Orlicz type category theorems for functional and differential equations

Józef Myjak

GDML_Books, (1983), p.

Résumé

CONTENTSIntroduction.....................................................................................................................................5  1. General results.........................................................................................................................7  1.1. Residual sets.........................................................................................................................7  1.2. Generic properties of abstract functional equations..............................................................8II. 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..............................................................................................................................26III. Non-expansive mappings in Banach spaces............................................................................28  3.1. Generic properties..............................................................................................................28  3.2. The density result...............................................................................................................30  3.3. Supplementary remarks......................................................................................................32IV. 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......................................................................................................................41V. 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..............................................................................................................................48VI. 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...................................................................................................................57VII. Functional integral equations.................................................................................................58  7.1. Notation and auxiliary lemmas.............................................................................................58  7.2. Category theorems..............................................................................................................61  7.3. Some generalizations..........................................................................................................63VIII. 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.....................................75References..................................................................................................................................79

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Zbl 0523.34070

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