Orlicz type category theorems for functional and differential equations

Myjak, Józef

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Orlicz type category theorems for functional and differential equations

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Józef Myjak

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Rozprawy Matematyczne tom/nr w serii: 206 wydano: 1983

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CONTENTS
Introduction.....................................................................................................................................5
  1. General results.........................................................................................................................7
  1.1. Residual sets.........................................................................................................................7
  1.2. Generic properties of abstract functional equations..............................................................8
II. Differential equations................................................................................................................13
  2.1. Continuous differential equations without existence............................................................13
  2.2 Existence, uniqueness and continuous dependence............................................................18
  2.3. Successive approximations..................................................................................................21
  2.4. Remarks..............................................................................................................................26
III. Non-expansive mappings in Banach spaces............................................................................28
  3.1. Generic properties..............................................................................................................28
  3.2. The density result...............................................................................................................30
  3.3. Supplementary remarks......................................................................................................32
IV. Asymptotic equilibria for accretive operators...........................................................................33
  4.1. Notation and preliminaries...................................................................................................33
  4.2. Lemmas...............................................................................................................................35
  4.3. Category theorem................................................................................................................37
  4.4 Application to the fixed-point theory......................................................................................38
  4.5 Further results......................................................................................................................41
V. Hyperbolic equations................................................................................................................43
  5.1. Notation...............................................................................................................................43
  5.2. Generic property of existence, uniqueness and continuous dependence...........................43
  5.3. Generic property of the convergence of successive approximations...................................45
  5.4. A density theorem................................................................................................................47
  5.5. Remarks..............................................................................................................................48
VI. Generic asymptotic stability.....................................................................................................50
  6.1. Stability and asymptotic stability of stationary equations.....................................................50
  6.2. Stability and asymptotic stability of non-stationary equations..............................................54
  6.3. Stability by the Lyapunov function method..........................................................................55
  6.4. Generic stability...................................................................................................................57
VII. Functional integral equations.................................................................................................58
  7.1. Notation and auxiliary lemmas.............................................................................................58
  7.2. Category theorems..............................................................................................................61
  7.3. Some generalizations..........................................................................................................63
VIII. Functional differential equations............................................................................................64
  8.1. Notation and preliminaries...................................................................................................64
  8.2. Existence of unlimited solutions...........................................................................................65
  8.3. Continuous dependence.....................................................................................................67
  8.4. Existence and uniqueness as a generic property................................................................72
  8.5. Convergence of successive approximations as a generic property.....................................75
References..................................................................................................................................79

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 206

Liczba stron

81

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CCVI

Daty

wydano

1983

Twórcy

autor

Józef Myjak

Bibliografia

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Książka - szczegóły

Orlicz type category theorems for functional and differential equations

Autorzy

Seria

Zawartość

Warianty tytułu

Abstrakty

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Copyright

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Bibliografia

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