Table of Contents
Preface vii
1 Introduction to Boundary Value Problems 1
1.1 Introduction 1
1.2 Disconjugacy 2
1.3 Disconjugacy for Equations of Arbitrary Order 18
Bibliography 23
2 Linear Problems and Green's Functions 25
2.1 Introduction to Linear ODE's 25
2.2 Linear BVP's 31
2.3 Green's Functions for Linear BVP's 33
2.4 Properties of the Green's Function 35
2.4.1 Second method for constructing the Green's function 36
2.4.2 Another approach using the properties 37
2.4.3 Green's functions for general linear homogeneous BVP's 40
Bibliography 47
3 Existence of Solutions I 49
3.1 Local Existence Theorems for Solutions of BVP's for the Nonlinear Equation, y" = f(x, y, y') 49
3.2 Linear Second Order Inequalities 63
3.3 Existence of Solutions of BVP's for Nonlinear ODE's 75
3.4 Existence of Solutions of BVP's for Nonlinear Higher Order ODE's 89
Bibliography 105
4 Existence of Solutions II 107
4.1 Existence of Solutions of BVP's Associated with y"' = f(x, y, y', y") 107
4.2 A Converse of Lemma 4.3 127
Bibliography 133
5 Solution Matching 135
5.1 Introduction 135
5.1.1 Uniqueness and existence of solutions by matching 136
5.1.2 Uniqueness and existence of solutions for a larger class of BVP's by matching 140
5.2 Five-Point Nonlocal BVP's for nth Order ODE's by Solution Matching 141
5.2.1 Uniqueness of solutions 142
5.2.2 Existence of solutions 144
Bibliography 147
6 Comparison of Smallest Eigenvalues 149
6.1 Comparison Theorems for 2-Point Conjugate Eigenvalue Problems 149
6.2 Eigenvalue Comparisons for Nonlocal BVP's 156
6.3 Solutions of Second Order Multi-Point BVP's 160
6.3.1 Preliminary lemmas 161
6.3.2 Existence results 168
Bibliography 185
7 BVP's for Functional Differential Equations 187
7.1 Existence of Solutions for BVP's for Second Order Functional Differential Equations 187
7.2 Periodic Solutions for First Order Functional Differential Equations 193
7.3 Periodic Solutions for Functional Differential Equations with Sign-Changing Nonlinearities 199
7.3.1 Preliminary results 200
7.3.2 Existence results 208
7.3.3 Proofs of the existence results 211
Bibliography 219
8 Positive Solutions 221
8.1 Positive Solutions for Nonlinear Eigenvalue Problems 221
8.1.1 Solutions in the cone 222
8.2 Multiplicity of Positive Solutions 226
8.3 Positive Solutions for Nonlinear Difference Equations 230
8.4 Multiple Symmetric Positive Solutions 237
8.5 Positive Solutions for a Singular Third Order BVP 241
8.5.1 A priori bounds on norms of solutions 245
8.5.2 Existence of positive solutions 247
8.6 Positive Solutions for Third Order BVP's with p-Laplacian 249
8.6.1 Main results 250
8.6.2 Proofs of the main results 252
Bibliography 267
9 Boundary Data Smoothness 271
9.1 Differentiation of Solutions with Respect to Boundary Conditions 271
Bibliography 281
10 Nodal Solutions of BVP's for ODE's 283
10.1 Nodal Solutions of Nonlinear BVP's with Separated BC's 283
10.1.1 Main results 284
10.1.2 Proofs of the main results 287
10.2 Nodal Solutions of Nonlocal BVP's - I 302
10.2.1 Main results 303
10.2.2 Proofs of the main results 307
10.3 Nodal Solutions of Nonlocal BVP's - II 311
10.3.1 Existence and nonexistence of nodal solutions 313
10.3.2 Dependence of nodal solutions on the problem 319
Bibliography 327
Index 329
