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Author: Uri Elias Publisher: Springer Science & Business Media ISBN: 9401725179 Category : Mathematics Languages : en Pages : 232
Book Description
Oscillation theory was born with Sturm's work in 1836. It has been flourishing for the past fifty years. Nowadays it is a full, self-contained discipline, turning more towards nonlinear and functional differential equations. Oscillation theory flows along two main streams. The first aims to study prop erties which are common to all linear differential equations. The other restricts its area of interest to certain families of equations and studies in maximal details phenomena which characterize only those equations. Among them we find third and fourth order equations, self adjoint equations, etc. Our work belongs to the second type and considers two term linear equations modeled after y(n) + p(x)y = O. More generally, we investigate LnY + p(x)y = 0, where Ln is a disconjugate operator and p(x) has a fixed sign. These equations enjoy a very rich structure and are the natural generalization of the Sturm-Liouville operator. Results about such equations are distributed over hundreds of research papers, many of them are reinvented again and again and the same phenomenon is frequently discussed from various points of view and different definitions of the authors. Our aim is to introduce an order into this plenty and arrange it in a unified and self contained way. The results are readapted and presented in a unified approach. In many cases completely new proofs are given and in no case is the original proof copied verbatim. Many new results are included.
Author: Dimit?r Ba?nov Publisher: World Scientific ISBN: 9789810211004 Category : Mathematics Languages : en Pages : 70
Book Description
In this book, the authors aim at expounding a sufficiently rich oscillation theory and asymptotic theory of operator-differential equations. This book will be of interest not only to mathematicians, but also to experts in other areas of science and technology due to the numerous applications of the results discussed in the book.
Author: Uri Elias Publisher: Springer Science & Business Media ISBN: 9401725179 Category : Mathematics Languages : en Pages : 232
Book Description
Oscillation theory was born with Sturm's work in 1836. It has been flourishing for the past fifty years. Nowadays it is a full, self-contained discipline, turning more towards nonlinear and functional differential equations. Oscillation theory flows along two main streams. The first aims to study prop erties which are common to all linear differential equations. The other restricts its area of interest to certain families of equations and studies in maximal details phenomena which characterize only those equations. Among them we find third and fourth order equations, self adjoint equations, etc. Our work belongs to the second type and considers two term linear equations modeled after y(n) + p(x)y = O. More generally, we investigate LnY + p(x)y = 0, where Ln is a disconjugate operator and p(x) has a fixed sign. These equations enjoy a very rich structure and are the natural generalization of the Sturm-Liouville operator. Results about such equations are distributed over hundreds of research papers, many of them are reinvented again and again and the same phenomenon is frequently discussed from various points of view and different definitions of the authors. Our aim is to introduce an order into this plenty and arrange it in a unified and self contained way. The results are readapted and presented in a unified approach. In many cases completely new proofs are given and in no case is the original proof copied verbatim. Many new results are included.
Author: Anatoly G. Kusraev Publisher: Springer Nature ISBN: 3030497631 Category : Mathematics Languages : en Pages : 337
Book Description
This volume features selected papers from The Fifteenth International Conference on Order Analysis and Related Problems of Mathematical Modeling, which was held in Vladikavkaz, Russia, on 15 - 20th July 2019. Intended for mathematicians specializing in operator theory, functional spaces, differential equations or mathematical modeling, the book provides a state-of-the-art account of various fascinating areas of operator theory, ranging from various classes of operators (positive operators, convolution operators, backward shift operators, singular and fractional integral operators, partial differential operators) to important applications in differential equations, inverse problems, approximation theory, metric theory of surfaces, the Hubbard model, social stratification models, and viscid incompressible fluids.
Author: Drumi D Bainov Publisher: World Scientific ISBN: 9814505250 Category : Mathematics Languages : en Pages : 218
Book Description
In this book, the authors aim at expounding a sufficiently rich oscillation theory and asymptotic theory of operator-differential equations. This book will be of interest not only to mathematicians, but also to experts in other areas of science and technology due to the numerous applications of the results discussed in the book.
Author: Joachim Weidmann Publisher: Springer ISBN: 3540479120 Category : Mathematics Languages : en Pages : 310
Book Description
These notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.
Author: I. Győri Publisher: Clarendon Press ISBN: Category : Mathematics Languages : en Pages : 392
Book Description
In recent years there has been a resurgence of interest in the study of delay differential equations motivated largely by new applications in physics, biology, ecology, and physiology. The aim of this monograph is to present a reasonably self-contained account of the advances in the oscillation theory of this class of equations. Throughout, the main topics of study are shown in action, with applications to such diverse problems as insect population estimations, logistic equations in ecology, the survival of red blood cells in animals, integro-differential equations, and the motion of the tips of growing plants. The authors begin by reviewing the basic theory of delay differential equations, including the fundamental results of existence and uniqueness of solutions and the theory of the Laplace and z-transforms. Little prior knowledge of the subject is required other than a firm grounding in the main techniques of differential equation theory. As a result, this book provides an invaluable reference to the recent work both for mathematicians and for all those whose research includes the study of this fascinating class of differential equations.
Author: Ondrej Dosly Publisher: Elsevier ISBN: 0080461239 Category : Mathematics Languages : en Pages : 533
Book Description
The book presents a systematic and compact treatment of the qualitative theory of half-lineardifferential equations. It contains the most updated and comprehensive material and represents the first attempt to present the results of the rapidly developing theory of half-linear differential equations in a unified form. The main topics covered by the book are oscillation and asymptotic theory and the theory of boundary value problems associated with half-linear equations, but the book also contains a treatment of related topics like PDE's with p-Laplacian, half-linear difference equations and various more general nonlinear differential equations.- The first complete treatment of the qualitative theory of half-linear differential equations.- Comparison of linear and half-linear theory.- Systematic approach to half-linear oscillation and asymptotic theory.- Comprehensive bibliography and index.- Useful as a reference book in the topic.
Author: Norio Yoshida Publisher: World Scientific Publishing Company ISBN: 9813107375 Category : Mathematics Languages : en Pages : 339
Book Description
This unique book is designed to provide the reader with an exposition of interesting aspects — encompassing both rudimentary and advanced knowledge — of oscillation theory of partial differential equations, which dates back to the publication in 1955 of a paper by Ph Hartman and A Wintner. The objective of oscillation theory is to acquire as much information as possible about the qualitative properties of solutions of differential equations through the analysis of laws governing the distribution of zeros of solutions as well as the asymptotic behavior of solutions of differential equations under consideration.This textbook on oscillation theory of partial differential equations is useful for both specialists and graduate students working in the field of differential equations. The book will also help to stimulate further progress in the study of oscillation theory and related subjects.
Author: Gerald Teschl Publisher: American Mathematical Society ISBN: 147047641X Category : Mathematics Languages : en Pages : 370
Book Description
This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm–Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincaré–Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.
Author: Werner O. Amrein Publisher: Springer Science & Business Media ISBN: 3764373598 Category : Mathematics Languages : en Pages : 348
Book Description
This is a collection of survey articles based on lectures presented at a colloquium and workshop in Geneva in 2003 to commemorate the 200th anniversary of the birth of Charles François Sturm. It aims at giving an overview of the development of Sturm-Liouville theory from its historical roots to present day research. It is the first time that such a comprehensive survey has been made available in compact form. The contributions come from internationally renowned experts and cover a wide range of developments of the theory. The book can therefore serve both as an introduction to Sturm-Liouville theory and as background for ongoing research. The volume is addressed to researchers in related areas, to advanced students and to those interested in the historical development of mathematics. The book will also be of interest to those involved in applications of the theory to diverse areas such as engineering, fluid dynamics and computational spectral analysis.
Author: Uri Elias
Author: Dimit?r Ba?nov
Author: Uri Elias
Author: Anatoly G. Kusraev
Author: Drumi D Bainov
Author: Joachim Weidmann
Author: I. Győri
Author: Ondrej Dosly
Author: Norio Yoshida
Author: Gerald Teschl
Author: Werner O. Amrein