Existence Theorems for Ordinary Differential Equations (Dover Books on Mathematics) by Francis J. Murray (PDF)

Description

Theorems stating the existence of an object—such as the solution to a problem or equation—are known as existence theorems. This text examines fundamental and general existence theorems, along with the Picard iterants, and applies them to properties of solutions and linear differential equations.The authors assume a basic knowledge of real function theory, and for certain specialized results, of elementary functions of a complex variable. They do not consider the elementary methods for solving certain special differential equations, nor advanced specialized topics; within these restrictions, they obtain a logically coherent discussion for students at a specific phase of their mathematical development. The treatment begins with a survey of fundamental existence theorems and advances to general existence and uniqueness theorems. Subsequent chapters explore the Picard iterants, properties of solutions, and linear differential equations.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Ordinary differential equations are one of the most important topics in applied mathematics. As such there is nothing in this book that is not key.This book manages to be both a brief and reasonably clear introduction to the theory of this field, providing proofs of some the most widely sited and used results in science and engineering as well as other areas of mathematics like differential geometry. It will not help you get better at solving differential equations, but it will help you better understand them and better understand when they have solutions, when those solutions are unique, and what additional properties those solutions might have based on the properties of the differential equations themselves.So for one thing, the title is too narrow for the contents.This book is divided into six chapters, and presents the theory in a logical and progressive order. The first chapter covers a basic existence theorem first in one dependent variable then in multiple dependent variables. The second chapter covers a general existence theorem based on the implicit function theorem which is proven at the outset of this chapter.Chapter three looks at uniqueness and introduces the famous Lipschitz condition in this connection. Chapter four explores Picard iterants and uses the Lipschitz condition to prove convergence of these iterated integrals thus providing an intuitive and useful theoretical tool for exploring questions of the continuity of solutions with respect to initial conditions and/or parameters, and these are the topics that round out chapter four.Chapter five studies additional properties of solutions beyond the ones treated thus far and some sufficient conditions for them. This chapter has by far the fussiest limit arguments in the book and it is also in this chapter that complex analysis first rears its head in connection with the analyticity of solutions.Chapter six concludes the book with a look at linear equations, the Jordan normal form of a linear transformation (defined and explained, but not proven), Green functions for ODE’s, the monodromy group, and a final section on linear equations with constant coefficients.This book is not without its faults. The Lipschitz condition is defined only for convex domains, and domains are generally assumed to be convex thereafter. But the importance of requiring convexity is never discussed. Ever. This is quite puzzling as most results are local results where convexity is guaranteed anyway.In chapter five, where the going gets tougher, the typo density increases, although there are relatively few typos throughout.In chapter six, the notation is awkward and the author proves several results using division. Division? In a vector space? The author apparently doesn’t even think this is worthy of comment, but for me it was a serious breach of mathematical good taste.Finally throughout the book there is very little effort made to visually separate the theorems from the main text, making it difficult to flip through and find results. I kept a highlighter handy while I read.Regardless of its faults though, this book is valuable. The scope is limited to only THE most important results for applications, and the proofs demonstrate some very nice uses of elementary analysis. As the author points out, the theory of differential equations was a prime motivating factor for the development of analysis in the first place. So for the student of analysis there are some great applications of many of the components of elementary analysis to be found in these pages. In particular, the fussy proofs in chapter five are well worth the effort as are the proofs of chapter four. The techniques used here can be generalized to other tricky situations.And while the notation is not always beautiful (particularly in the last chapter), it is at least consistent.A good first book on a vital mathematical subject for readers with a background in elementary real and complex analysis as well as finite dimensional linear algebra. Worthwhile for mathematicians, scientists, and engineers looking for deeper understanding of this important area. Also worthwhile as preparation for deeper study in ODE’s.

⭐This 1954 book

⭐by Murray and Miller is very useful to learn the basics concerning existence, uniqueness and sensitivity for systems of ODEs.This book works systematically through the various issues, giving details that are usually skimmed over in modern books in the interests of making courses short and sweet. (Sometimes a more detailed study is better in the long run.)* Peano existence for a single ODE.* Peano existence for ODE systems.* Uniqueness for single ODEs and systems, using integrals.* Picard existence (using integrals) for ODEs and systems of ODEs.* Sensitivity with respect to initial conditions and parameters.* Linear systems of ODEs.The earlier Peano approach has some advantages over the later Picard approach. So it’s good that the book commences with that. But the Picard approach is what suits most applications nowadays. Both approaches are clearly presented.I’m particularly interested in the linear systems because they are directly applicable to parallel transport for affine connections on differentiable manifolds (and linear connections on differentiable vector bundles). Every book I have looked at on differential geometry states (most often without proof) that vector fields have integral curves and that the connection on a differentiable fibre bundle can be integrated to obtain parallel transport along curves. Since it is a “classical result”, proofs and background theory are rarely presented, and even most analysis books merely skim over this subject as quickly as possible.This book is brief and concise. Not a page is wasted. It was originally published by New York University Press. Maybe they were having a funding crisis at the time and couldn’t afford to let their authors waste space. The result, anyway, is a truly valuable book on the classic theory at an impressively low price.

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