations

Author: Françoise Demengel

Publisher: Springer Science & Business Media

Published: 2012-01-24

Total Pages: 480

ISBN-13: 1447128079

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The theory of elliptic boundary problems is fundamental in analysis and the role of spaces of weakly differentiable functions (also called Sobolev spaces) is essential in this theory as a tool for analysing the regularity of the solutions. This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem. The main purpose of the book is to provide a tool for graduate and postgraduate students interested in partial differential equations, as well as a useful reference for researchers active in the field. Prerequisites include a knowledge of classical analysis, differential calculus, Banach and Hilbert spaces, integration and the related standard functional spaces, as well as the Fourier transformation on the Schwartz space. There are complete and detailed proofs of almost all the results announced and, in some cases, more than one proof is provided in order to highlight different features of the result. Each chapter concludes with a range of exercises of varying levels of difficulty, with hints to solutions provided for many of them.