Nonlinear Interpolation And Boundary Value Problems

by Paul W Eloe, Johnny L Henderson

Overview

This book is devoted to the study of boundary value problems for nonlinear ordinary differential equations and focuses on questions related to the study of nonlinear interpolation. In 1967, Andrzej Lasota and Zdzisław Opial showed that, under suitable hypotheses, if solutions of a second-order nonlinear differential equation passing through two distinct points are unique, when they exist, then, in fact, a solution passing through two distinct points does exist. That result, coupled with the pioneering work of Philip Hartman on what was then called unrestricted n-parameter families, has stimulated 50 years of development in the study of solutions of boundary value problems as nonlinear interpolation problems.The purpose of this book is two-fold. First, the results that have been generated in the past 50 years are collected for the first time to produce a comprehensive and coherent treatment of what is now a well-defined area of study in the qualitative theory of ordinary differential equations. Second, methods and technical tools are sufficiently exposed so that the interested reader can contribute to the study of nonlinear interpolation.

Product Details

ISBN-13:	9789814733472
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/24/2016
Series:	Trends In Abstract And Applied Analysis , #2
Pages:	248
Product dimensions:	6.10(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface vii

1 Uniqueness Implies Uniqueness 1

1.1 Some preliminaries 1

1.2 Conjugate boundary value problems: for m > k, uniqueness of m-point implies uniqueness of k-point 9

1.3 Conjugate boundary value problems: for m < k, uniqueness of m-point implies uniqueness of k-point 16

1.4 Right focal boundary value problems: for m > r, uniqueness of m-point implies uniqueness of r-point 26

1.5 Right focal boundary value problems: for m < r, uniqueness of m-point implies uniqueness of r-point 35

2 Uniqueness Implies Existence 41

2.1 Conjugate boundary value problems: for n = 2, k = 2, uniqueness of 2-point implies existence of 2-point 41

2.2 Conjugate boundary value problems: for n = 3, uniqueness of 3-point implies existence of 2-point and 3-point 43

2.3 Conjugate boundary value problems: nth order 53

2.4 Right focal boundary value problems: 2-point, uniqueness implies existence 61

2.5 Right focal boundary value problems: r-point, uniqueness implies existence 65

3 Nonlocal Boundary Value Problems: Uniqueness and Existence 73

3.1 Nonlocal problems: uniqueness implies uniqueness, I 74

3.2 Nonlocal problems: uniqueness implies existence, I 78

3.3 Nonlocal problems: uniqueness implies uniqueness, II 84

3.4 Nonlocal problems: uniqueness implies existence, II 113

4 Boundary Value Problems for Finite Difference Equations 139

4.1 Conjugate boundary value problems: uniqueness implies existence 140

4.2 Focal boundary value problems: uniqueness implies existence 149

4.3 "Between" boundary value problems: uniqueness implies existence 163

4.4 Lidstone boundary value problems: uniqueness implies existence 173

5 Boundary Value Problems for Dynamic Equations on Time Scales 181

5.1 Conjugate boundary value problems: uniqueness implies existence 184

5.2 Right focal boundary value problems: uniqueness implies existence 215

5.3 Nonlocal boundary value problems: uniqueness implies existence 222

5.4 Additional remarks 224

6 Postscript 225

Bibliography 227

Index 235