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E-raamat: Asymptotic Integration of Differential and Difference Equations

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  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2129
  • Ilmumisaeg: 26-May-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319182483
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Asymptotic Integration of Differential and Difference EquationsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2129
  • Ilmumisaeg: 26-May-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319182483
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This book presents the theory of asymptotic integration for both linear differential and difference equations. This type of asymptotic analysis is based on some fundamental principles by Norman Levinson. While he applied them to a special class of differential equations, subsequent work has shown that the same principles lead to asymptotic results for much wider classes of differential and also difference equations.

After discussing asymptotic integration in a unified approach, this book studies how the application of these methods provides several new insights and frequent improvements to results found in earlier literature. It then continues with a brief introduction to the relatively new field of asymptotic integration for dynamic equations on time scales.

Asymptotic Integration of Differential and Difference Equations is a self-contained and clearly structured presentation of some of the most important results in asymptotic integration and the techniques used in this field. It will appeal to researchers in asymptotic integration as well to non-experts who are interested in the asymptotic analysis of linear differential and difference equations. It will additionally be of interest to students in mathematics, applied sciences, and engineering. Linear algebra and some basic concepts from advanced calculus are prerequisites.

Arvustused

This very readable book gives a nice presentation of the theory of asymptotic integration for both linear differential and linear difference equations . The book provides a very deep insight into the theory of asymptotic integrations of linear differential and difference equations. Excellently written, it has a place in the bookcase of every mathematician, engineer or student, and is also of value for non-experts interested in the rudiments and applications of this theory. (Josef Diblík, Mathematical Reviews, May, 2016)

This book is a self-contained and clearly structured presentation of important results in asymptotic integration and the techniques which are used in this field. I have really enjoyed reading it. This text appeals to (non-)experts who are interested in asymptotic behavior of solutions to differential and differenceequations. It can be of interest to students in mathematics, applied sciences, and engineering. For anyone who works in asymptotic integration, this monograph is a must. (Pavel Rehak, zbMATH 1331.34001, 2016)

1 Introduction, Notation, and Background
1(10)
1.1 Introduction
1(3)
1.2 Relationship to Other Asymptotic Theories
4(1)
1.3 Common Asymptotic Terminology
5(1)
1.4 Notation and Background
6(5)
2 Asymptotic Integration for Differential Systems
11(58)
2.1
Chapter Overview
11(2)
2.2 Ordinary Dichotomies, L1-Perturbations, and Levinson's Fundamental Theorem
13(7)
2.3 Sharpness of Dichotomy and Growth Conditions
20(3)
2.4 Weak Dichotomies
23(5)
2.5 Perturbations of Systems with an Exponential Dichotomy
28(10)
2.6 Perturbations of Constant Systems
38(3)
2.7 Estimate of the Error Term o(1)
41(5)
2.8 (h, k)-Dichotomies
46(3)
2.9 Coupled Dichotomy-Growth Conditions
49(5)
2.10 Asymptotic Equivalences of Linear Systems
54(5)
2.11 Other Asymptotic Integration Results
59(10)
2.11.1 A Row-Wise Generalization of Levinson's Fundamental Theorem
59(3)
2.11.2 A Row-Wise Lp Result
62(4)
2.11.3 Uniform Asymptotic Integration
66(3)
3 Asymptotic Representation for Solutions of Difference Systems
69(50)
3.1
Chapter Overview
69(1)
3.2 Ordinary Dichotomies and l1-Perturbations
70(11)
3.3 Sharpness of Dichotomy and Growth Conditions
81(2)
3.4 Weak Dichotomies
83(4)
3.5 Perturbations of Systems with a "Column-Wise Exponential Dichotomy"
87(9)
3.6 Perturbations of Constant Systems
96(6)
3.7 Estimate of the Error Term o(1)
102(4)
3.8 Asymptotic Equivalence
106(2)
3.9 (h, k)-Dichotomies for Difference Systems
108(2)
3.10 Noninvertible Systems
110(9)
4 Conditioning Transformations for Differential Systems
119(60)
4.1
Chapter Overview
119(2)
4.2 Conditioning Transformations and Approximate Equations
121(2)
4.3 Reduction to Eigenvalues
123(20)
4.3.1 Asymptotically Constant Systems
124(6)
4.3.2 Eastham's Generalization
130(10)
4.3.3 Higher Derivatives
140(3)
4.4 Lp-Perturbations with p > 1
143(5)
4.5 Special Dichotomies and Triangular Perturbations
148(4)
4.6 Conditionally Integrable Perturbations
152(22)
4.6.1 Some Initial Remarks
152(2)
4.6.2 A Theorem of Wintner and Extensions
154(5)
4.6.3 Conditionally Integrable Perturbations of Constant Systems
159(6)
4.6.4 Conditions for Non-resonance
165(6)
4.6.5 Some Further Results Involving Averaging
171(3)
4.7 Preliminary Transformations
174(5)
5 Conditioning Transformations for Difference Systems
179(30)
5.1
Chapter Overview
179(1)
5.2 Conditioning Transformations and Approximate Equations
180(2)
5.3 Reduction to Eigenvalues
182(6)
5.4 lp-Perturbations for p > 1
188(5)
5.5 Special Dichotomies and Triangular Perturbations
193(2)
5.6 Conditionally Convergent Perturbations
195(9)
5.7 Preliminary Transformations for Difference Equations
204(5)
6 Perturbations of Jordan Differential Systems
209(24)
6.1
Chapter Overview
209(1)
6.2 One Single Jordan Block
210(7)
6.3 Multiple Jordan Blocks
217(5)
6.4 Weak Dichotomies
222(2)
6.5 Block-Triangular Perturbations
224(4)
6.6 An Early Result on L1-Perturbations of Certain Jordan Matrices
228(5)
7 Perturbations of Jordan Difference Systems
233(4)
8 Applications to Classes of Scalar Linear Differential Equations
237(58)
8.1
Chapter Overview
237(2)
8.2 Perturbations of Constant Differential Equations
239(3)
8.3 Second-Order Linear Differential Equations
242(13)
8.3.1 General Remarks
242(2)
8.3.2 Liouville--Green Approximations
244(4)
8.3.3 Olver-Type Error Estimates
248(7)
8.4 Asymptotic Comparison Results for Second-Order Equations
255(11)
8.4.1 A General Comparison Result
255(2)
8.4.2 Nonoscillatory Second-Order Differential Equations
257(3)
8.4.3 Special Cases
260(6)
8.5 Asymptotic Factorization of a Differential Operator
266(4)
8.6 Perturbations of the Harmonic Linear Oscillator
270(12)
8.6.1 Perturbations of Harmonic Oscillators Involving Conditional Integrability
272(7)
8.6.2 Almost Every where Asymptotic Integration
279(3)
8.7 Some Liouville--Green Formulas for Third-Order Differential Equations
282(3)
8.8 Second-Order Matrix Equations
285(10)
9 Applications to Classes of Scalar Linear Difference Equations
295(74)
9.1
Chapter Overview
295(2)
9.2 Asymptotically Constant Scalar Equations
297(12)
9.3 Liouville--Green Results of Second-Order Linear Scalar Difference Equations
309(2)
9.4 Some Special Second-Order Linear Difference Equations
311(47)
9.4.1 l1-Perturbations and Liouville--Green--Olver Error Estimates
311(14)
9.4.2 Coefficients with Regular Growth
325(9)
9.4.3 Perturbations with Averaged Growth Conditions
334(8)
9.4.4 Perturbations Having Regular Variation
342(10)
9.4.5 Another Class of Perturbations with Bounded Variation
352(6)
9.5 Asymptotic Factorization of a Difference Operator
358(7)
9.6 Special Classes and Applications to Jacobi Operators
365(4)
10 Asymptotics for Dynamic Equations on Time Scales
369(24)
10.1 Introduction
369(4)
10.2 Levinson's Fundamental Theorem
373(5)
10.3 Weak Dichotomies
378(1)
10.4 Reduction to Eigenvalues
379(2)
10.5 Lp-Perturbations with p > 1
381(2)
10.6 Conditionally Integrable Perturbations
383(10)
References 393(8)
Index 401
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