Basic topological structures of ordinary differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
1998
|
| Series: | Mathematics and its applications
432 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 516 S. graph. Darst. |
| ISBN: | 0792349512 |
Staff View
MARC
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| 100 | 1 | |a Filippov, V. V. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Basic topological structures of ordinary differential equations |c by V. V. Filippov |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 1998 | |
| 300 | |a XIV, 516 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Mathematics and its applications |v 432 | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Differential inclusions | |
| 650 | 4 | |a Topology | |
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Record in the Search Index
| _version_ | 1804127085897187328 |
|---|---|
| adam_text | Table of Contents
PREFACE xi
1 TOPOLOGICAL AND METRIC SPACES 1
1 Sets, mappings 1
2 Topological spaces 8
3 Some notions related to a topology. 12
4 Metric spaces 14
5 Limits 17
6 Compactness and completeness 23
7 Continuous mappings 27
8 Zorn s lemma and well ordered sets 33
2 SOME PROPERTIES OF TOPOLOGICAL, METRIC AND
EUCLIDEAN SPACES 37
1 Tychonoff topology of products 37
2 Continuous bijections. Homeomorphisms. Metrizability. . . 42
3 Some properties of a metric 46
4 Some properties of Euclidean, locally compact, and normed
spaces 48
5 Remarks on multi valued mappings 54
6 Connectedness 56
7 Plane regions 61
8 Degree of a mapping of a circle into itself. 63
3 SPACES OF MAPPINGS AND SPACES OF COMPACT
SUBSETS 71
1 Metric and norm of uniform convergence 71
2 Compact open topology 73
3 Vietoris topology 77
4 Hausdorff metric 79
5 Space of partial mappings 83
6 Compactness in the space of partial mappings 87
7 Space C,{M) 89
v
vi
8 Alexander s lemma and its consequences 92
4 DERIVATION AND INTEGRATION 95
1 Measure of a set on the line 95
2 Measurable functions 104
3 Derivation of non decreasing functions 108
4 Derivative with respect to a set 116
5 Absolutely continuous functions 118
6 Derivation of absolutely continuous functions 123
7 Lebesgue integral 125
8 Density points and approximate derivatives 138
9 Generalized absolutely continuous functions and the Denjoy
integral 139
10 Mean Value theorem 144
11 Change of variable in an integral and derivation of a
composite function 145
12 Measurable multi valued mappings 152
13 Multi valued mappings defined on products 156
5 WEAK TOPOLOGY ON THE SPACE Lx AND
DERIVATION OF CONVERGENT SEQUENCES 163
1 Continuous linear functionals 163
2 Hahn Banach theorem 166
3 Convex sets and linear functionals 169
4 Weakly convergent sequences 171
5 Spaces Lx and L^ 173
6 Common representation of a linear functional on the space
U 175
7 Space of measurable sets 179
8 Weak compactness in the space Li 180
9 Derivation of a convergent equi absolutely continuous sequence
of functions 189
6 BASIC PROPERTIES OF SOLUTION SPACES 191
1 Ordinary differential equations 191
2 Solutions of differential inclusion y € F(t, y) 192
3 The inequality y {t) ^ p(t) 195
4 Some properties of solution spaces 197
5 Cauchy problem. Existence and uniqueness theorems. . . . 199
6 Maximal extensions of solutions 204
7 Continuity of the dependence of solutions on initial values. 207
vii
7 CONVERGENT SEQUENCES OF SOLUTION
SPACES 209
1 Continuity of the dependence of solutions of equations on
parameters of right hand sides 209
2 Properties of convergent space sequences 215
3 Equicontinuity condition 219
4 Estimates of set location 232
5 Estimates of the upper limit of a space sequence 235
6 Additional remarks 242
7 R(U) as a topological space 243
8 PEANO, CARATHEODORY AND DAVY
CONDITIONS 247
1 Kneser condition 247
2 Caratheodory and Davy conditions 253
3 Localization principle 262
4 Absolutely continuous solutions 264
5 Continuously differentiate solutions. Peano and Picard
conditions 266
6 General approach to existence theorems 269
7 Majorants of right hand sides 271
9 COMPARISON THEOREM 277
1 Existence of upper solutions in the scalar case 278
2 Scorza Dragoni property. 280
3 Comparison theorem: scalar case 282
4 Comparison theorem: n dimensional case 284
5 Localness of the property in question 288
10 CHANGES OF VARIABLES, MORPHISMS AND
MAXIMAL EXTENSIONS 291
1 Change of variables: general remarks 291
2 Change of variables in equations and inclusions 295
3 Maximal extensions. Some generalizations 304
4 Convergent space sequences again 309
5 Morphisms 313
6 One more remark about the continuity of the dependence
of solutions on the right hand side 316
7 Uniqueness theorem 321
viii
11 SOME METHODS OF INVESTIGATION
OF EQUATIONS 327
1 Linear equations (systems) 327
2 Linear equation of order n 334
3 Modification of the right hand side. Equations with discontinuous
right hand sides 335
4 Simplest singularities. Existence of solutions 339
5 Collage of spaces 342
6 Simplest singularities. Approximations 344
7 Optimization 350
8 Control 352
12 EQUATIONS AND INCLUSIONS WITH COMPLICATED
DISCONTINUITIES IN THE SPACE VARIABLES 357
1 Sufficient conditions for equi absolute continuity 358
2 Continuity of the dependence of solutions on the right hand
side. First step 368
3 Existence theorems. First step 373
4 Continuity of the dependence of solutions on the right hand
side. Second step: complication of singularities 378
5 Existence theorem. Second step 384
13 EQUATIONS AND INCLUSIONS OF SECOND ORDER.
CAUCHY PROBLEM THEORY 389
1 Cauchy problem theory for differential inclusions of second
order 389
2 Relatively weakly compact families of majorants 394
3 Equation x + f(x)x + h(t, x) = 0 402
14 EQUATIONS AND INCLUSIONS OF SECOND ORDER.
PERIODIC SOLUTIONS, DIRICHLET PROBLEM 409
1 First remark on existence of periodic solutions 409
2 Second remark on existence of periodic solutions 415
3 Upper and lower solutions of the inclusion x G F(t, x). . . 431
15 BEHAVIOR OF SOLUTIONS 441
1 Autonomous and asymptotically autonomous spaces. . . . 441
2 Limit sets 446
3 Some geometric properties of solution spaces 449
4 Periodic spaces 452
5 Limit passages in the space RC(U) 453
6 First approximation. Asymptotic stability of a stationary
point 457
ix
7 Asymptotic estimates 460
8 Behavior of a solution space as ||y|| —» oo 466
16 TWO DIMENSIONAL SYSTEMS 469
1 Existence of a stationary point 469
2 Indices 480
3 Calculation of index by the right hand side 486
4 Poincare Bendixson theorem 493
5 Neighborhoods of stationary points in the plane 500
REFERENCES 507
INDEX 510
NOTATION 515
|
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| id | DE-604.BV012447739 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:27:45Z |
| institution | BVB |
| isbn | 0792349512 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008446971 |
| oclc_num | 38144349 |
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| owner_facet | DE-824 DE-703 DE-11 |
| physical | XIV, 516 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Kluwer Acad. Publ. |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Filippov, V. V. Verfasser aut Basic topological structures of ordinary differential equations by V. V. Filippov Dordrecht [u.a.] Kluwer Acad. Publ. 1998 XIV, 516 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 432 Differential equations Differential inclusions Topology Topologische Struktur (DE-588)4332615-8 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Topologische Struktur (DE-588)4332615-8 s DE-604 Mathematics and its applications 432 (DE-604)BV008163334 432 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008446971&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Filippov, V. V. Basic topological structures of ordinary differential equations Mathematics and its applications Differential equations Differential inclusions Topology Topologische Struktur (DE-588)4332615-8 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4332615-8 (DE-588)4020929-5 |
| title | Basic topological structures of ordinary differential equations |
| title_auth | Basic topological structures of ordinary differential equations |
| title_exact_search | Basic topological structures of ordinary differential equations |
| title_full | Basic topological structures of ordinary differential equations by V. V. Filippov |
| title_fullStr | Basic topological structures of ordinary differential equations by V. V. Filippov |
| title_full_unstemmed | Basic topological structures of ordinary differential equations by V. V. Filippov |
| title_short | Basic topological structures of ordinary differential equations |
| title_sort | basic topological structures of ordinary differential equations |
| topic | Differential equations Differential inclusions Topology Topologische Struktur (DE-588)4332615-8 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Differential equations Differential inclusions Topology Topologische Struktur Gewöhnliche Differentialgleichung |
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