《ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION》求取 ⇩

Ⅰ.Preliminaries1

1.Preliminaries1

2.Basic theorems2

3.Smooth approximations6

4.Change of integration variables7

Notes7

Ⅱ.Existence8

1.The Picard-Lindelof theorem8

2.Peano's existence theorem10

3.Extension theorem12

4.H.Kneser's theorem15

5.Example of nonuniqueness18

Notes23

Ⅲ.Differential inequalities and uniqueness24

1.Gronwall's inequality24

2.Maximal and minimal solutions25

3.Right derivatives26

4.Differential inequalities26

5.A theorem of Wintner29

6.Uniqueness theorems31

7.van Kampen's uniqueness theorem35

8.Egress points and Lyapunov functions37

9.Successive approximations40

Notes44

Ⅳ.Linear differential equations45

1.Linear systems45

2.Variation of constants48

3.Reductions to smaller systems49

4.Basic inequalities54

5.Constant coefficients57

6.Floquet theory60

7.Adjoint systems62

8.Higher order linear equations63

9.Remarks on changes of variables68

APPENDIX.ANALYTIC LINEAR EQUATIONS70

10.Fundamental matrices70

11.Simple singularities73

12.Higher order equations84

13.A nonsimple singularity87

Notes91

Ⅴ.Dependence on initial conditions and parameters93

1.Preliminaries93

2.Continuity94

3.Differentiability95

4.Higher order differentiability100

5.Exterior derivatives101

6.Another differentiability theorem104

7.S- and L-Lipschitz continuity107

8.Uniqueness theorem109

9.A lemma110

10.Proof of Theorem 8.1111

11.Proof of Theorem 6.1113

12.First integrals114

Notes116

Ⅵ.Total and partial differential equations117

PART Ⅰ.A THEOREM OF FROBENIUS117

1.Total differential equations117

2.Algebra of exterior forms120

3.A theorem of Frobenius122

4.Proof of Theorem 3.1124

5.Proof of Lemma 3.1127

6.The system （1.1）128

PART Ⅱ.CAUCHY'S METHOD OF CHARACTERISTICS131

7.A nonlinear partial differential equation131

8.Characteristics135

9.Existence and uniqueness theorem137

10.Haar's lemma and uniqueness139

Notes142

Ⅶ.The Poincare-Bendixson theory144

1.Autonomous systems144

2.Umlaufsatz146

3.Index of a stationary point149

4.The Poincare-Bendixson theorem151

5.Stability of periodic solutions156

6.Rotation points158

7.Foci, nodes, and saddle points160

8.Sectors161

9.The general stationary point166

10.A second order equation174

APPENDIX.POINCARE-BENDIXSON THEORY ON 2-MANIFOLDS182

11.Preliminaries182

12.Analogue of the Poincare-Bendixson theorem185

13.Flow on a closed curve190

14.Flow on a torus195

Notes201

Ⅷ.Plane stationary points202

1.Existence theorems202

2.Characteristic directions209

3.Perturbed linear systems212

4.More general stationary point220

Notes227

Ⅸ.Invariant manifolds and linearizations228

1.Invariant manifolds228

2.The maps Tt231

3.Modification of F（ξ）232

4.Normalizations233

5.Invariant manifolds of a map234

6.Existence of invariant manifolds242

7.Linearizations244

8.Linearization of a map245

9.Proof of Theorem 7.1250

10.Periodic solution251

11.Limit cycles253

APPENDIX.SMOOTH EQUIVALENCE MAPS256

12.Smooth linearizations256

13.Proof of Lemma 12.1259

14.Proof of Theorem 12.2261

APPENDIX.SMOOTHNESS OF STABLE MANIFOLDS271

Notes271

Ⅹ.Perturbed linear systems273

1.The case E = 0273

2.A topological principle278

3.A theorem of Wazewski280

4.Preliminary lemmas283

5.Proof of Lemma 4.1290

6.Proof of Lemma 4.2291

7.Proof of Lemma 4.3292

8.Asymptotic integrations.Logarithmic scale294

9.Proof of Theorem 8.2297

10.Proof of Theorem 8.3299

11.Logarithmic scale （continued）300

12.Proof of Theorem 11.2303

13.Asymptotic integration304

14.Proof of Theorem 13.1307

15.Proof of Theorem 13.2310

16.Corollaries and refinements311

17.Linear higher order equations314

Notes320

Ⅺ,Linear second order equations322

1.Preliminaries322

2.Basic facts325

3.Theorems of Sturm333

4.Sturm-Liouville boundary value problems337

5.Number of zeros344

6.Nonoscillatory equations and principal solutions350

7.Nonoscillation theorems362

8.Asymptotic integrations.Elliptic cases369

9.Asymptotic integrations.Nonelliptic cases375

APPENDIX.DISCONJUGATE SYSTEMS384

10.Disconjugate systems384

11.Generalizations396

Notes401

Ⅻ.Use of implicit function and fixed point theorems404

PART Ⅰ.PERIODIC SOLUTIONS407

1.Linear equations407

2.Nonlinear problems412

PART Ⅱ.SECOND ORDER BOUNDARY VALUE PROBLEMS418

3.Linear problems418

4.Nonlinear problems422

5.A priori bounds428

PART Ⅲ.GENERAL THEORY435

6.Basic facts435

7.Green's functions439

8.Nonlinear equations441

9.Asymptotic integration445

Notes447

ⅩⅢ.Dichotomies for solutions of linear equations450

PART Ⅰ.GENERAL THEORY451

1.Notations and definitions451

2.Preliminary lemmas455

3.The operator T461

4.Slices of ||Py（t）||465

5.Estimates for ||y（t）||470

6.Applications to first order systems474

7.Applications to higher order systems478

8.P（B, D）-manifolds483

PART Ⅱ.ADJOINT EQUATIONS484

9.Associate spaces484

10.The operator T'486

11.Individual dichotomies486

12.P'-admissible spaces for T'490

13.Applications to differential equations493

14.Existence of PD-solutions497

Notes498

ⅩⅣ.Miscellany on monotony500

PART Ⅰ.MONOTONE SOLUTIONS500

1.Small and large solutions500

2.Monotone solutions506

3.Second order linear equations510

4.Second order linear equations （continuation）515

PART Ⅱ.A PROBLEM IN BOUNDARY LAYER THEORY519

5.The problem519

6.The case λ ＞ 0520

7.The case λ ＜ 0525

8.The case λ = 0531

9.Asymptotic behavior534

PART Ⅲ.GLOBAL ASYMPTOTIC STABILITY537

10.Global asymptotic stability537

11.Lyapunov functions539

12.Nonconstant G540

13.On Corollary 11.2545

14.On “J（y）x · x ≤ 0 if x · f （y） = 0”548

15.Proof of Theorem 14.2550

16.Proof of Theorem 14.1554

Notes554

HINTS FOR EXERCISES557

REFERENCES581

INDEX607