nach oben

2003 | Buch

Kapitel lesen Erstes Kapitel lesen

Advances in Dynamic Equations on Time Scales

herausgegeben von: Martin Bohner, Allan Peterson

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

The development of time scales is still in its infancy, yet as inroads are made, interest is gathering steam. Of a great deal of interest are methods being intro duced for dynamic equations on time scales, which now explain some discrepancies that have been encountered when results for differential equations and their dis crete counterparts have been independently considered. The explanations of these seeming discrepancies are incidentally producing unifying results via time scales methods. The study of dynamic equations on time scales is a fairly new subject, and research in this area is rapidly growing. It has been created in order to unify continuous and discrete analysis, and it allows a simultaneous treatment of dif ferential and difference equations, extending those theories to so-called dynamic equations. An introduction to this subject is given in Dynamic Equations on Time Scales: An Introduction with Applications (MARTIN BOHNER and ALLAN PETER SON, Birkhauser, 2001 [86]). The current book is designed to supplement this introduction and to offer access to the vast literature that has already emerged in this field. It consists of ten chapters, written by an international team of 21 experts in their areas, thus providing an overview of the recent advances in the theory on time scales. We want to emphasize here that this book is not just a collection of papers by different authors.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to the Time Scales Calculus

Abstract

In this chapter we introduce some basic concepts concerning the calculus on time scales that one needs to know to read this book. Most of these results will be stated without proof. Proofs can be found in the book by Bohner and Peterson [86]. A time scale is an arbitrary nonempty closed subset of the real numbers. Thus

$$ \mathbb{R},\mathbb{Z},\mathbb{N},\mathbb{N}_0 , $$

i.e., the real numbers, the integers, the natural numbers, and the nonnegative integers are examples of time scales, as are

$$ [0,1] \cup [2,3],[0,1] \cup \mathbb{N} $$

, and the Cantor set, while

$$ \mathbb{Q},\mathbb{R}\backslash \mathbb{Q},\mathbb{C}(0,1), $$

i.e., the rational numbers, the irrational numbers, the complex numbers, and the open interval between 0 and 1, are not time scales. Throughout this book we will denote a time scale by the symbol $ \mathbb{T} $ . We assume throughout that a time scale $ \mathbb{T} $ has the topology that it inherits from the real numbers with the standard topology.

Martin Bohner, Gusein Guseinov, Allan Peterson

Chapter 2. Some Dynamic Equations

Abstract

In this chapter we consider several dynamic equations and present methods on how to solve these equations. Among them are linear equations of higher order, Euler-Cauchy equations of higher order, logistic equations (or Verhulst equations), Bernoulli equations, Riccati equations, and Clairaut equations.

Elvan Akin-Bohner, Martin Bohner

Chapter 3. Nabla Dynamic Equations

Abstract

If $ \mathbb{T} $ has a right-scattered minimum m, define $ \mathbb{T}_\kappa : = \mathbb{T} - \{ m\} $ ; otherwise, set $ \mathbb{T}_\kappa = \mathbb{T} $ . The backwards graininess $ \nu :\mathbb{T}_\kappa \to \mathbb{R}_0^ + $ is defined by

$$ \nu (t) = t - \rho (t). $$

For $ f:\mathbb{T} \to \mathbb{R} $ and $ t \in \mathbb{T}_\kappa $ , define the nabla derivative [42] of f at t, denoted f ^∇(t), to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t such that

$$ |f(\rho (t)) - f(s) - f^\nabla (t)(\rho (t) - s)| \leqslant \varepsilon |\rho (t) - s) $$

for all s € U. For $ \mathbb{T} = \mathbb{R} $ , we have f ^∇=f′, the usual derivative, and for $ \mathbb{T} = \mathbb{Z} $ we have the backward difference operator, f ^∇(t)=∇f(t):=f(t)-f(t-1). Note that the nabla derivative is the alpha derivative when α = p. Many of the results in this chapter can be generalized to the alpha derivative case. Many of the results in this chapter can be found in [35, 37].

Douglas Anderson, John Bullock, Lynn Erbe, Allan Peterson, HoaiNam Tran

Chapter 4. Second Order Self-Adjoint Equations with Mixed Derivatives

Abstract

In this chapter, we are concerned with the second order self-adjoint dynamic equation (p(t)x ^∆)^∇+q(t)x=0 on a time scale. When $ \mathbb{T} = \mathbb{R} $ , this reduces to the usual self-adjoint differential equation, (p(t)x′)′+q(t)x=0.

Kirsten Messer

Chapter 5. Riemann and Lebesgue Integration

Abstract

In [86, Section 1.4], the concept of integration on time scales is defined by means of an antiderivative (or pre-antiderivative) of a function and is called the Cauchy integral (we remark that in [191, p. 255] such an integral is named as the Newton integral).

Martin Bohner, Gusein Guseinov

Chapter 6. Lower and Upper Solutions of Boundary Value Problems

Abstract

The method of lower and upper solutions, coupled with the monotone iterative technique provides an effective and flexible mechanism that offers theoretical as well as constructive existence results for nonlinear problems in a closed set which is generated by the lower and upper solutions. The lower and upper solutions serve as bounds for solutions which are improved by a monotone iterative process. The ideas imbedded in this technique have proved to be of immense value and have played an important rôle in unifying a variety of nonlinear problems. A comprehensive introduction to the monotone iterative techniques is given by Ladde, Lakshmikantham and Vatsala in [196].

Elvan Akin-Bohner, Ferhan Merdivenci Atici, Billûr Kaymakçalan

Chapter 7. Positive Solutions of Boundary Value Problems

Abstract

At the time of this book, there is considerable research devoted to positive solutions for boundary value problems in each of the areas of partial differential equations, ordinary differential equations, and finite difference equations. In view of the unification theory brought forth by dynamic equations on time scales, it is natural that some research has emerged in terms of positive solutions for boundary value problems for dynamic equations on time scales. In this chapter, we present a synopsis of some of this recent work in a number of venues, including positive solutions that arise as eigenfunctions in a positive cone associated with eigenvalue comparison results, existence of at least one positive solution for a nonlinear dynamic equation as an application of the Guo-Krasnosel’skii fixed point theorem, existence of at least two positive solutions as dual applications of the Guo-Krasnosel’ski fixed point theorem or in some cases of the Avery-Henderson fixed point theorem. Finally, we conclude with results concerning the existence of at least three positive solutions via applications of either the Leggett-Williams fixed point theorem or a generalization of this theorem due to Avery.

Douglas Anderson, Richard Avery, John Davis, Johnny Henderson, William Yin

Chapter 8. Disconjugacy and Higher Order Dynamic Equations

Abstract

In this chapter, we introduce the study of disconjugacy of nth order dynamic equations on time scales. Disconjugacy of ordinary differential equations is thoroughly studied and has a rich history. Much of what we develop in this chapter has been presented for ordinary differential equations in Coppel’s often cited monograph [100]. The analogous theory for forward difference equations was developed by Philip Hartman [154] in a landmark paper which has generated so much activity in the study of difference equations.

Paul Eloe

Chapter 9. Boundary Value Problems on Infinite Intervals: A Topological Approach

Abstract

The aim of this chapter is twofold. First we wish to survey most of the fixed point theorems available in the literature for compact operators defined on Fréchet spaces. In particular we present the three “most applicable” results from the literature in Section 9.2. The first result is the well-known Schauder-Tychonoff theorem, the second, a Furi-Pera type result and the third, a fixed point result based on a diagonalization argument. Applications of these fixed point theorems to differential and difference equations can be found in a recent book of Agarwal and O’Regan [17]. Our second aim is to survey the results in the literature concerning time scale problems on infinite intervals. Only a handful of results are known, and the theory we present in Section 9.3 is based on the diagonalization approach in Section 9.2; this approach seems to give the most general and natural results. In Section 9.4 we consider linear systems on infinite intervals.

Ravi Agarwal, Martin Bohner, Donal O’Regan

Chapter 10. Symplectic Dynamic Systems

Abstract

This chapter continues from [86, Chapter 7] the study of symplectic dynamic systems of the form (S)

$$ z^\Delta = S(t)z $$

on time scales. In particular, we investigate the relationship between the nonoscillatory properties (no focal points) of certain conjoined bases of (S), the solvability of the corresponding Riccati matrix dynamic equation, and the positivity of the associated quadratic functional. Furthermore, we establish Sturmian separation and comparison theorems. As applications of the transformation theory of symplectic dynamic systems, we study trigonometric and hyperbolic symplectic systems, and the Prüfer transformation.

Ondřej Došlý, Stefan Hilger, Roman Hilscher

Backmatter

Titel: Advances in Dynamic Equations on Time Scales
herausgegeben von: Martin Bohner
Allan Peterson
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-8230-9
Print ISBN: 978-1-4612-6502-3
DOI: https://doi.org/10.1007/978-0-8176-8230-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to the Time Scales Calculus

Chapter 2. Some Dynamic Equations

Chapter 3. Nabla Dynamic Equations

Chapter 4. Second Order Self-Adjoint Equations with Mixed Derivatives

Chapter 5. Riemann and Lebesgue Integration

Chapter 6. Lower and Upper Solutions of Boundary Value Problems

Chapter 7. Positive Solutions of Boundary Value Problems

Chapter 8. Disconjugacy and Higher Order Dynamic Equations

Chapter 9. Boundary Value Problems on Infinite Intervals: A Topological Approach

Chapter 10. Symplectic Dynamic Systems

Backmatter