Author | Hale, Jack K. author |
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Title | Theory of Functional Differential Equations [electronic resource] / by Jack K. Hale |
Imprint | New York, NY : Springer New York, 1977 |
Edition | 2 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-9892-2 |
Descript | X, 366 p. online resource |
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 systemsโmaximal 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 solutionsโcritical 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 $$\dot x(t) = - \alpha x(t - 1)[1 - {x̂2}(t)]$$ -- 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