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
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A directed-edge-reinforced random walk on graphs is considered. Criteria for the walk to end up in a limit cycle are given. Asymptotic stability of some neural networks is shown.