Difference equations and inequalities theory, methods, and applications
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| Format: | Buch |
| Sprache: | English |
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Dekker
2000
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| Ausgabe: | 2. ed., rev. and expanded |
| Schriftenreihe: | Pure and applied mathematics
228 |
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| 245 | 1 | 0 | |a Difference equations and inequalities |b theory, methods, and applications |c Ravi P. Agarwal |
| 250 | |a 2. ed., rev. and expanded | ||
| 264 | 1 | |a New York u.a. |b Dekker |c 2000 | |
| 300 | |a XIII, 971 S. | ||
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| 650 | 7 | |a Équations aux différences - Étude et enseignement (supérieur) |2 ram | |
| 650 | 4 | |a Difference equations | |
| 650 | 4 | |a Inequalities (Mathematics) | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102/MAT 390f 2001 A 5176 |
|---|---|
| DE-BY-TUM_katkey | 1157950 |
| DE-BY-TUM_media_number | 040010569265 |
| _version_ | 1816711948036734976 |
| adam_text | Titel: Difference equations and inequalities
Autor: Agarwal, Ravi P
Jahr: 2000
Contents
Preface to the Second Edition v
Preface to the First Edition v
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
1.9. Problems 34
1.10. Notes 44
1.11. References 45
Chapter 2 Linear Initial Value Problems
2.1. Introduction 49
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
2.16. Problems ^7
2.17. Notes J]3
2.18. References
vii
viii
Contents
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 i 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
3.19. Problems 167
3.20. Notes 180
3.21. References 181
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
4.7. Problems 219
4.8. Notes 229
4.9. References 230
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
Contents
IX
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
5.16. Problems 308
5.17. Notes 326
5.18. References 328
Chapter 6 Qualitative Properties of Solutions of
Higher Order Difference Equations
6.1. General Properties of Solutions of
(6.1.1) p(k)u(k + 1) + p(k - 1 )u(k - 1) = q(k)u(h) 335
6.2. Boundedness of Solutions of (6.1.1) 337
6.3. Recessive and Dominant Solutions of (6.1.1) 342
6.4. Oscillation and Nonoscillation for (6.1.1) 348
6.5. Riccati Type Transformations for (6.1.1) 349
6.6. Riccati Type Transformations for
(6.6.1) A(p(fc)Au(fc)) + r{k)u{k + 1) = 0 355
6.7. Olver Type Comparison Results 362
6.8. Sturm Type Comparison Results 365
6.9. Variety of Properties of Solutions of
(6.9.1) p(k)z(k + 1) + p(k — )z(k — 1)
= q(k)z(k) + r(k) 367
6.10. Variety of Properties of Solutions of
(6.10.1) A2u{k - 1) + pityiPik) = 0 370
6.11. Oscillation and Nonoscillation for
(6.11.1) A(r(k)Au(k)) + f(k)F{u(k)) = 0 378
6.12. Asymptotic Behavior of Solutions of
(6.12.1) A(r{k)Au{k)) + f(k)F(u(k)) = g{k) 384
6.13. 12 and cq Solutions of
(6.13.1) A2u(k) + f(k,u(k)) = 0 387
6.14. Oscillation and Nonoscillation for
(6.14.1) Alu(k) = f(k,u(k),A0u(k)) 391
X
Contents
6.15. Oscillation and Nonoscillation for
(6.15.1) A(r(k)Au(k)) + f{k)F(k,u(k), Au(k))
— g(k,u(k), Au(k)) 393
6.16. Variety of Properties of Solutions of
(6.16.1) AAu(k — 2) = p(k)u(k) 398
6.17. Asymptotic Behavior of Solutions of
(6.17.1) Anu{k) + /(*,u{k),Au(k) - • ¦, An~lu(k)) = 0 403
6.18. Asymptotic Behavior, Oscillation and
Nonoscillation for
(6.18.1) Anu(k) + h(k)F(k,u(k), Au(k), • ¦ ¦, An-1u(A;))
= g{k, u{k), Au(fc), • • •, An_1u(fc)) 408
6.19. Oscillation and Nonoscillation for
(6.19.1) Anu(k) + fi{k)Fi{u(k),Au(k), ¦ ¦ •,
An~1u(k))=0 414
6.20. Oscillation and Nonoscillation for
(6.20.1) u(k + 1) - u(k) + p(k)u(k — m) = 0 420
6.21. Oscillation and Nonoscillation for
(6.21.1)5 AQu(fc)+^r,i/iW^(%W))=° 425
6.22. Oscillation and Nonoscillation for
(6.22.1) A(p{k)(Au(k))a) + q(k + l)/(u(fc + 1)) = 0 427
6.23. Oscillation and Nonoscillation for
(6.23.1) u(t) — u(t — t) + p(t)u(t — cr) = 0 436
6.24. Problems 443
6.25. Notes 473
6.26. References 474
Chapter 7 Qualitative Properties of Solutions of
Neutral Difference Equations
7.1. Oscillation and Nonoscillation for
(7.1.1) A(u(k) + pu(k — r)) + q(k)u(k — ct) = 0 485
7.2. Existence and Asymptotic Behavior
of Nonoscillatory Solutions of (7.1.1) 493
7.3. Oscillation and Comparison Theorems for (7.1.1) 509
7.4. Global Asymptotic Stability Criterion for (7.1.1) 512
7.5. Oscillation and Nonoscillation for
(7.5.1) A (u(k)+pu(k—r))- -q(k)u(k — oi)—h(k)u(k—02) = 0 517
7.6. Oscillation and Nonoscillation for
(7.6.1) A (u(k) + pu(k - t)) + q(k)u(k - a) = F(k) 519
7.7. Oscillation and Nonoscillation for
(7.7.1) A (u(k) - p(k)u(k - r)) + q(k)u(k - r) = 0 521
Contents
xi
7.8. Oscillation and Nonoscillation for
(7.8.1) A (r(k)u(k) - p(k)u(k - r)) + q(k)u(k ~ a(k)) - 0 527
7.9. Oscillatory and Asymptotic Behavior for
(7.9.1) A (u(k) + p(k)u(k - r)) + q(k)f(u(k - a)) — 0 531
7.10. Oscillation and Nonoscillation for
(7.10.1)5 A(u(k) + p(k)u(k + 5r)) - q(k)f(u(a(k))) — F(k) 544
7.11. Oscillation and Asymptotic Behavior for
(7.11.1) A2(u(k) + p(k)u(k - r)) + q(k)f(u(k -f 1 — o)) = 0 548
7.12. Classification of Solutions for
(7.12.1) A(a(k)A(u(k) + p(k)u(k — r)))
+ q(k + 1 )f(u(k + 1 — cr)) = 0 556
7.13. Existence of Solutions for (7.12.1) 565
7.14. Oscillation of Mixed Difference Equations 572
7.15. Oscillation and Nonoscillation for
(7.15.1) An(u(k) + p(k)u(k - r)) + q(k)f(u(k — a)) = 0 583
7.16. Oscillation and Nonoscillation for
(7.16.1) A (a(k)An~l (u(k) - p(k)u(k - r)))
+ 5q{k)f(u{a{k))) = 0 590
7.17. Problems 593
7.18. Notes 620
7.19. References 621
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
8.10. Problems 674
8.11. Notes 676
8.12. References 677
Chapter 9 Boundary Value Problems for
Nonlinear Systems
9.1. Preliminary Results from Analysis 681
9.2. Existence and Uniqueness 684
xii
Contents
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
9.13. Problems 734
9.14. Notes 739
9.15. References 739
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
10.13. Problems 778
10.14. Notes 789
10.15. References 790
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
Contents
xiii
11.7. Problems 839
11.8. Notes 841
11.9. References 842
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) A3u(k) + Xa(k)f(u(k)) = 0, k e N(2, K + 2)
(12.10.2) u(0) = u(l) = u(K + 3) = 0 873
12.11. Problems 882
12.12. Notes 896
12.13. References 897
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
13.10. Problems 941
13.11. Notes 944
13.12. References 945
Author Index 919
Subject Index 963
|
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| id | DE-604.BV013390122 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-25T17:16:24Z |
| institution | BVB |
| isbn | 0824790073 |
| language | English |
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| physical | XIII, 971 S. |
| publishDate | 2000 |
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| series | Pure and applied mathematics |
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| spellingShingle | Agarwal, Ravi P. 1947- Difference equations and inequalities theory, methods, and applications Pure and applied mathematics Differentiaalvergelijkingen gtt Inégalités (mathématiques) ram Équations aux différences - Étude et enseignement (supérieur) ram Difference equations Inequalities (Mathematics) Differenzengleichung (DE-588)4012264-5 gnd Ungleichung (DE-588)4139098-2 gnd |
| subject_GND | (DE-588)4012264-5 (DE-588)4139098-2 |
| title | Difference equations and inequalities theory, methods, and applications |
| title_auth | Difference equations and inequalities theory, methods, and applications |
| title_exact_search | Difference equations and inequalities theory, methods, and applications |
| title_full | Difference equations and inequalities theory, methods, and applications Ravi P. Agarwal |
| title_fullStr | Difference equations and inequalities theory, methods, and applications Ravi P. Agarwal |
| title_full_unstemmed | Difference equations and inequalities theory, methods, and applications Ravi P. Agarwal |
| title_short | Difference equations and inequalities |
| title_sort | difference equations and inequalities theory methods and applications |
| title_sub | theory, methods, and applications |
| topic | Differentiaalvergelijkingen gtt Inégalités (mathématiques) ram Équations aux différences - Étude et enseignement (supérieur) ram Difference equations Inequalities (Mathematics) Differenzengleichung (DE-588)4012264-5 gnd Ungleichung (DE-588)4139098-2 gnd |
| topic_facet | Differentiaalvergelijkingen Inégalités (mathématiques) Équations aux différences - Étude et enseignement (supérieur) Difference equations Inequalities (Mathematics) Differenzengleichung Ungleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009133899&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000001885 |
| work_keys_str_mv | AT agarwalravip differenceequationsandinequalitiestheorymethodsandapplications |