Theory of Functional Differential Equations

1 Linear differential difference equations.- 1.1 Differential and difference equations.- 1.2 Retarded differential difference equations.- 1.3 Exponential estimates of x(?, f).- 1.4 The characteristic equation.- 1.5 The fundamental solution.- 1.6 The variation-of-constants formula.- 1.7 Neutral differential difference equations.- 1.8 Supplementary remarks.- 2 Retarded functional differential equations : basic theory.- 2.1 Definition.- 2.2 Existence, uniqueness, and continuous dependence.- 2.3 Continuation of solutions.- 2.4 Differentiability of solutions.- 2.5 Backward continuation.- 2.6 Caratheodory conditions.- 2.7 Supplementary remarks.- 3 Properties of the solution map.- 3.1 Finite- or infinite-dimensional problem?.- 3.2 Equivalence classes of solutions.- 3.3 Exponential decrease for linear systems.- 3.4 Unique backward extensions.- 3.5 Range in ?n.- 3.6 Compactness and representation.- 3.7 Supplementary remarks.- 4 Autonomous and periodic processes.- 4.1 Processes.- 4.2 Invariance.- 4.3 Discrete systems-maximal compact invariant sets.- 4.4 Fixed points of discrete dissipative processes.- 4.5 Stability and maximal invariant sets in processes.- 4.6 Periodic trajectories of ?-periodic processes.- 4.7 Convergent systems.- 4.8 Supplementary remarks.- 5 Stability theory.- 5.1 Definitions.- 5.2 The method of Liapunov functional.- 5.3 Liapunov functional for autonomous systems.- 5.4 Razumikhin-type theorems.- 5.5 Supplementary remarks.- 6 General linear systems.- 6.1 Global existence and exponential estimates.- 6.2 Variation-of-constants formula.- 6.3 The formal adjoint equation.- 6.4 The true adjoint.- 6.5 Boundary-value problems.- 6.6 Stability and boundedness.- 6.7 Supplementary remarks.- 7 Linear autonomous equations.- 7.1 The semigroup and infinitesimal generator.- 7.2 Spectrum of the generator-decomposition of C.- 7.3 Decomposing C with the formal adjoint equation.- 7.4 Estimates on the complementary subspace.- 7.5 An example.- 7.6 The decomposition in the variation-of-constants formula.- 7.7 Supplementary remarks.- 8 Linear periodic systems.- 8.1 General theory.- 8.2 Decomposition.- 8.3 Supplementary remarks.- 9 Perturbed linear systems.- 9.1 Forced linear systems.- 9.2 Bounded, almost-periodic, and periodic solutions; stable and unstable manifolds.- 9.3 Periodic solutions-critical cases.- 9.4 Averaging.- 9.5 Asymptotic behavior.- 9.6 Boundary-value problems.- 9.7 Supplementary remarks.- 10 Behavior near equilibrium and periodic orbits for autonomous equations.- 10.1 The saddle-point property near equilibrium.- 10.2 Nondegenerate periodic orbits.- 10.3 Hyperbolic periodic orbits.- 10.4 Supplementary remarks.- 11 Periodic solutions of autonomous equations.- 11.1 Hopf bifurcation.- 11.2 A periodicity theorem.- 11.3 Range of the period.- 11.4 The equation $$\dot x(t) = - \alpha x(t - 1)[1 + x(t)]$$.- 11.5 The equation $$\dot x(t) = - \alpha x(t - 1)[1 - {x^2}(t)]$$.- 11.6 The equation $$\ddot x(t) + f(x(t))\dot x(t) + g(x(t - r)) = 0$$.- 11.7 Supplementary remarks.- 12 Equations of neutral type.- 12.1 Definition of a neutral equation.- 12.2 Fundamental properties.- 12.3 Linear autonomous D operators.- 12.4 Stable D operators.- 12.5 Strongly stable D operators.- 12.6 Properties of equations with stable D operators.- 12.7 Stability theory.- 12.8 General linear equations.- 12.9 Stability of autonomous perturbed linear systems.- 12.10 Linear autonomous and periodic equations.- 12.11 Nonhomogeneous linear equations.- 12.12 Supplementary remarks.- 13 Global theory.- 13.1 Generic properties of retarded equations.- 13.2 The set of global solutions.- 13.3 Equations on manifolds : definitions.- 13.4 Retraded equations on compact manifolds.- 13.5 Further properties of the attractor.- 13.6 Supplementary remarks.- Appendix Stability of characteristic equations.