.

0 (hodnocen0 x )

BK

2nd ed.

Philadelphia : SIAM, c2002

xx, 612 s. : il. ; 23 cm

ISBN 978-0-898715-10-1 (brož.)

Classics in applied mathematics ; 38l

Obsahuje bibliografii na s. 581-606 a rejstřík

000235830

Contents // Foreword to the Classics Edition xv // Preface to the First Edition xvii // Preface to the Second Edition xviii // Errata xix // I. Preliminaries...I // 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-Lindelöf 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 // HI. Differential inequalities and uniqueness...24 // 1. Gronwalls 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. 5- 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 // . 144 // VII. The Poincaré-Bendixson theory ... // 1. Autonomous systems, 144 // 2. Umlaufsatz, 146 // 3. Index of a stationary point, 149 // 4. The Poincaré-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. Poincaré-Bendixson theory on 2-manifolds, 182 // 11. Preliminaries, 182 // 12. Analogue of the Poincaré-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 ?*, 231 // 3. Modification of F(f), 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 L // 273 // X. Perturbed linear systems... // 1. The case Ł = 0, 273 // 2. A topological principle, 278 // 3. A theorem of Wažewski, 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 // ??. Use of implicit function and fixed point theorems ... 404 // Part I. Periodic solutions, 407 // 1. Linear equations, 407 // 2. Nonlinear problems, 412 // Part IL Second order boundary value problems, 418 // 3. Linear problems, 418 // 4. Nonlinear problems, 422 // 5. A priori bounds, 428 // Part HI. General theory, 435 // 6. Basic facts, 435 // 7. Green’s functions, 439 // 8. Nonlinear equations, 441 // 9. Asymptotic integration, 445 Notes, 447 // ??. Dichotomies for solutions of linear equations...450 // Part L General theory, 451 // 1. Notations and definitions, 451 // 2. Preliminary lemmas, 455 // 3. The operator T, 461 // 4. Slices of ||Py(/)||, 465 // 5. Estimates for ||y(f)ll, 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 // XIY. 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 A > 0, 520 // 7. The case A < 0, 525 // 8. The case A = 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 “?(?)? • x  0 if ?: ?/(?) = 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