Skip to main content

2015 | Buch

Asymptotic Integration of Differential and Difference Equations

insite
SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction, Notation, and Background
Abstract
This book is concerned with the problem of determining the asymptotic behavior of solutions of non-autonomous systems of linear differential and linear difference equations. It has been observed from the work by Poincaré and Perron that there is a very close and symbiotic relationship between many results for differential and difference equations, and we wish to further demonstrate this by treating the asymptotic theories here in parallel. In Chaps. 2, 4, 6, and 8 we will discuss topics related to asymptotic behavior of solutions of differential equations, and in Chaps. 3, 5, 7, and 9 some corresponding results for difference equations. In Chap. 10 we will show how some of these results can be simultaneously treated within the framework of so-called dynamic equations on time scales.
Sigrun Bodine, Donald A. Lutz
Chapter 2. Asymptotic Integration for Differential Systems
Abstract
In this chapter we are concerned with the following general problem: If we are given a linear system
$$\displaystyle{ y' = \left [A(t) + R(t)\right ]y,\qquad t \geq t_{0}, }$$
(2.1)
and “know” a fundamental solution matrix X(t) for the (unperturbed) system x′ = A(t)x, how “small” should the perturbation R(t) be so that we can determine an asymptotic behavior for solutions of (2.1)? This question is intentionally vague because depending upon the particular circumstances, there are many possible answers.
Sigrun Bodine, Donald A. Lutz
Chapter 3. Asymptotic Representation for Solutions of Difference Systems
Abstract
In this chapter we first consider systems of linear difference equations
$$\displaystyle{ y(n + 1) = A(n)y(n),\qquad n \geq n_{0}, }$$
(3.1)
which are in what we call l 1-diagonal form
$$\displaystyle{ y(n + 1) = [\varLambda (n) + R(n)]y(n),\qquad n \geq n_{0} }$$
(3.2)
and establish results similar to those in Sects. 2.​22.​4. This includes discrete versions of Coppel’s theorem and Levinson’s fundamental theorem from Sect. 2.​2 for l 1-diagonal systems, and some results for weak dichotomies.
Sigrun Bodine, Donald A. Lutz
Chapter 4. Conditioning Transformations for Differential Systems
Abstract
In this chapter we will consider linear systems of the form x  = A(t)x and discuss various procedures which may be used for transforming such a system (if possible) into an L-diagonal form, so that the theorems in Chap. 2 could be used to obtain an asymptotic representation for solutions.
Sigrun Bodine, Donald A. Lutz
Chapter 5. Conditioning Transformations for Difference Systems
Abstract
In this chapter we will consider linear difference systems of the form \(x(n + 1) = A(n)x(n)\), where det A(n) ≠ 0 for all n ≥ n 0. Various procedures will be discussed (similar to those in the preceding chapter) for bringing such a system (if possible) into what we have called an L-diagonal form, so that the results of Chap. 3 may be used.
Sigrun Bodine, Donald A. Lutz
Chapter 6. Perturbations of Jordan Differential Systems
Abstract
Whereas in Chaps. 2 and 4, we studied the asymptotic behavior of solutions of perturbations of diagonal systems of differential equations, we are now interested in the asymptotic behavior of solutions of systems of the form
$$\displaystyle{ y^{{\prime}} = \left [J(t) + R(t)\right ]y(t)t \geq t_{ 0}, }$$
(6.1)
where J(t) is now in Jordan form and R(t) is again a perturbation. Early results on perturbations of constant Jordan blocks include works by Dunkel [50] and Hartman–Wintner [73]. The focus here is an approach, developed by Coppel and Eastham, to reduce perturbed Jordan systems to a situation where Levinson’s fundamental theorem can be applied.
Sigrun Bodine, Donald A. Lutz
Chapter 7. Perturbations of Jordan Difference Systems
Abstract
In this brief chapter, we only consider perturbations of systems of difference equations with a single non-singular Jordan block. That is, we consider
$$\displaystyle{ y(n+1) = \left [\lambda I + N + R(n)\right ]y(n),\qquad \lambda \neq 0,\qquad N = \left (\begin{array}{cccc} 0&1&& \\ &0 &\ddots & \\ & &\ddots&1\\ & & &0 \end{array} \right )\,,\qquad n \geq n_{0}. }$$
(7.1)
Following the approach taken in Sect. 6.​2, the next theorem can be considered as a discrete counterpart of Corollary 6.​2, and its proof is parallel to the proof given in Theorem 6.​1.
Sigrun Bodine, Donald A. Lutz
Chapter 8. Applications to Classes of Scalar Linear Differential Equations
Abstract
In this chapter we consider various classes of dth-order (d ≥ 2) linear homogeneous equations \(\displaystyle{ y^{(d)} + a_{ 1}(t)y^{(d-1)} +\,\ldots \, +a_{ d}(t)y = 0, }\)
Sigrun Bodine, Donald A. Lutz
Chapter 9. Applications to Classes of Scalar Linear Difference Equations
Abstract
In this chapter we are interested in scalar dth-order linear difference equations (also called linear recurrence relations) of the form \(\displaystyle{ y(n + d) = c_{1}(n)y(n) +\, \cdots +\, c_{d}(n)y(n + d - 1),\qquad n \in \mbox{ $\mathbb{N}$}, }\)
Sigrun Bodine, Donald A. Lutz
Chapter 10. Asymptotics for Dynamic Equations on Time Scales
Abstract
In the foregoing chapters, one sees that many asymptotic results for linear differential and difference equations as well as the methods used to derive them are closely related. So it is natural to ask for a framework which would encompass both sets of results as well as including some generalizations. One possibility for doing this has been discussed by Spigler and Vianello [143].
Sigrun Bodine, Donald A. Lutz
Backmatter
Metadaten
Titel
Asymptotic Integration of Differential and Difference Equations
verfasst von
Sigrun Bodine
Donald A. Lutz
Copyright-Jahr
2015
Electronic ISBN
978-3-319-18248-3
Print ISBN
978-3-319-18247-6
DOI
https://doi.org/10.1007/978-3-319-18248-3