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The present book builds upon an earlier work of J. Hale, "Theory of
Func tional Differential Equations" published in 1977. We have
tried to maintain the spirit of that book and have retained
approximately one-third of the material intact. One major change
was a complete new presentation of lin ear systems (Chapters 6 9)
for retarded and neutral functional differential equations. The
theory of dissipative systems (Chapter 4) and global at tractors
was completely revamped as well as the invariant manifold theory
(Chapter 10) near equilibrium points and periodic orbits. A more
complete theory of neutral equations is presented (see Chapters 1,
2, 3, 9, and 10). Chapter 12 is completely new and contains a guide
to active topics of re search. In the sections on supplementary
remarks, we have included many references to recent literature,
but, of course, not nearly all, because the subject is so
extensive. Jack K. Hale Sjoerd M. Verduyn Lunel Contents
Preface............................................................
v Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1. Linear
differential difference equations . . . . . . . . . . . . . . 11 .
. . . . . 1.1 Differential and difference equations. . . . . . . .
. . . . . . . . . . . . 11 . . . . . . . . 1.2 Retarded
differential difference equations. . . . . . . . . . . . . . . . 13
. . . . . . . 1.3 Exponential estimates of x( cents, f) . . . . . .
. . . . . . . . . . . . . . . 15 . . . . . . . . . . 1.4 The
characteristic equation . . . . . . . . . . . . . . . . . . . . . .
. . 17 . . . . . . . . . . . . 1.5 The fundamental solution. . . .
. . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . .
. . 1.6 The variation-of-constants
formula............................. 23 1. 7 Neutral differential
difference equations . . . . . . . . . . . . . . . . . 25 . . . . .
. . 1.8 Supplementary remarks. . . . . . . . . . . . . . . . . . .
. . . . . . . . 34 . . . . . . . . . . . . . 2. Functional
differential equations: Basic theory . . . . . . . . 38 . . 2.1
Definition of a retarded equation. . . . . . . . . . . . . . . . .
. . . . . 38 . . . . . . . . . 2.2 Existence, uniqueness, and
continuous dependence . . . . . . . . . . 39 . . . 2.3 Continuation
of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 44
. . . . . . . . . . . ."
The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.
This monograph presents necessary and sufficient conditions for
completeness of the linear span of eigenvectors and generalized
eigenvectors of operators that admit a characteristic matrix
function in a Banach space setting. Classical conditions for
completeness based on the theory of entire functions are further
developed for this specific class of operators. The classes of
bounded operators that are investigated include trace class and
Hilbert-Schmidt operators, finite rank perturbations of Volterra
operators, infinite Leslie operators, discrete semi-separable
operators, integral operators with semi-separable kernels, and
period maps corresponding to delay differential equations. The
classes of unbounded operators that are investigated appear in a
natural way in the study of infinite dimensional dynamical systems
such as mixed type functional differential equations, age-dependent
population dynamics, and in the analysis of the Markov semigroup
connected to the recently introduced zig-zag process.
This monograph presents necessary and sufficient conditions for
completeness of the linear span of eigenvectors and generalized
eigenvectors of operators that admit a characteristic matrix
function in a Banach space setting. Classical conditions for
completeness based on the theory of entire functions are further
developed for this specific class of operators. The classes of
bounded operators that are investigated include trace class and
Hilbert-Schmidt operators, finite rank perturbations of Volterra
operators, infinite Leslie operators, discrete semi-separable
operators, integral operators with semi-separable kernels, and
period maps corresponding to delay differential equations. The
classes of unbounded operators that are investigated appear in a
natural way in the study of infinite dimensional dynamical systems
such as mixed type functional differential equations, age-dependent
population dynamics, and in the analysis of the Markov semigroup
connected to the recently introduced zig-zag process.
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