Higher Order Boundary Value Problems On Unbounded Domains: Types Of Solutions, Functional Problems And Applications

by Feliz Manuel Minhos, Hugo Alexandre Sacristao Carrasco

Overview

This volume provides a comprehensive overview on different types of higher order boundary value problems defined on the half-line or on the real line (Sturm-Liouville and Lidstone types, impulsive, functional and problems defined by Hammerstein integral equations). It also includes classical and new methods and techniques to deal with the lack of compactness of the related operators.The reader will find a selection of original and recent results in this field, conditions to obtain solutions with particular qualitative properties, such as homoclinic and heteroclinic solutions and its relation with the solutions of Lidstone problems on all the real line.Each chapter contains applications to real phenomena, to classical equations or problems, with a common denominator: they are defined on unbounded intervals and the existing results in the literature are scarce or proven only numerically in discrete cases.The last part features some higher order functional problems, which generalize the classical two-point or multi-point boundary conditions, to more comprehensive data where an overall behavior of the unknown functions and their derivatives is involved.

Product Details

ISBN-13:	9789813209909
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/13/2017
Series:	Trends In Abstract And Applied Analysis , #5
Pages:	216
Product dimensions:	5.90(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface ix

Introduction xi

Part I Boundary Value Problems on the Half-Line 1

Introduction 3

1 Third-Order Boundary Value Problems 5

1.1 Introduction 5

1.2 Definitions and auxiliary results 6

1.3 Existence and localization result 11

1.4 Example 17

2 General nth-Order Problems 19

2.1 Introduction 19

2.2 Preliminary results 20

2.3 Existence and localization result 22

2.4 Example 23

3 Impulsive Problems on the Half-Line with Infinite Impulse Moments 25

3.1 Introduction 25

3.2 Definitions and preliminary results 26

3.3 Main result 28

3.4 Example 36

Part II Homoclinic Solutions and Lidstone Problems 39

Introduction 41

4 Homoclinic Solutions for Second-Order Problems 43

4.1 Introduction 43

4.2 Preliminaries 44

4.3 Existence and localization of homoclinics 47

4.4 Example of a discontinuous BVP 52

4.5 Duffing equation 53

4.6 Forced cantilever beam equation with damping 54

5 Homoclinic Solutions to Fourth-Order Problems 57

5.1 Introduction 57

5.2 Definitions and auxiliary results 59

5.3 Existence results 61

5.4 Example 65

5.5 Bernoulli-Euler-v. Karman problem 66

5.6 Extended Fisher-Kolmogorov and Swift-Hohenberg problems 67

6 Lidstone Boundary Value Problems 71

6.1 Introduction 71

6.2 Auxiliary definitions and Green's functions 73

6.3 Existence result 75

6.4 An infinite beam resting on granular foundations 78

Part III Heteroclinic Solutions and Hammerstein Equations 81

Introduction 83

7 Heteroclinic Solutions for Semi-linear Problems (i) 85

7.1 Introduction 85

7.2 Definitions and preliminary results 87

7.3 Existence of heteroclinics 89

7.4 Example 94

8 Heteroclinic Solutions for Semi-linear Problems (ii) 97

8.1 Introduction 97

8.2 Auxiliary results 99

8.3 Existence of heteroclinics solutions 102

8.4 Examples 108

9 Heteroclinic Solutions for Semi-linear Problems (iii) 111

9.1 Introduction 111

9.2 Existence results 113

9.3 Example 124

9.4 Singular ø-Laplacian equations 125

10 Hammerstein Integral Equations with Sign-Changing Kernels 127

10.1 Introduction 127

10.2 Main result 129

10.3 Application to fourth-order BVPs and infinite beams 131

Part IV Functional Boundary Value Problems 133

Introduction 135

11 Second-Order Functional Problems 139

11.1 Introduction 139

11.2 Definitions and auxiliary results 140

11.3 Existence and localization results 142

11.4 Example 149

11.5 Emden-Fowler equation 149

12 Third-Order Functional Problems 153

12.1 Introduction 153

12.2 Definitions and a priori bounds 154

12.3 Existence and localization results 159

12.4 Falkner-Skan equation 168

13 ø-Laplacian Equations with Functional Boundary Conditions 171

13.1 Introduction 171

13.2 Preliminary results 173

13.3 Existence and localization result 181

13.4 Examples 186

Bibliography 189

Index 199