Difference equations and inequalities : theory, methods, and applications / Ravi P. Agarwal.

Format:

Book

Author/Creator:

Agarwal, Ravi P.

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

Monographs and textbooks in pure and applied mathematics ; 228.

Monographs and textbooks in pure and applied mathematics ; 228

Language:

English

Subjects (All):

Difference equations.

Inequalities (Mathematics).

Physical Description:

xiii, 971 pages ; 24 cm.

Edition:

Second edition, revised and expanded.

Place of Publication:

New York : Marcel Dekker, [2000]

Contents:

Chapter 1 Preliminaries

1.1. Notations 1

1.2. Difference Equations 2

1.3. Initial Value Problems 4

1.4. Some Examples: Initial Value Problems 6

1.5. Boundary Value Problems 11

1.6. Some Examples: Boundary Value Problems 13

1.7. Some Examples: Real World Phenomena 22

1.8. Finite Difference Calculus 26

Chapter 2 Linear Initial Value Problems

2.2. Preliminary Results from Algebra 49

2.3. Linear Dependence and Independence 54

2.4. Matrix Linear Systems 55

2.5. Variation of Constants Formula 58

2.6. Green's Matrix 59

2.7. Adjoint Systems 60

2.8. Systems with Constant Coefficients 62

2.9. Periodic Linear Systems 69

2.10. Almost Periodic Linear Systems 72

2.11. Higher Order Linear Equations 74

2.12. Method of Generating Functions 80

2.13. Bernoulli's Method 85

2.14. Poincare's and Perron's Theorems 87

2.15. Regular and Singular Perturbations 91

Chapter 3 Miscellaneous Difference Equations

3.1. Clairaut's Equation 117

3.2. Euler's Equation 118

3.3. Riccati's Equation 120

3.4. Bernoulli's Equation 125

3.5. Verhulst's Equation 126

3.6. Best Discrete Approximations: Harmonic Oscillator Equation 128

3.7. Duffing's Equation 130

3.8. van der Pol's Equation 135

3.9. Hill's Equation 142

3.10. Mathieu's Equation 147

3.11. Weierstrass' Elliptic Equations 148

3.12. Volterra's Equations 150

3.13. Elementary Partial Difference Equations: Riccati's Extended Form 152

3.14. Wave Equation 158

3.15. FitzHugh-Nagumo's Equation 159

3.16. Korteweg-de Vries' Equation 160

3.17. Modified KdV Equation 161

3.18. Lagrange's Equations 162

Chapter 4 Difference Inequalities

4.1. Gronwall Inequalities 184

4.2. Nonlinear Inequalities 193

4.3. Inequalities Involving Differences 200

4.4. Finite Systems of Inequalities 205

4.5. Opial Type Inequalities 209

4.6. Wirtinger Type Inequalities 214

Chapter 5 Qualitative Properties of Solutions of Difference Systems

5.1. Dependence on Initial Conditions and Parameters 234

5.2. Asymptotic Behavior of Linear Systems 238

5.3. Asymptotic Behavior of Nonlinear Systems 247

5.4. Concepts of Stability 250

5.5. Stability of Linear Systems 255

5.6. Stability of Nonlinear Systems 262

5.7. Nonlinear Variation of Constants 269

5.8. Dichotomies 272

5.9. Lyapunov's Direct Method for Autonomous Systems 281

5.10. Lyapunov's Direct Method for Non-Autonomous Systems 289

5.11. Stability of Discrete Models in Population Dynamics 292

5.12. Converse Theorems 298

5.13. Total Stability 303

5.14. Practical Stability 305

5.15. Mutual Stability 306

Chapter 6 Qualitative Properties of Solutions of Higher Order Difference Equations

Chapter 7 Qualitative Properties of Solutions of Neutral Difference Equations

Chapter 8 Boundary Value Problems for Linear Systems

8.1. Existence and Uniqueness 629

8.2. Method of Complementary Functions 634

8.3. Method of Particular Solutions 643

8.4. Method of Adjoints 643

8.5. Method of Chasing 651

8.6. Method of Imbedding: First Formulation 656

8.7. Method of Imbedding: Second Formulation 662

8.8. Method of Sweep 666

8.9. Miller's and Olver's Algorithms 671

Chapter 9 Boundary Value Problems for Nonlinear Systems

9.1. Preliminary Results from Analysis 681

9.2. Existence and Uniqueness 684

9.3. Approximate Picard's Iterates 691

9.4. Oscillatory State 695

9.5. Stopping Criterion 698

9.6. Application to the Perturbation Method 700

9.7. Monotone Convergence 703

9.8. Periodic Boundary Value Problems 707

9.9. Newton's Method 714

9.10. Approximate Newton's Method 719

9.11. Initial-Value Methods 724

9.12. Invariant Imbedding Method 731

Chapter 10 Miscellaneous Properties of Solutions of Higher Order Linear Difference Equations

10.1. Disconjugacy 745

10.2. Right and Left Disconjugacy 746

10.3. Adjoint Equations 750

10.4. Right and Left Disconjugacy for the Adjoint Equation 751

10.5. Right Disfocality 752

10.6. Eventual Disconjugacy and Right Disfocality 754

10.7. A Classification of Solutions 755

10.8. Interpolating Polynomials 757

10.9. Green's Functions 762

10.10. Inequalities and Equalities for Green's Functions 772

10.11. Maximum Principles 774

10.12. Error Estimates in Polynomial Interpolation 775

Chapter 11 Boundary Value Problems for Higher Order Difference Equations

11.1. Existence and Uniqueness 795

11.2. Picard's and Approximate Picard's Methods 803

11.3. Quasilinearization and Approximate Quasilinearization 809

11.4. Monotone Convergence 819

11.5. Initial-Value Methods 824

11.6. Uniqueness Implies Existence 833

Chapter 12 Sturm-Liouville Problems and Related Inequalities

12.1. Sturm-Liouville Problems 844

12.2. Eigenvalue Problems for Symmetric Matrices 848

12.3. Matrix Formulation of Sturm-Liouville Problems 850

12.4. Symmetric, Antisymmetric and Periodic Boundary Conditions 852

12.5. Discrete Fourier Series 856

12.6. Wirtinger Type Inequalities 859

12.7. Generalized Wirtinger Type Inequalities 865

12.8. Generalized Opial Type Inequalities 867

12.9. Comparison Theorems for Eigenvalues 869

12.10. Positive Solutions of (12.10.1) [Delta][superscript 3]u(k) + [lambda]a(k)f(u(k)) = 0, k [set membership] N(2, K + 2) (12.10.2) u(0) = u(1) = u(K + 3) = 0 873

Chapter 13 Difference Inequalities in Several Independent Variables

13.1. Discrete Riemann's Function 901

13.2. Linear Inequalities 905

13.3. Wendroff Type Inequalities 908

13.4. Nonlinear Inequalities 911

13.5. Inequalities Involving Partial Differences 914

13.6. Multidimensional Linear Inequalities 921

13.7. Multidimensional Nonlinear Inequalities 925

13.8. Convolution Type Inequalities 930

13.9. Opial and Wirtinger Type Inequalities in Two Variables 933.

Notes:

Includes bibliographical references and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0824790073

OCLC:

42921306