《Ordinary differential equations》

Ⅰ.Preliminaries1

1.Preliminaries，1

2.Basic theorems，2

3.Smooth approximations，6

4.Change of integration variables，7

Notes，7

Ⅱ.Existence8

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 uniqueness24

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 equations45

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 parameters93

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 equations117

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 theory144

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 points202

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 linearizations228

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 systems273

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 equations322

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 theorems404

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 equations450

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