Ordinary differential equations / Philip Hartman.

Format:

Book

Author/Creator:

Hartman, Philip, 1915-

Contributor:

Phi Beta Kappa Library Trust Fund.

Series:

Classics in applied mathematics ; 38.

Classics in applied mathematics ; 38

Language:

English

Subjects (All):

Differential equations.

Physical Description:

xx, 612 pages : illustrations ; 24 cm.

Place of Publication:

Philadelphia : Society for Industrial and Applied Mathematics, 2002.

Contents:

2. Basic theorems 2

3. Smooth approximations 6

4. Change of integration variables 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

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

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

V. Dependence on initial conditions and parameters 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

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

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

12. Analogue of the Poincare-Bendixson theorem 185

13. Flow on a closed curve 190

14. Flow on a torus 195

VIII. Plane stationary points 202

1. Existence theorems 202

2. Characteristic directions 209

3. Perturbed linear systems 212

4. More general stationary point 220

IX. Invariant manifolds and linearlizations 228

1. Invariant manifolds 228

2. The maps T[superscript t] 231

3. Modification of F([xi]) 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

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

XI. Linear second order equations 322

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

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

XIII. Dichotomies for solutions of linear equations 450

Part I. General theory 451

2. Preliminary lemmas 455

3. The operator T 461

4. Slices of [double vertical line]Py(t)[double vertical line] 465

5. Estimates for [double vertical line]y(t)[double vertical line] 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 P D-solutions 497

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

6. The case [lambda] > 0 520

7. The case [lambda] < 0 525

8. The case [lambda] = 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 [less than or equal] 0 if x . f(y) = 0" 548

15. Proof of Theorem 14.2 550

16. Proof of Theorem 14.1 554.

Notes:

Previously published: 2nd ed. Boston : Birkhäuser, 1982. Originally published: Baltimore, Md., 1973.

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Phi Beta Kappa Library Trust Fund.

ISBN:

0898715105

OCLC:

48817308