Ordinary differential equations in Rn: problems and methods
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| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Book |
| Language: | English Italian |
| Published: |
New York ; Berlin ; Heidelberg ; Tokyo
Springer-Verlag
1984
|
| Series: | Applied mathematical sciences
39 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | EST: Equazioni differenziali ordinarie in Rn <engl.>. - Aus d. Ital. übers. |
| Physical Description: | xii, 385 Seiten |
| ISBN: | 0387907238 3540907238 |
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MARC
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| 245 | 1 | 0 | |a Ordinary differential equations in Rn |b problems and methods |c L. C. Piccinini ; G. Stampacchia ; G. Vidossich ; translated by A. LoBello |
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Record in the Search Index
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| adam_text | Contents
Page
CHAPTER I EXISTENCE AND UNIQUENESS FOR THE INITIAL VALUE PROBLEM
UNDER THE HYPOTHESIS OF LIPSCHITZ 1
1. General Results 1
1.1 Definitions 1
1.2 Geometrical Interpretation 3
1.3 Functions Satisfying a Lipschitz Condition 4
1.4 Existence Theorem 5
1.5 Uniqueness Theorem 10
1.6 Continuous Dependence on Initial Conditions
and Parameters 12
1.7 Interval of Definition and Extension of
Solutions 13
1.8 Gronwall s Lemma 17
1.9 Application of Gronwall s Lemma to the
Cauchy Problem 19
2. Qualitative Properties of Solutions 23
2.1 Differentiability of Solutions 23
2.2 Analyticity of the Solutions 24
3. Solutions as Functions of the Initial Data 29
3.1 Differentiability with Respect to the
Parameter 30
3.2 Differentiability with Respect to the Initial
Point 32
3.3 Higher Order Differentiability and Analyticity 39
3.4 Remark about a More General Point of View 40
4. Systems of Equations as Particular Transformations
Between Function Spaces 40
4.1 Review of Metric Spaces 41
4.2 Review of Banach Spaces 47
4.3 The Cauchy Problem and Fixed Points of Certain
Transformations in Banach Spaces 52
5. Exercises 56
5.1 Variables Separable Equations 56
5.2 Equations Reducible to Separable Equations 57
5.3 Linear Equations of the First Order 59
5.4 Linear Equations of Order Higher than the First
with Constant Coefficients 59
5.5 Euler Equations 61
5.6 Envelopes and Differential Equations 61
5.7 Various Exercises 63
5.8 Selected Exercises 64
6. Bibliographical Notes 75
ix
Page
CHAPTER II LINEAR SYSTEMS 79
1. Elements of Linear Algebra 79
1.1 Matrices and Eigenvalues 79
1.2 Linear Operators Between Banach Spaces 84
1.3 Canonical Form of Matrices 88
1.4 Spectrum and Eigenvalues of a Linear
Operator 93
1.5 Limits of Operators 94
2. Linear Systems of Ordinary Differential Equations 97
2.1 Formal Solution of Linear Systems 97
2.2 Fundamental Systems of Solutions and
Adjoint Systems 99
2.3 Nonhomogeneous Systems 102
3. Operational Calculus 103
3.1 Analytic Functions of Operators 103
3.2 Linear Systems with Constant Coefficients 110
4. Linear Finite Differences Equations 118
4.1 Homogeneous Linear Finite Differences
Equations 118
4.2 Nonhomogeneous Linear Finite Differences
Equations 124
5. Examples 124
6. Bibliography 131
CHAPTER III EXISTENCE AND UNIQUENESS FOR THE CAUCHY PROBLEM
UNDER THE CONDITION OF CONTINUITY 132
1. Existence Theorem 133
1.1 Characterization of Compact Sets of
Continuous Functions: Ascoli s Theorem 133
1.2 Local Existence 140
1.3 Global Existence 146
2. The Peano Phenomenon 155
2.1 Approximation of all Solutions to a Given
Cauchy Problem 156
2.2 Maximal and Minimal Solutions. The
Peano Phenomenon 159
2.3 The Peano Phenomenon for Systems 162
2.4 Maximal Solutions, Differential Inequalities,
and Global Existence 168
3. Questions of Uniqueness 171
3.1 Continuous Dependence 171
3.2 Uniqueness Theorems 178
3.3 How Many Differential Equations Have the
Uniqueness Property? 183
4. Elements of G Convergence 187
4.1 Introduction 187
4.2 G Convergence for Equations Satisfying the
Lipschitz Condition 187
4.3 Homogenization 189
4.4 G Compactness 196
4.5 G Convergence and the Peano Phenomenon 198
5. Bibliographical Notes 200
Page
CHAPTER IV BOUNDARY VALUE PROBLEMS 207
1. Continuous Mappings on Euclidean Spaces 207
1.1 The Topological Degree 208
1.2 The Theorems of Brouwer and Miranda 213
2. Geometric Boundary Value Problems 217
2.1 The Boundary Value Problems of Picard and
Nicoletti 217
2.2 A Geometrical Formulation of the Boundary
Value Problem 224
2.3 Some Applications of the Geometric
Formulation 233
3. Sturm Liouville Problems: Eigenvalues and
Existence and Uniqueness Theorems 236
3.1 Eigenvalues and Eigenfunctions 236
3.2 Priifer s Change of Variables 239
3.3 Existence and Properties of the Eigenvalues 247
3.4 Applications to Questions of Uniqueness for
Problems Involving Nonlinear Equations 251
3.5 Application to the Existence of Solutions
for Problems Involving Nonlinear Equations 254
3.6 Further Properties of Eigenvalues and
Eigenfunctions 261
4. Periodic Solutions 265
4.1 The Case of First Order Equations 265
4.2 The Case of Second Order Equations 268
4.3 The Case of Systems 272
4.4 On the Structure of Periodic Solutions 279
5. Functional Boundary Value Problems 284
5.1 Linear Functional Problems 286
5.2 Nonlinear Functional Problems 296
6. Bibliographical Notes 299
CHAPTER V QUESTIONS OF STABILITY 311
1. Stability of the Solutions of Linear Systems 312
1.1 Definition of Stability 312
1.2 Stability for Autonomous Linear Systems 314
1.3 Autonomous Linear Systems of the Second Order 317
1.4 Certain Stability Problems for Nonautonomous
Linear Systems 325
2. Some Methods for the Determination of the Stability
of Nonlinear Systems 332
2.1 Definitions 332
2.2 Liapunov s Method 337
2.3 The Fixed Point Method: Asymptotic
Equivalence 346
2.4 Olech s Method 352
2.5 The Method of the Logarithmic Norm 358
2.6 Invariant Sets 361
3. Some Applications 365
3.1 Problems in Biology and Chemistry 365
3.2 Problems in Automatic Control Theory 370
Page
CHAPTER V (cont.)
4. The Method of Runge and Kutta 373
4.1 The Fourth Order Runge Kutta Algorithm 374
4.2 Practical Use of the Runge Kutta Method 376
5. Bibliographical Notes 378
INDEX 382
|
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| author | Piccinini, Livio Clemente 1943- Stampacchia, Guido 1922-1978 Vidossich, Giovanni |
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| isbn | 0387907238 3540907238 |
| language | English Italian |
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| physical | xii, 385 Seiten |
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| spelling | Piccinini, Livio Clemente 1943- (DE-588)1130662314 aut Equazioni differenziali ordinarie in Rn Ordinary differential equations in Rn problems and methods L. C. Piccinini ; G. Stampacchia ; G. Vidossich ; translated by A. LoBello New York ; Berlin ; Heidelberg ; Tokyo Springer-Verlag 1984 xii, 385 Seiten txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 39 EST: Equazioni differenziali ordinarie in Rn <engl.>. - Aus d. Ital. übers. Ecuaciones diferenciales Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s Differentialgleichung (DE-588)4012249-9 s Stampacchia, Guido 1922-1978 (DE-588)1089367198 aut Vidossich, Giovanni aut Lo Bello, Anthony 1947- (DE-588)129030880 trl Erscheint auch als Online-Ausgabe 978-1-4612-5188-0 Applied mathematical sciences 39 (DE-604)BV000005274 39 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000143536&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Piccinini, Livio Clemente 1943- Stampacchia, Guido 1922-1978 Vidossich, Giovanni Ordinary differential equations in Rn problems and methods Applied mathematical sciences Ecuaciones diferenciales Differentialgleichung (DE-588)4012249-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4044779-0 (DE-588)4020929-5 |
| title | Ordinary differential equations in Rn problems and methods |
| title_alt | Equazioni differenziali ordinarie in Rn |
| title_auth | Ordinary differential equations in Rn problems and methods |
| title_exact_search | Ordinary differential equations in Rn problems and methods |
| title_full | Ordinary differential equations in Rn problems and methods L. C. Piccinini ; G. Stampacchia ; G. Vidossich ; translated by A. LoBello |
| title_fullStr | Ordinary differential equations in Rn problems and methods L. C. Piccinini ; G. Stampacchia ; G. Vidossich ; translated by A. LoBello |
| title_full_unstemmed | Ordinary differential equations in Rn problems and methods L. C. Piccinini ; G. Stampacchia ; G. Vidossich ; translated by A. LoBello |
| title_short | Ordinary differential equations in Rn |
| title_sort | ordinary differential equations in rn problems and methods |
| title_sub | problems and methods |
| topic | Ecuaciones diferenciales Differentialgleichung (DE-588)4012249-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Ecuaciones diferenciales Differentialgleichung Partielle Differentialgleichung Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000143536&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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