Contents
1 Linear Algebraic Aspects 1
1.1 Linear Symplectic Spaces 1
1.2 Symplectic Matrices 3
1.3 Lagrangian Subspaces 12
1.4 Linear Hamiltonian Systems 17
1.5 Eigenvalues of Symplectic Matrices 19
2 A Brief Introduction to Index Functions 23
2.1 Maslov Type Index i1(��) 24
2.2 w-Jndex iw(��) 28
3 Relative Morse Index 35
3.1 Relative Index via Galerkin Approximation Sequences 35
3.2 Relative Morse Index via Orthogonal Projections 40
3.3 Morse Index via Dual Methods 41
3.3.1 The Definition of Index Pair in Case 1 and 2 42
3.3.2 The Definition of Index Pair in Case 3 48
3.4 Saddle Point Reduction for the General Cases 49
4 The P-Index Theory 55
4.1 P-Index Theory 55
4.2 Relative Index via Saddle Point Reduction Method 66
4.3 Galerkin Approximation for the (P�� (w)-Boundary Problem of Hamiltonian Systems 69
4.4 (P�� w)-Index Theory from Analytical Point of View 76
4.5 Bott-Type Formula for the Maslov Type P-Index 79
4.6 Iteration Theory for P-Index 87
4.6.1 Splitting Numbers 87
4.6.2 Abstract Precise Iteration Formulas 89
4.6.3 Iteration Inequalities 91
5 The L-Index Theory 95
5.1 Definition of L-Index 95
5.1.1 The Properties of the L-Indices 101
5.1.2 The Relations of iL(��)and i1(��) 107
5.1.3 L-Index for General Symplectic Paths 111
5.2 The (L�� Z��)-Index Theory 117
5.3 Understanding the Index ip(��) in View of the Lagrangian Index i(��) 121
5.4 The Relation with the Morse Index in Calculus Variations 122
5.5 Saddle Point Reduction Formulas 127
5.6 Galerkin Approximation Formulas for L-Index 134
5.7 Dual L-Index Theory for Linear Hamiltonian Systems 140
5.8 The (L��w)-Index Theory 144
5.9 The Bott Formulas of L-Index 147
5.10 Iteration Inequalities of L-Index 156
5.10.1 Precise Iteration Index Formula 156
5.10.2 Iteration Inequalities 158
6 Maslov Type Index for Lagrangian Paths 161
6.1 Lagrangian Paths 161
6.2 Maslov Type Index for a Pair of Lagrangian Paths 164
6.3 Hormander Index Theory 171
7 Revisit of Maslov Type Index for Symplectic Paths 177
7.1 Maslov Type Index for Symplectic Paths 177
7.2 The cy-Index Function for P-Index 179
7.3 The Concavity of Symplectic Paths and (e�� Lq�� Li)-Signature 180
7.4 The Mixed (L0�� Li)-Concavity 211
8 Applications of -Index 219
8.1 The Existence of P-Solution of Nonlinear Hamiltonian Systems 219
8.2 The Existence of Periodic Solutions for Delay Differential Equations 222
8.2.1 -Boundary Problem of a Hamiltonian System 222
8.2.2 Delay Differential Systems 224
8.2.3 Poisson Structure 226
8.2.4 First Order Delay Hamiltonian Systems 228
8.2.5 Second Order Delay Hamiltonian Systems 230
8.2.6 Background and Related Works 231
8.2.7 Main Results 231
8.3 The Minimal Period Problem for P-Symmetric Solutions 236
9 Applications of L-Index 253
9.1 The Existence of L-Solutions of Nonlinear Hamiltonian Systems 253
9.2 The Minimal Period Problem for Brake Solutions 261
9.3 Brake Subharmonic Solutions of First Order Hamiltonian Systems 270
10 Multiplicity of Brake Orbits on a Fixed Energy Surface 275
10.1 Brake Orbits of Nonlinear Hamiltonian Systems 275
10.1.1 Seifert Conjecture 277
10.1.2 Some Related Results Since 1948 278
10.1.3 Some Consequences of Theorem 1.2 and Further Arguments 279
10.2 Proofs of Theorems 1.2 and 1.9 280
11 The Existence and Multiplicity of Solutions of Wave Equations 293
11.1 Variational Setting and Critical Point Theories 293
11.1.1 Critical Point Theorems in Case 1 and Case 2 293
11.1.2 Critical Point Theorems in Case 3 302
11.2 Applications: The Existence and Multiplicity of Solutions for Wave Equations 304
11.2.1 One Dimensional Wave Equations 304
11.2.2 n-Dimensional Wave Equations 314
Bibliography 319
Index 331