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.Normal izations， 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
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＝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