Functional Differential Operators and Equations / Edition 1

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Overview

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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Editorial Reviews

From the Publisher
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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Product Details

  • ISBN-13: 9780792356240
  • Publisher: Springer Netherlands
  • Publication date: 4/30/1999
  • Series: Mathematics and Its Applications (closed) Series , #473
  • Edition description: 1999
  • Edition number: 1
  • Pages: 454
  • Product dimensions: 1.00 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface
Ch. I Functional analysis preliminaries 1
1.1 Banach spaces and bounded linear operators 1
1.2 Diagrams 7
1.3 Lower norms 14
1.4 Banach algebras 20
1.5 Spaces of continuous and measurable functions 31
1.6 Infinite matrices 44
1.7 Tensor products 61
1.8 Conjugate spaces 81
Ch. II The initial value problem 97
2.1 The causal invertibility 98
2.2 The causal spectrum 106
2.3 Spaces of smooth functions and distributions 112
2.4 The unique solubility 127
2.5 The evolutionary solubility 136
2.6 Criteria for evolutionary solubility 140
Ch. III Stability 150
3.1 Algebraic preliminaries 150
3.2 Input-output stability: discrete time 157
3.3 Input-output stability: continuous time 162
3.4 Exponential stability: discrete time 171
3.5 Exponential stability: continuous time 179
3.6 Exponential dichotomy: continuous time 196
Ch. IV Shift invariant operators and equations 209
4.1 Algebras of bounded measures 209
4.2 The Fourier transform 221
4.3 The Laplace transform 239
4.4 Convolution operators 247
4.5 Invertibility and causal invertibility 267
Ch. V Operators with varying coefficients 285
5.1 Multiplication operators 285
5.2 Difference operators 296
5.3 Smoothing operators 307
5.4 Integral operators 315
5.5 Operators with locally fading memory 322
5.6 Operators with continuous coefficients 333
Ch. VI Differential difference equations 343
6.1 The local Fredholm property 343
6.2 The solubility for derivatives 355
6.3 The independence from the choice of a norm 366
6.4 Green's function 372
6.5 Almost periodic operators 382
Comments 403
Bibliography 411
Index 429
Notation Index 433
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