Topics On Stability And Periodicity In Abstract Differential Equations:
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2008
|
| Series: | Series on concrete and applicable mathematics
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 208 S. |
| ISBN: | 9789812818232 |
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Record in the Search Index
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| adam_text | Titel: Topics on stability and periodicity in abstract differential equations
Autor: Liu, James Hetao
Jahr: 2008
Contents
Preface v
1. Preliminaries 1
1.1 Banach Spaces and Linear Operators............ 1
1.1.1 Banach Spaces.................... 1
1.1.2 Linear Operators .................. 2
1.1.3 Spectral Theory of Linear (Closed) Operators . . 3
1.2 Strongly Continuous Semigroups of Operators....... 7
1.2.1 Definition and Basic Properties.......... 7 •
1.2.2 Compact Semigroups and Analytic Strongly Con-
tinuous Semigroups................. 12
1.2.3 Spectral Mapping Theorems............ 14
1.2.4 Commuting Operators ............... 17
1.3 Spectral Theory........................ 19
1.3.1 Introduction..................... 19
1.3.2 Spectrum of a Bounded Function ......... 19
1.3.3 Uniform Spectrum of a Bounded Function .... 24
1.3.4 Almost Periodic Functions............. 26
1.3.5 Sprectrum of an Almost Periodic Function .... 29
1.3.6 A Spectral Criterion for Almost Periodicity of a
Function....................... 30
1.3.7 Almost Automorphic Functions.......... 31
2. Stability and Exponential Dichotomy 39
2.1 Perron Theorem....................... 39
2.2 Evolution Semigroups and Perron Theorem........ 47
viii Topics on Stability and Periodicity in Abstract Differential Equations
2.3 Stability Theory....................... 51
2.3.1 Exponential Stability................ 51
2.3.2 Strong Stability................... 54
2.4 Comments and Further Reading Guide........... 58
2.4.1 Further Reading Guide............... 58
2.4.2 Comments...................... 59
3. Almost Periodic Solutions 61
3.1 Evolution Semigroups Periodic Equations........ 61
3.1.1 An Example..................... 61
3.1.2 Evolution Semigroups................ 63
3.1.3 The Finite Dimensional Case............ 64
3.1.4 The Infinite Demensional Case........... 65
3.1.5 Almost Periodic Solutions and Applications ... 68
3.2 Sums of Commuting operators............... 81
3.2.1 Invariant Function Spaces ............. 81
3.2.2 Differential Operator d/dt — A and Notions of Ad-
missibility...................... 83
3.2.3 Admissibility for Abstract Ordinary Differential
Equations...................... 86
3.2.4 Higher Order Differential Equations........ 89
3.2.5 Abstract Functional Differential Equations .... 96
3.2.6 Examples and Applications............. 98
3.3 Decomposition Theorem .................. 103
3.3.1 Spectral Decomposition............... 107
3.3.2 Spectral Criteria For Almost Periodic Solutions . 114
3.4 Comments and Further Reading Guide........... 118
3.4.1 Further Reading Guide............... 118
3.4.2 Comments...................... 119
4. Almost Automorphic Solutions 121
4.1 The Inhomogeneous Linear Equation............ 121
4.2 Method of Invariant Subspaces and Almost Automorphic
Solutions of Second-Order Differential Equations..... 127
4.3 Existence of Almost Automorphic Solutions to Semilinear
Differential Equations.................... 131
4.4 Method of Sums of Commuting Operators and Almost Au-
tomorphic Functions..................... 135
Contents ix
4.5 Almost Automorphic Solutions of Second Order Evolution
Equations........................... 139
4.5.1 Mild Solutions of Inhomogeneous Second Order
Equations ...................... 140
4.5.2 Operators A..................... 141
4.5.3 Nonlinear Equations................. 145
4.6 The Equations x =f(t,x)................... 146
4.7 Comments and Further Reading Guide........... 151
5. Nonlinear equations 153
5.1 Periodic Solutions of Nonlinear equations......... 153
5.1.1 Nonlinear Equations Without Delay........ 153
5.1.2 Nonlinear Equations With Finite Delay...... 162
5.1.3 Nonlinear Equations With Infinite Delay..... 166
5.1.4 Non-Densely Defined Equations.......... 180
5.2 Evolution Semigroups and Almost Periodic Solutions . . . 183
5.2.1 Evolution Semigroups................ 183
5.2.2 Almost periodic solutions.............. 186
5.3 Comments and Further Reading Guide........... 190
5.3.1 Further Reading Guide............... 190
5.3.2 Comments...................... 191
Appendix 193
A.I Lipschitz Operators ..................... 193
A.2 Fixed Point Theorems.................... 195
A.3 Invariant Subspaces ..................... 197
A.4 Semilinear Evolution Equations............... 198
Bibliography 201
Index 207
|
| adam_txt |
Titel: Topics on stability and periodicity in abstract differential equations
Autor: Liu, James Hetao
Jahr: 2008
Contents
Preface v
1. Preliminaries 1
1.1 Banach Spaces and Linear Operators. 1
1.1.1 Banach Spaces. 1
1.1.2 Linear Operators . 2
1.1.3 Spectral Theory of Linear (Closed) Operators . . 3
1.2 Strongly Continuous Semigroups of Operators. 7
1.2.1 Definition and Basic Properties. 7 •
1.2.2 Compact Semigroups and Analytic Strongly Con-
tinuous Semigroups. 12
1.2.3 Spectral Mapping Theorems. 14
1.2.4 Commuting Operators . 17
1.3 Spectral Theory. 19
1.3.1 Introduction. 19
1.3.2 Spectrum of a Bounded Function . 19
1.3.3 Uniform Spectrum of a Bounded Function . 24
1.3.4 Almost Periodic Functions. 26
1.3.5 Sprectrum of an Almost Periodic Function . 29
1.3.6 A Spectral Criterion for Almost Periodicity of a
Function. 30
1.3.7 Almost Automorphic Functions. 31
2. Stability and Exponential Dichotomy 39
2.1 Perron Theorem. 39
2.2 Evolution Semigroups and Perron Theorem. 47
viii Topics on Stability and Periodicity in Abstract Differential Equations
2.3 Stability Theory. 51
2.3.1 Exponential Stability. 51
2.3.2 Strong Stability. 54
2.4 Comments and Further Reading Guide. 58
2.4.1 Further Reading Guide. 58
2.4.2 Comments. 59
3. Almost Periodic Solutions 61
3.1 Evolution Semigroups Periodic Equations. 61
3.1.1 An Example. 61
3.1.2 Evolution Semigroups. 63
3.1.3 The Finite Dimensional Case. 64
3.1.4 The Infinite Demensional Case. 65
3.1.5 Almost Periodic Solutions and Applications . 68
3.2 Sums of Commuting operators. 81
3.2.1 Invariant Function Spaces . 81
3.2.2 Differential Operator d/dt — A and Notions of Ad-
missibility. 83
3.2.3 Admissibility for Abstract Ordinary Differential
Equations. 86
3.2.4 Higher Order Differential Equations. 89
3.2.5 Abstract Functional Differential Equations . 96
3.2.6 Examples and Applications. 98
3.3 Decomposition Theorem . 103
3.3.1 Spectral Decomposition. 107
3.3.2 Spectral Criteria For Almost Periodic Solutions . 114
3.4 Comments and Further Reading Guide. 118
3.4.1 Further Reading Guide. 118
3.4.2 Comments. 119
4. Almost Automorphic Solutions 121
4.1 The Inhomogeneous Linear Equation. 121
4.2 Method of Invariant Subspaces and Almost Automorphic
Solutions of Second-Order Differential Equations. 127
4.3 Existence of Almost Automorphic Solutions to Semilinear
Differential Equations. 131
4.4 Method of Sums of Commuting Operators and Almost Au-
tomorphic Functions. 135
Contents ix
4.5 Almost Automorphic Solutions of Second Order Evolution
Equations. 139
4.5.1 Mild Solutions of Inhomogeneous Second Order
Equations . 140
4.5.2 Operators A. 141
4.5.3 Nonlinear Equations. 145
4.6 The Equations x'=f(t,x). 146
4.7 Comments and Further Reading Guide. 151
5. Nonlinear equations 153
5.1 Periodic Solutions of Nonlinear equations. 153
5.1.1 Nonlinear Equations Without Delay. 153
5.1.2 Nonlinear Equations With Finite Delay. 162
5.1.3 Nonlinear Equations With Infinite Delay. 166
5.1.4 Non-Densely Defined Equations. 180
5.2 Evolution Semigroups and Almost Periodic Solutions . . . 183
5.2.1 Evolution Semigroups. 183
5.2.2 Almost periodic solutions. 186
5.3 Comments and Further Reading Guide. 190
5.3.1 Further Reading Guide. 190
5.3.2 Comments. 191
Appendix 193
A.I Lipschitz Operators . 193
A.2 Fixed Point Theorems. 195
A.3 Invariant Subspaces . 197
A.4 Semilinear Evolution Equations. 198
Bibliography 201
Index 207 |
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| isbn | 9789812818232 |
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| spelling | Topics On Stability And Periodicity In Abstract Differential Equations James H Liu ... New Jersey [u.a.] World Scientific 2008 IX, 208 S. txt rdacontent n rdamedia nc rdacarrier Series on concrete and applicable mathematics 6 Abstrakte Differentialgleichung (DE-588)4141144-4 gnd rswk-swf Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd rswk-swf Abstrakte Differentialgleichung (DE-588)4141144-4 s Asymptotisches Lösungsverhalten (DE-588)4134367-0 s DE-604 Liu, James H. Sonstige oth Series on concrete and applicable mathematics 6 (DE-604)BV035419910 6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016674310&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Topics On Stability And Periodicity In Abstract Differential Equations Series on concrete and applicable mathematics Abstrakte Differentialgleichung (DE-588)4141144-4 gnd Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd |
| subject_GND | (DE-588)4141144-4 (DE-588)4134367-0 |
| title | Topics On Stability And Periodicity In Abstract Differential Equations |
| title_auth | Topics On Stability And Periodicity In Abstract Differential Equations |
| title_exact_search | Topics On Stability And Periodicity In Abstract Differential Equations |
| title_exact_search_txtP | Topics On Stability And Periodicity In Abstract Differential Equations |
| title_full | Topics On Stability And Periodicity In Abstract Differential Equations James H Liu ... |
| title_fullStr | Topics On Stability And Periodicity In Abstract Differential Equations James H Liu ... |
| title_full_unstemmed | Topics On Stability And Periodicity In Abstract Differential Equations James H Liu ... |
| title_short | Topics On Stability And Periodicity In Abstract Differential Equations |
| title_sort | topics on stability and periodicity in abstract differential equations |
| topic | Abstrakte Differentialgleichung (DE-588)4141144-4 gnd Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd |
| topic_facet | Abstrakte Differentialgleichung Asymptotisches Lösungsverhalten |
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