Functional Differential Operators and Equations

By: U.G. Kurbatov

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Hardcover | 30 April 1999

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456 Pages

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23.5 x 15.88 x 2.54

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This book deals with linear functional differential equations and operator theory methods for their investigation.
The main topics are: the equivalence of the input-output stability of the equation Lx = &mathsf; and the invertibility of the operator L in the class of casual operators; the equivalence of input-output and exponential stability; the equivalence of the dichotomy of solutions for the homogeneous equation Lx = 0 and the invertibility of the operator L; the properties of Green's function; the independence of the stability of an equation from the norm on the space of solutions; shift invariant functional differential equations in Banach space; the possibility of the reduction of an equation of neutral type to an equation of retarded type; special full subalgebras of integral and difference operators, and operators with unbounded memory; and the analogue of Fredholm's alternative for operators with almost periodic coefficients where one-sided invertibility implies two-sided invertibility.
Audience: This monograph will be of interest to students and researchers working in functional differential equations and operator theory and is recommended for graduate level courses.

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U.G. Kurbatov

Functional Differential Operators and Equations

"This book provides a thorough and rigorous presentation of the use of operator theory methods to study the linear theory of functional-differential equations . . . It is a practically self-contained book by means of which students and researchers working in functional analysis can be introduced to the abstract linear theory of functional-differential equations."--MATHEMATICAL REVIEWS

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Preface
Functional analysis preliminaries	p. 1
Banach spaces and bounded linear operators	p. 1
Diagrams	p. 7
Lower norms	p. 14
Banach algebras	p. 20
Spaces of continuous and measurable functions	p. 31
Infinite matrices	p. 44
Tensor products	p. 61
Conjugate spaces	p. 81
The initial value problem	p. 97
The causal invertibility	p. 98
The causal spectrum	p. 106
Spaces of smooth functions and distributions	p. 112
The unique solubility	p. 127
The evolutionary solubility	p. 136
Criteria for evolutionary solubility	p. 140
Stability	p. 150
Algebraic preliminaries	p. 150
Input-output stability: discrete time	p. 157
Input-output stability: continuous time	p. 162
Exponential stability: discrete time	p. 171
Exponential stability: continuous time	p. 179
Exponential dichotomy: continuous time	p. 196
Shift invariant operators and equations	p. 209
Algebras of bounded measures	p. 209
The Fourier transform	p. 221
The Laplace transform	p. 239
Convolution operators	p. 247
Invertibility and causal invertibility	p. 267
Operators with varying coefficients	p. 285
Multiplication operators	p. 285
Difference operators	p. 296
Smoothing operators	p. 307
Integral operators	p. 315
Operators with locally fading memory	p. 322
Operators with continuous coefficients	p. 333
Differential difference equations	p. 343
The local Fredholm property	p. 343
The solubility for derivatives	p. 355
The independence from the choice of a norm	p. 366
Green's function	p. 372
Almost periodic operators	p. 382
Comments	p. 403
Bibliography	p. 411
Index	p. 429
Notation Index	p. 433
Table of Contents provided by Blackwell. All Rights Reserved.

Functional Differential Operators and Equations

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Hardcover

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