Ⅰ.Preliminaries 1
1.Preliminaries 1
2.Basic theorems 2
3.Smooth approximations 6
4.Change of integration variables 7
Notes 7
Ⅱ.Existence 8
1.The Picard-Lindelof theorem 8
2.Peano's existence theorem 10
3.Extension theorem 12
4.H.Kneser's theorem 15
5.Example of nonuniqueness 18
Notes 23
Ⅲ.Differential inequalities and uniqueness 24
1.Gronwall's inequality 24
2.Maximal and minimal solutions 25
3.Right derivatives 26
4.Differential inequalities 26
5.A theorem of Wintner 29
6.Uniqueness theorems 31
7.van Kampen's uniqueness theorem 35
8.Egress points and Lyapunov functions 37
9.Successive approximations 40
Notes 44
Ⅳ.Linear differential equations 45
1.Linear systems 45
2.Variation of constants 48
3.Reductions to smaller systems 49
4.Basic inequalities 54
5.Constant coefficients 57
6.Floquet theory 60
7.Adjoint systems 62
8.Higher order linear equations 63
9.Remarks on changes of variables 68
APPENDIX.ANALYTIC LINEAR EQUATIONS 70
10.Fundamental matrices 70
11.Simple singularities 73
12.Higher order equations 84
13.A nonsimple singularity 87
Notes 91
Ⅴ.Dependence on initial conditions and parameters 93
1.Preliminaries 93
2.Continuity 94
3.Differentiability 95
4.Higher order differentiability 100
5.Exterior derivatives 101
6.Another differentiability theorem 104
7.S- and L-Lipschitz continuity 107
8.Uniqueness theorem 109
9.A lemma 110
10.Proof of Theorem 8.1 111
11.Proof of Theorem 6.1 113
12.First integrals 114
Notes 116
Ⅵ.Total and partial differential equations 117
PART Ⅰ.A THEOREM OF FROBENIUS 117
1.Total differential equations 117
2.Algebra of exterior forms 120
3.A theorem of Frobenius 122
4.Proof of Theorem 3.1 124
5.Proof of Lemma 3.1 127
6.The system （1.1） 128
PART Ⅱ.CAUCHY'S METHOD OF CHARACTERISTICS 131
7.A nonlinear partial differential equation 131
8.Characteristics 135
9.Existence and uniqueness theorem 137
10.Haar's lemma and uniqueness 139
Notes 142
Ⅶ.The Poincare-Bendixson theory 144
1.Autonomous systems 144
2.Umlaufsatz 146
3.Index of a stationary point 149
4.The Poincare-Bendixson theorem 151
5.Stability of periodic solutions 156
6.Rotation points 158
7.Foci, nodes, and saddle points 160
8.Sectors 161
9.The general stationary point 166
10.A second order equation 174
APPENDIX.POINCARE-BENDIXSON THEORY ON 2-MANIFOLDS 182
11.Preliminaries 182
12.Analogue of the Poincare-Bendixson theorem 185
13.Flow on a closed curve 190
14.Flow on a torus 195
Notes 201
Ⅷ.Plane stationary points 202
1.Existence theorems 202
2.Characteristic directions 209
3.Perturbed linear systems 212
4.More general stationary point 220
Notes 227
Ⅸ.Invariant manifolds and linearizations 228
1.Invariant manifolds 228
2.The maps Tt 231
3.Modification of F（ξ） 232
4.Normalizations 233
5.Invariant manifolds of a map 234
6.Existence of invariant manifolds 242
7.Linearizations 244
8.Linearization of a map 245
9.Proof of Theorem 7.1 250
10.Periodic solution 251
11.Limit cycles 253
APPENDIX.SMOOTH EQUIVALENCE MAPS 256
12.Smooth linearizations 256
13.Proof of Lemma 12.1 259
14.Proof of Theorem 12.2 261
APPENDIX.SMOOTHNESS OF STABLE MANIFOLDS 271
Notes 271
Ⅹ.Perturbed linear systems 273
1.The case E = 0 273
2.A topological principle 278
3.A theorem of Wazewski 280
4.Preliminary lemmas 283
5.Proof of Lemma 4.1 290
6.Proof of Lemma 4.2 291
7.Proof of Lemma 4.3 292
8.Asymptotic integrations.Logarithmic scale 294
9.Proof of Theorem 8.2 297
10.Proof of Theorem 8.3 299
11.Logarithmic scale （continued） 300
12.Proof of Theorem 11.2 303
13.Asymptotic integration 304
14.Proof of Theorem 13.1 307
15.Proof of Theorem 13.2 310
16.Corollaries and refinements 311
17.Linear higher order equations 314
Notes 320
Ⅺ, Linear second order equations 322
1.Preliminaries 322
2.Basic facts 325
3.Theorems of Sturm 333
4.Sturm-Liouville boundary value problems 337
5.Number of zeros 344
6.Nonoscillatory equations and principal solutions 350
7.Nonoscillation theorems 362
8.Asymptotic integrations.Elliptic cases 369
9.Asymptotic integrations.Nonelliptic cases 375
APPENDIX.DISCONJUGATE SYSTEMS 384
10.Disconjugate systems 384
11.Generalizations 396
Notes 401
Ⅻ.Use of implicit function and fixed point theorems 404
PART Ⅰ.PERIODIC SOLUTIONS 407
1.Linear equations 407
2.Nonlinear problems 412
PART Ⅱ.SECOND ORDER BOUNDARY VALUE PROBLEMS 418
3.Linear problems 418
4.Nonlinear problems 422
5.A priori bounds 428
PART Ⅲ.GENERAL THEORY 435
6.Basic facts 435
7.Green's functions 439
8.Nonlinear equations 441
9.Asymptotic integration 445
Notes 447
ⅩⅢ.Dichotomies for solutions of linear equations 450
PART Ⅰ.GENERAL THEORY 451
1.Notations and definitions 451
2.Preliminary lemmas 455
3.The operator T 461
4.Slices of ||Py（t）|| 465
5.Estimates for ||y（t）|| 470
6.Applications to first order systems 474
7.Applications to higher order systems 478
8.P（B, D）-manifolds 483
PART Ⅱ.ADJOINT EQUATIONS 484
9.Associate spaces 484
10.The operator T' 486
11.Individual dichotomies 486
12.P'-admissible spaces for T' 490
13.Applications to differential equations 493
14.Existence of PD-solutions 497
Notes 498
ⅩⅣ.Miscellany on monotony 500
PART Ⅰ.MONOTONE SOLUTIONS 500
1.Small and large solutions 500
2.Monotone solutions 506
3.Second order linear equations 510
4.Second order linear equations （continuation） 515
PART Ⅱ.A PROBLEM IN BOUNDARY LAYER THEORY 519
5.The problem 519
6.The case λ ＞ 0 520
7.The case λ ＜ 0 525
8.The case λ = 0 531
9.Asymptotic behavior 534
PART Ⅲ.GLOBAL ASYMPTOTIC STABILITY 537
10.Global asymptotic stability 537
11.Lyapunov functions 539
12.Nonconstant G 540
13.On Corollary 11.2 545
14.On “J（y）x · x ≤ 0 if x · f （y） = 0” 548
15.Proof of Theorem 14.2 550
16.Proof of Theorem 14.1 554
Notes 554
HINTS FOR EXERCISES 557
REFERENCES 581
INDEX 607