A treatise on finite difference inequalities that have important applications to theories of various classes of finite difference and sum- difference equations.

Preface

iii

Introduction

1

(9)

Linear finite difference inequalities

10

(79)

Introduction

10

(1)

Basic finite difference inequalities

10

(8)

Further finite difference inequalities

18

(7)

Pachpatte's inequalities I

25

(11)

Pachpatte's inequalities II

36

(9)

Inequalities with iterated sums

45

(9)

Applications

54

(24)

Perturbation of linear difference equations

54

(4)

Perturbation of nonlinear difference equations

58

(6)

Higher order difference equations

64

(5)

Stochastic difference equations

69

(9)

Miscellaneous inequalities

78

(11)

Notes

88

(1)

Nonlinear finite difference inequalities I

89

(112)

Introduction

89

(1)

Basic Comparison theorems

90

(13)

Fundamental finite difference inequalities

103

(21)

Pachpatte's inequalities I

124

(9)

Pachpatte's inequalities II

133

(15)

Inequalities involving iterated sums

148

(10)

Applications

158

(32)

Stability of finite difference systems

159

(4)

Behavior of solutions of sum-difference and difference equations

163

(9)

Bounds for solutions of certain difference equations

172

(9)

Perturbed sum-difference equations

181

(9)

Miscellaneous inequalities

190

(11)

Notes

199

(2)

Nonlinear finite difference inequalities II

201

(97)

Introduction

201

(1)

Finite difference inequalities

202

(9)

Pachpatte's inequalities I

211

(13)

Pachpatte's inequalities II

224

(19)

Pachpatte's inequalities III

243

(8)

Inequalities involving iterated sums

251

(17)

Applications

268

(14)

Volterra type sum-difference equations

269

(3)

Properties of solutions of certain sum-difference equations

272

(3)

Second order difference equations

275

(4)

Bounds on the solutions of certain sum-difference and difference equations

279

(3)

Miscellaneous inequalities

282

(16)

Notes

297

(1)

Linear multidimensional finite difference inequalities

298

(88)

Introduction

298

(1)

Basic inequalities in two independent variables

299

(11)

Further inequalities in two independent variables

310

(12)

Pachpatte's inequalities I

322

(13)

Inequalities in many variables

335

(9)

Pachpatte's inequalities II

344

(10)

Applications

354

(17)

Finite difference equations in two independent variables

354

(4)

Perturbations of finite difference equations in two independent variables

358

(7)

Estimates on the solutions of certain sum-difference equations

365

(2)

Fourth order finite difference equations in two independent variables

367

(4)

Miscellaneous inequalities

371

(15)

Notes

384

(2)

Nonlinear multidimensional finite difference inequalities

386

(109)

Introduction

386

(1)

Fundamental inequalities in two independent variables

387

(17)

General inequalities in two independent variables

404

(16)

Pachpatte's inequalities I

420

(16)

Pachpatte's inequalities II

436

(10)

Multidimensional inequalities

446

(14)

Applications

460

(13)

Estimates on the solutions of sum-difference equations

460

(4)

Bounds on the solutions of certain difference equations

464

(4)

Higher order finite difference equations

468

(3)

Sum-difference equations in two independent variables

471

(2)

Miscellaneous inequalities

473

(22)

Notes

493

(2)

References

495

(14)

Author Index

509

(2)

Subject Index

511

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