Difference equations in normed spaces : stability and oscillations / M.I. Gil'.

Format:

Book

Author/Creator:

Gilʹ, M. I. (Mikhail Iosifovich)

Series:

North-Holland mathematics studies ; 206.

North-Holland mathematics studies ; 206

Language:

English

Subjects (All):

Difference equations.

Normed linear spaces.

Physical Description:

1 online resource (379 p.)

Edition:

1st ed.

Place of Publication:

Amsterdam ; Oxford : Elsevier, 2007.

Language Note:

English

Summary:

Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete. They also appear in the applications of discretization methods for differential, integral and integro-differential equations. The application of the theory of difference equations is rapidly increasing to various fields, such as numerical analysis, control theory, finite mathematics, and computer sciences. This book is devoted to linear and nonlinear difference equations in a normed space.Deals systematically with differe

Contents:

Cover; Copyright Page; Preface; Table of Contents; Chapter 1 Definitions and Preliminaries; 1.1 Banach and Hilbert spaces; 1.2 Examples of normed spaces; 1.3 Linear operators; 1.4 Examples of difference equations; 1.5 Stability notions; 1.6 The comparison principle; 1.7 Liapunov functions; 1.8 Ordered spaces and Banach lattices; 1.9 The Abstract Gronwall Lemma; 1.10 Discrete inequalities in a Banach lattice; Chapter 2 Classes of Operators; 2.1 Classification of spectra; 2.2 Compact operators in a Hilbert space; 2.3 Compact matrices; 2.4 Integral operators

Chapter 3 Functions of Finite Matrices3.1 Matrix-valued functions; 3.2 Estimates for the resolvent; 3.3 Examples; 3.4 Estimates for regular matrix functions; 3.5 Proof of Theorem 3.2.4; 3.6 Proofs of Theorems 3.2.1 and 3.2.3; 3.7 Proof of Theorem 3.4.1; 3.8 Non-Euclidean norms of powers of matrices; 3.9 Absolute values of matrix functions; 3.10 Proof of Theorem 3.9.1; Chapter 4 Norm Estimates for Operator Functions; 4.1 Regular operator functions; 4.2 Functions of Hilbert-Schmidt operators; 4.3 Operators with Hilbert-Schmidt powers; 4.4 Resolvents of Neumann-Schatten operators

4.5 Functions of quasi-Hermitian operators4.6 Functions of quasiunitary operators; 4.7 Auxiliary results; 4.8 Equalities for eigenvalues; 4.9 Proofs of Theorems 4.2.1, 4.2.2 and 4.4.1; Chapter 5 Spectrum Perturbations; 5.1 Roots of algebraic equations; 5.2 Roots of functional equations; 5.3 Spectral variations; 5.4 Perturbations of Hilbert-Schmidt operators; 5.5 Perturbations of Neumann - Schatten operators; 5.6 Perturbations of quasi-Hermitian operators; 5.7 Perturbations of finite matrices; Chapter 6 Linear Equations with Constant Operators; 6.1 Homogeneous equations in a Banach space

6.2 Nonhomogeneous equations with constant operators6.3 Perturbations of autonomous equations; 6.4 Equations with Hilbert-Schmidt operators; 6.5 Equations with Neumann-Schatten operators; 6.6 Equations with non-compact operators; 6.7 Equations in finite dimensional spaces; 6.8 Z-transform; 6.9 Exponential dichotomy; 6.10 Equivalent norms in a Banach space; Chapter 7 Liapunov's Type Equations; 7.1 Solutions of Liapunov's type equations; 7.2 Bounds for solutions of Liapunov's type equations; 7.3 Equivalent norms in a Hilbert space; 7.4 Particular cases; Chapter 8 Bounds for Spectral Radiuses

8.1 Preliminary results8.2 Hille - Tamarkin matrices; 8.3 Proof of Theorem 8.2.1; 8.4 Lower bounds for spectral radiuses; 8.5 Finite matrices; 8.6 General operator and block matrices; 8.7 Operator matrices "close" to triangular ones; 8.8 Proof of Theorem 8.7.1; 8.9 Operator matrices with normal entries; 8.10 Scalar integral operators; 8.11 Matrix integral operators; Chapter 9 Linear Equations with Variable Operators; 9.1 Evolution operators; 9.2 Stability conditions; 9.3 Perturbations of evolution operators; 9.4 Equations "close" to autonomous; 9.5 Linear equations with majorants

Chapter 10 Linear Equations with Slowly Varying Coefficients

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

1-280-75184-3

9786610751846

0-08-046935-3

OCLC:

469399157