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

Format:

Book

Author/Creator:

Gilʹ, M. I. (Mikhail Iosifovich)

Contributor:

Louis A. Duhring Fund.

Series:

North-Holland mathematics studies ; 206.

North-Holland mathematics studies ; 206

Language:

English

Subjects (All):

Difference equations.

Normed linear spaces.

Physical Description:

xvi, 362 pages ; 25 cm.

Edition:

First edition.

Place of Publication:

Amsterdam ; Boston : Elsevier, 2007.

Contents:

1.1 Banach and Hilbert spaces 1

1.2 Examples of normed spaces 3

1.3 Linear operators 4

1.4 Examples of difference equations 6

1.5 Stability notions 8

1.6 The comparison principle 9

1.7 Liapunov functions 12

1.8 Ordered spaces and Banach lattices 15

1.9 The Abstract Gronwall Lemma 16

1.10 Discrete inequalities in a Banach lattice 17

2 Classes of Operators 21

2.1 Classification of spectra 21

2.2 Compact operators in a Hilbert space 23

2.3 Compact matrices 25

2.4 Integral operators 28

3 Functions of Finite Matrices 33

3.1 Matrix-valued functions 33

3.2 Estimates for the resolvent 34

3.4 Estimates for regular matrix functions 37

3.5 Proof of Theorem 3.2.4 38

3.6 Proofs of Theorems 3.2.1 and 3.2.3 40

3.7 Proof of Theorem 3.4.1 47

3.8 Non-Euclidean norms of powers of matrices 51

3.9 Absolute values of matrix functions 53

3.10 Proof of Theorem 3.9.1 54

4 Norm Estimates for Operator Functions 57

4.1 Regular operator functions 57

4.2 Functions of Hilbert-Schmidt operators 58

4.3 Operators with Hilbert-Schmidt powers 59

4.4 Resolvents of Neumann-Schatten operators 61

4.5 Functions of quasi-Hermitian operators 62

4.6 Functions of quasiunitary operators 64

4.7 Auxiliary results 66

4.8 Equalities for eigenvalues 70

4.9 Proofs of Theorems 4.2.1, 4.2.2 and 4.4.1 71

5 Spectrum Perturbations 75

5.1 Roots of algebraic equations 75

5.2 Roots of functional equations 76

5.3 Spectral variations 78

5.4 Perturbations of Hilbert-Schmidt operators 80

5.5 Perturbations of Neumann - Schatten operators 80

5.6 Perturbations of quasi-Hermitian operators 81

5.7 Perturbations of finite matrices 83

6 Linear Equations with Constant Operators 85

6.1 Homogeneous equations in a Banach space 85

6.2 Nonhomogeneous equations with constant operators 86

6.3 Perturbations of autonomous equations 87

6.4 Equations with Hilbert-Schmidt operators 89

6.5 Equations with Neumann-Schatten operators 91

6.6 Equations with non-compact operators 93

6.7 Equations in finite dimensional spaces 94

6.8 Z-transform 96

6.9 Exponential dichotomy 99

6.10 Equivalent norms in a Banach space 101

7 Liapunov's Type Equations 105

7.1 Solutions of Liapunov's type equations 105

7.2 Bounds for solutions of Liapunov's type equations 107

7.3 Equivalent norms in a Hilbert space 108

7.4 Particular cases 110

8 Bounds for Spectral Radiuses 113

8.1 Preliminary results 113

8.2 Hille - Tamarkin matrices 114

8.3 Proof of Theorem 8.2.1 117

8.4 Lower bounds for spectral radiuses 118

8.5 Finite matrices 120

8.6 General operator and block matrices 123

8.7 Operator matrices "close" to triangular ones 124

8.8 Proof of Theorem 8.7.1 126

8.9 Operator matrices with normal entries 128

8.10 Scalar integral operators 129

8.11 Matrix integral operators 137

9 Linear Equations with Variable Operators 143

9.1 Evolution operators 143

9.2 Stability conditions 144

9.3 Perturbations of evolution operators 145

9.4 Equations "close" to autonomous 149

9.5 Linear equations with majorants 151

10 Linear Equations with Slowly Varying Coefficients 153

10.1 The freezing method 153

10.2 Proof of Theorem 10.1.1 155

10.3 Equations in Hilbert spaces 156

10.4 Equations in Euclidean spaces 158

10.5 Applications of the Liapunov type equation 159

10.6 Proofs of Lemma 10.5.1 and Theorem 10.5.2 160

11 Nonlinear Equations with Autonomous Linear Parts 163

11.1 Solution estimates 163

11.2 Proof of Theorem 11.1.1 164

11.3 Stability and boundedness 166

11.4 Stability and instability by linear approximation 167

11.5 Equations with Hilbert-Schmidt operators 170

11.6. l[superscript 2]-norms of solutions 170

12 Nonlinear Equations with Time-Variant Linear Parts 173

12.1 Equations with general linear parts 173

12.2 Proof of Theorem 12.1.1 176

12.3 Slowly varying equations in a Banach space 177

12.4 Proof of Theorem 12.3.1 178

12.5 Slowly varying equations in a Hilbert space 180

12.6 The finite dimensional case 182

12.7 Equations in ordered spaces 183

12.8 Perturbations of nonlinear equations 184

13 Higher Order Linear Difference Equations 187

13.1 Homogeneous time-invariant equations 187

13.2 Nonhomogeneous time-invariant equations 189

13.3 Nonautonomous equations 191

13.4 l[superscript 2]-norms of solutions 193

13.5 Positive solutions of linear equations 196

13.6 Proof of Theorem 13.5.1 196

14 Nonlinear Higher Order Difference Equations 201

14.1 General higher order equations 201

14.2 The Lur'e type equations 203

14.3 Proof of Theorem 14.2.1 204

14.4 Equations in Euclidean spaces 206

14.5 The Aizerman type problem 207

14.6 Proofs of Theorem 14.5.2 and Lemma 14.5.3 209

14.7 Positive solutions of nonlinear equations 212

14.8 Proof of Theorem 14.7.1 212

15 Input-to-State Stability 215

15.1 General equations 215

15.2 Equations with time-variant linear parts 217

15.3 Equations with bounded nonlinearities 217

15.4 Input version of the Aizerman type problem 218

15.5 Proof of Theorem 15.4.1 221

16 Periodic Solutions of Difference Equations and Orbital Stability 223

16.1 Linear autonomous equations 223

16.2 Linear nonautonomous equations 224

16.3 Semilinear autonomous equations 225

16.4 Semilinear nonautonomous equations 227

16.5 Essentially nonlinear equations 229

16.6 Equations with linear majorants 232

16.7 Equations in a Euclidean space 234

16.8 Positive periodic solutions 235

16.9 Orbital stability 236

17 Discrete Volterra Equations in Banach Spaces 239

17.1 Linear Volterra equations 239

17.2 Nonlinear recurrence equations 241

17.3 Proof of Theorem 17.2.1 242

17.4 Convolution type Volterra equations 244

17.5 Linear perturbations of convolution equations 246

17.6 Nonlinear convolution type equations 247

17.7 Operator pencils in a Hilbert space 248

17.8 Pencils with Hilbert-Schmidt off-diagonals 251

17.9 Neumann-Schatten pencils 252

17.10 Stability conditions 253

17.11 Volterra equations in l[superscript 2](C) 255

17.12 Multiplicative representations of solutions 257

17.13 Proof of Theorem 17.12.1 258

18 Convolution type Volterra Difference Equations in Euclidean Spaces and their Perturbations 261

18.1 Conditions in terms of determinants 261

18.2 Finite order entire matrix pencils 263

18.3 Variations of characteristic values 265

18.4 Proof of Theorem 18.3.1 268

18.5 Polynomial matrix pencils 271

18.6 Conditions in terms of characteristic values 272

19 Stieltjes Differential Equations 275

19.2 Scalar linear Stieltjes equations 276

19.3 A Gronwall-like inequality 279

19.4 The [Mu]-exponential matrix 280

19.5 Estimates for the [Mu]-exponential matrix 282

19.6 Stability and boundedness 283

19.7 Existence and uniqueness of solutions 286

19.8 Dependence on time integrators 287

20 Volterra - Stieltjes Equations 291

20.2 Solution estimates 293

20.3 Proof of Theorem 20.2.1 294

20.4 The continuous case 296

21 Difference Equations with Continuous Time 299

21.2 Linear equations 300

21.3 Nonlinear equations 301

21.4 Proof of Theorem 21.3.1 302

21.5 Stability and boundedness 303

21.6 Equations in finite dimensional spaces 305

22 Steady States of Difference Equations 307

22.1 Spaces with generalized norms 307

22.2 Positive steady states 310

22.3 Proof of Theorem 22.2.1 311

22.4 Equations in l[superscript 2] 314

22.5 Equations in space C[0,1] 316

22.6 Finite systems of scalar equations 319

Appendix A Functions of Non-Compact Operators 325

23.2 Properties of Volterra operators 326

23.3 Resolvents of P-triangular operators 328

23.4 Representations of noncompact operators 331

23.5 Proof of Theorem 4.5.1 332

23.6 Proof of Theorem 4.5.5 333

23.7 Proof of Theorem 4.5.3 334

23.8 Representations of regular functions 336

23.9 Proofs of Theorems 4.6.1 and 4.6.3 339.

Notes:

Includes bibliographical references (pages 347-358) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Louis A. Duhring Fund.

ISBN:

9780444527134

0444527133

OCLC:

74967021

Online:

Publisher description