This book concerns index theory for linear Hamiltonian systems and multiple solutions for asymptotically linear Hamiltonian systems, as well as some related problems.There are two excellent books on Hamiltonian systems: Convexity Methods in Hamiltonian Mechanics by I. Ekeland and Index Theory for Symplectic Paths with Applications by Y. Long. Periodic solutions are the main topics in those books, wher,eas non-periodic solutions will be investigated in most parts of this book. Most contents are from my own research works during the past twenty years and I will try to write this book following the style of the famous book Minimax Methods in Critical Point Theory with Applications to Differential Equations by Paul H. Rabinowitz. An overview is given of the subject matter in Chapter 1 and a detailed study is carried out in the Chapters that follow.
I began to study the existence of solutions for Duffing equations(the simplest

1 An Overview 1

2 Duffing Equations(I) 5

2.1 Lazer-Leach's theorem 5

2.2 A classification theory 9

2.3 A generalization of Lazer-Leach's theorem 13

2.4 Landesman-Lazer's condition 16

2.5 Nontrivial solutions 21

2.6 Sturm-Liouville BVPs 23

3 Duffing Equations(II) 30

3.1 Positive linear Duffing equations 30

3.2 Associated Leray-Schauder degrees 33

3.3 Asymptotically positive linear Duffing equations 36

3.4 Limiting cases 38

3.5 Proof of Theorem 3.4.1 43

3.6 Proof of Theorem 3.4.2 46

3.7 Open questions 50

4 One-dimensional p-Laplacian Equations 51

4.1 p-triangle functions 51

4.2 A classification theory 52

4.3 Associated Leray-Schauder degrees 56

4.4 Solutions of asymptotically homogeneous equations 57

4.5 Related problems 60

5 Second Order Hamiltonian Systems 61

5.1 Index theory 61

5.2 Relative Morse index and topological degree 66

5.3 Existence of solutions 69

5.4 Multiple solutions for symmetric Hamiltonian systems 73

5.5 Three solution theorems 77

6 First Order Hamiltonian Systems 84

6.1 Index theory 84

6.2 μ-index and relative Morse index 86

6.3 Existence of solutions 92

6.4 Multiple solutions for symmetric Hamiltonian systems 95

6.5 Ekeland's index and Long's index 98

7 Operator Equations(I) 105

7.1 Defini t ions for index and nullity 105

7.2 Properties for index and nullity 108

7.3 Solutions of operator equations 114

7.4 Multiple solutions for symmetric operator equations 118

7.5 Three solution theorems 121

8 Operator Equations(II) 126

8.1 Index theory 126

8.2 P-index 129

8.3 Ekeland 's type of index theory 136

8.4 Existence of solutions 140

8.5 Multiple solutions 141

8.6 A new reduced functional 143

8.7 The Morse index theory for a"(u* ) l46

8.8 Proofs of Theorems 8.5.1 and 8.5.2 153

Bibliography 159

Index 164

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