Theory of Functional Differential Equations, Volume 3, Part 1Since the publication of my lecture notes, Functional Differential Equations in the Applied Mathematical Sciences series, many new developments have occurred. As a consequence, it was decided not to make a few corrections and additions for a second edition of those notes, but to present a more compre hensive theory. The present work attempts to consolidate those elements of the theory which have stabilized and also to include recent directions of research. The following chapters were not discussed in my original notes. Chapter 1 is an elementary presentation of linear differential difference equations with constant coefficients of retarded and neutral type. Chapter 4 develops the recent theory of dissipative systems. Chapter 9 is a new chapter on perturbed systems. Chapter 11 is a new presentation incorporating recent results on the existence of periodic solutions of autonomous equations. Chapter 12 is devoted entirely to neutral equations. Chapter 13 gives an introduction to the global and generic theory. There is also an appendix on the location of the zeros of characteristic polynomials. The remainder of the material has been completely revised and updated with the most significant changes occurring in Chapter 3 on the properties of solutions, Chapter 5 on stability, and Chapter lOon behavior near a periodic orbit. |
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a)da attracts compact sets Ax(t Banach space boundary-value problem bounded set bounded variation Bx(t Chapter characteristic equation characteristic multiplier compact set continuous function continuously differentiable convex Corollary decomposition defined definition derivative difference equations differential difference equations discrete dynamical system discuss eigenvalues Equation 1.1 equivalent existence exponential finite fixed point following result formal adjoint formal adjoint equation functional differential equations Furthermore given Hale hypotheses implies Inequality initial data initial value integral k₁ Lemma Liapunov functionals linear operator linear systems manifold map T(t matrix neighborhood NFDE NFDE(D obtain ordinary differential equations P₁ periodic orbit periodic solution point dissipative precompact proof of Theorem properties RFDE RFDE(ƒ S₂ satisfies Equation scalar Section solution of Equation solution operator subset Suppose t₁ Theorem 3.1 theory uniformly asymptotically stable unique solution variation-of-constants formula w-periodic process x₁