Difference equations and inequalities: theory, methods, and applications
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Dekker
1992
|
| Series: | Pure and applied mathematics
155 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 777 S. |
| ISBN: | 0824786769 |
Staff View
MARC
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| 100 | 1 | |a Agarwal, Ravi P. |d 1947- |e Verfasser |0 (DE-588)112040187 |4 aut | |
| 245 | 1 | 0 | |a Difference equations and inequalities |b theory, methods, and applications |c Ravi P. Agarwal |
| 264 | 1 | |a New York u.a. |b Dekker |c 1992 | |
| 300 | |a XIII, 777 S. | ||
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| 490 | 1 | |a Pure and applied mathematics |v 155 | |
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Record in the Search Index
| _version_ | 1804118853136941056 |
|---|---|
| adam_text | Contents
Preface v
CHAPTER 1 Preliminaries 1
1.1 Notations 1
1.2 Difference Equations 2
1.3 Initial Value Problems 4
1.4 Some Examples: Initial Value Problems 7
1.5 Boundary Value Problems 12
1.6 Some Examples: Boundary Value Problems 14
1.7 Finite Difference Calculus 23
1.8 Problems 33
1.9 Notes 42
1.10 References 42
CHAPTER 2 Linear Initial Value Problems 46
2.1 Introduction 47
2.2 Preliminary Results from Algebra 47
2.3 Linear Dependence and Independence 53
2.4 Matrix Linear Systems 55
2.5 Variation of Constants Formula 57
2.6 Green s Matrix 58
2.7 Adjoint Systems 59
2.8 Systems with Constant Coefficients 60
2.9 Periodic Linear Systems 68
2.10 Higher Order Linear Equations 71
2.11 Method of Generating Functions 77
2.12 Bernoulli s Method 82
2.13 Poincare s and Perron s Theorems 84
2.14 Regular and Singular Perturbations 88
2.15 Problems 95
2.16 Notes 108
2.17 References 109
vii
viii Contents
CHAPTER 2 Miscellaneous Difference Equations 111
3.1 Clairaut s Equation 112
3.2 Euler s Equation 113
3.3 Riccati s Equation 115
3.4 Bernoulli s Equation 118
3.5 Verhulst s Equation 119
3.6 Best Discrete Approximations: Harmonic
Oscillator Equation 121
3.7 Duffing s Equation 123
3.8 van der Pol s Equation 129
3.9 Hill s Equation 137
3.10 Mathieu s Equation 142
3.11 Weierstrass Elliptic Equations 143
3.12 Volterra s Equations 145
3.13 Elementary Partial Difference Equations:
Riccati s Extended Form 147
3.14 Wave Equation 154
3.15 FitzHugh Nagumo s Equation 156
3.16 Korteweg de Vries Equation 157
3.17 Modified KdV Equation 157
3.18 Lagrange s Equations 158
3.19 Problems 164
3.20 Notes 177
3.21 References 178
CHAPTER 4 Difference Inequalities 181
4.1 Gronwall Inequalities 182
4.2 Nonlinear Inequalities 190
4.3 Inequalities Involving Differences 197
4.4 Finite Systems of Inequalities 202
4.5 Opial s Type Inequalities 206
4.6 Problems 210
4.7 Notes 217
4.8 References 218
Contents ix
CHAPTER ^ Qualitative Properties of Solutions
of Difference Systems 221
5.1 Dependence on Initial Conditions and Parameters 222
5.2 Asymptotic Behaviour of Linear Systems 226
5.3 Asymptotic Behaviour of Nonlinear Systems 237
5.4 Concepts of Stability 239
5.5 Stability of Linear Systems 245
5.6 Stability of Nonlinear Systems 254
5.7 Nonlinear Variation of Constants 261
5.8 Dichotomies 264
5.9 Lyapunov s Direct Method for Autonomous Systems 275
5.10 Lyapunov s Direct Method for Non Autonomous Systems 283
5.11 Stability of Discrete Models in Population Dynamics 287
5.12 Converse Theorems 294
5.13 Total Stability 300
5.14 Practical Stability 303
5.15 Mutual Stability 304
5.16 Problems 306
5.17 Notes 313
5.18 References 315
CHAPTER 6 Qualitative Properties of Solutions
of Higher Order Difference Equations 322
6.1 General Properties of Solutions of
(6.1.1) p(k)u(k+l)+p(k l)u(k l)=q(k)u(k) 323
j 6.2 Boundedness of Solutions of (6.1.1) 325
6.3 Recessive and Dominant Solutions of (6.1.1) 329
6.4 Oscillation and Nonoscillation for (6.1.1) 336
6.5 Riccati Type Transformations for (6.1.1) 337
6.6 Riccati Type Transformations for
(6.6.1) A(p(k)Au(k))+r(k)u(k+l) 0 343
6.7 Olver s Type Comparison Results 350
6.8 Sturm s Type Comparison Results 353
x Contents
6.9 Variety of Properties of Solutions of
(6.9.1) p(k)z(k+l)+p(k l)z(k l)
=q(k)z(k)+r(k) 356
6.10 Variety of Properties of Solutions of
(6.10.1) A2u(k l)+p(k)u7(k)=0 360
6.11 Oscillation and Nonoscillation for
(6.11.1) A(r(k)Au(k))+f(k)F(u(k)) 0 369
6.12 Asymptotic Behaviour of Solutions of
(6.12.1) A(r(k)Au(k))+f(k)F(u(k))=g(k) 375
6.13 i. and c. solutions of
(6.13.1) A2u(k)+f(k,u(k))=0 378
6.14 Oscillation and Nonoscillation for
(6.14.1) A2u(k)=f(k,u(k),Aflu(k)) 382
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)) 384
6.16 Variety of Properties of Solutions of
(6.16.1) A4u(k 2)=p(k)u(k) 388
6.17 Asymptotic Behaviour of Solutions of
(6.17.1) Anu(k)+f(k,u(k),Au(k) An~1u(k)) 0 396
6.18 Asymptotic Behaviour, Oscillation and
Nonoscillation for
(6.18.1) Anu(k)+h(k)F(k,u(k),Au(k) An 1u(k))
g(k,u(k) ,Au(k), . . . .A ^utk)) 400
6.19 Oscillation and Nonoscillation for
m
(6.19.1) Anu(k)+ I f.(k)F.(u(k),Au(k)
i=l X X
An u(k)) 0 406
6.20 Oscillation and Nonoscillation for
(6.20.1) u(k+l) u(k)+p(k)u(k m)=0 413
6.21 Oscillation and Nonoscillation for
m
(6.21.1) A u(k)+S I f,(k)F (u(g,(k))) 0 418
6.22 Problems 422
6.23 Notes 436
6.24 References 437
Contents xi
CHAPTER 7 Boundary Value Problems for Linear Systems 444
7.1 Existence and Uniqueness 444
7.2 Method of Complementary Functions 449
7.3 Method of Particular Solutions 459
7.4 Method of Adjoints 460
7.5 Method of Chasing 469
7.6 Method of Imbedding: First Formulation 475
7.7 Method of Imbedding: Second Formulation 481
7.8 Method of Sweep 486
7.9 Miller s and Olver s Algorithms 492
7.10 Problems 495
7.11 Notes 498
7.12 References 499
CHAPTER 8 Boundary Value Problems for Nonlinear Systems 504
8.1 Preliminary Results from Analysis 505
8.2 Existence and Uniqueness 507
8.3 Approximate Picard s Iterates 515
8.4 Oscillatory State 520
8.5 Stopping Criterion 522
8.6 Application to the Perturbation Method 524
8.7 Monotone Convergence 528
8.8 Newton s Method 533
8.9 Approximate Newton s Method 539
8.10 Initial Value Methods 544
8.11 Invariant Imbedding Method 551
8.12 Problems 554
8.13 Notes 560
8.14 References 561
CHAPTER 9 Miscellaneous Properties of Solutions of Higher
Order Linear Difference Equations 566
9.1 Disconjugacy 567
9.2 Right and Left Disconjugacy 568
xii Contents
9.3 Adjoint Equations 572
9.4 Right and Left Disconjugacy for the Adjoint Equation 574
9.5 Right Disfocality 575
9.6 Eventual Disconjugacy and Right Disfocality 578
9.7 A Classification of Solutions 578
9.8 Interpolating Polynomials 581
9.9 Green s Functions 585
9.10 Inequalities and Equalities for Green s Functions 595
9.11 Maximum Principles 597
9.12 Error Estimates in Polynomial Interpolation 598
9.13 Notes 600
9.14 References 601
CHAPTER 10 Boundary Value Problems for Higher
Order Difference Equations 606
10.1 Existence and Uniqueness 607
10.2 Picard s and Approximate Picard s Methods 614
10.3 Quasilinearization and Approximate
Quasilinearization 622
10.4 Monotone Convergence 631
10.5 Initial Value Methods 637
10.6 Uniqueness Implies Existence 648
10.7 Problems 656
10.8 Notes 659
10.9 References 659
CHAPTER 11 Sturm Liouville Problems and Related
Inequalities 662
11.1 Sturm Liouville Problems 662
11.2 Eigenvalue Problems for Symmetric Matrices 667
11.3 Matrix Formulation of Sturm Liouville Problems 669
11.4 Symmetric, Antisymmetric and Periodic Boundary
Conditions 670
11.5 Discrete Fourier Series 675
11.6 Wirtinger s Type Inequalities 677
Contents xiii
11.7 Generalized Wirtinger s Type Inequalities 684
11.8 Generalized Opial s Type Inequalities 686
11.9 Comparison Theorems for Eigenvalues 688
11.10 Problems 693
11.11 Notes 704
11.12 References 705
CHAPTER 12 Difference Inequalities in Several
Independent Variables 709
12.1 Discrete Riemann s Function 709
12.2 Linear Inequalities 713
12.3 Wendroff s Type Inequalities 716
12.4 Nonlinear Inequalities 720
12.5 Inequalities Involving Partial Differences 723
12.6 Multidimensional Linear Inequalities 731
12.7 Multidimensional Nonlinear Inequalities 734
12.8 Convolution Type Inequalities 741
12.9 Opial s and Wirtinger s Type Inequalities
in Two Variables 744
12.10 Problems 752
12.11 Notes 755
12.12 References 756
Name Index 760
Subject Index 769
|
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| id | DE-604.BV004733951 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:16:53Z |
| institution | BVB |
| isbn | 0824786769 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002912222 |
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| physical | XIII, 777 S. |
| publishDate | 1992 |
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| spelling | Agarwal, Ravi P. 1947- Verfasser (DE-588)112040187 aut Difference equations and inequalities theory, methods, and applications Ravi P. Agarwal New York u.a. Dekker 1992 XIII, 777 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 155 Ungleichung (DE-588)4139098-2 gnd rswk-swf Differenzengleichung (DE-588)4012264-5 gnd rswk-swf Differenzengleichung (DE-588)4012264-5 s DE-188 Ungleichung (DE-588)4139098-2 s Pure and applied mathematics 155 (DE-604)BV000001885 155 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002912222&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Agarwal, Ravi P. 1947- Difference equations and inequalities theory, methods, and applications Pure and applied mathematics Ungleichung (DE-588)4139098-2 gnd Differenzengleichung (DE-588)4012264-5 gnd |
| subject_GND | (DE-588)4139098-2 (DE-588)4012264-5 |
| 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 | Ungleichung (DE-588)4139098-2 gnd Differenzengleichung (DE-588)4012264-5 gnd |
| topic_facet | Ungleichung Differenzengleichung |
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