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2001 | Buch

The Time Scales Calculus Kapitel lesen Erstes Kapitel lesen

Dynamic Equations on Time Scales

An Introduction with Applications

verfasst von: Martin Bohner, Allan Peterson

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

On becoming familiar with difference equations and their close re­ lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that for ordinary differential equations. [HUGH L. TURRITTIN, My Mathematical Expectations, Springer Lecture Notes 312 (page 10), 1973] A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. BELL, Men of Mathematics, Simon and Schuster, New York (page 13/14), 1937] The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis [159] in 1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete analysis. This book is an intro­ duction to the study of dynamic equations on time scales. Many results concerning differential equations carryover quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equa­ tions and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals.

Inhaltsverzeichnis

Frontmatter
Chapter 1. The Time Scales Calculus
Abstract
A time scale (which is a special case of a measure chain, see Chapter 8) 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, Allan Peterson
Chapter 2. First Order Linear Equations
Abstract
Definition 2.1. Suppose \( f:\mathbb{T} \times \mathbb{R}^2 \to \mathbb{R}. \) Then the equation
$$ y^\Delta = f(t,y,y^\sigma ) $$
(2.1)
is called a first order dynamic equation, sometimes also a differential equation. If
$$ f(t,y,y^\sigma ) = f_1 (t)y + f_2 (t) or f(t,y,y^\sigma ) = f_1 (t)y^\sigma + f_2 (t) $$
for functions f1 and f2, then (2.1) is called a linear equation. A function \( y: \mathbb{T} \to \mathbb{R} \) is called a solution of (2.1) if
$$ y^\Delta (t) = f(t,y(t),y(\sigma (t))) is satisfied for all t \in \mathbb{T}^\kappa . $$
Martin Bohner, Allan Peterson
Chapter 3. Second Order Linear Equations
Abstract
In this section we consider the second order linear dynamic equation
$$ y^{\Delta \Delta } + p(t)y^\Delta + q(t)y = f(t), $$
where we assume that p, q, f ∊ Crd. If we introduce the operator L2 : C rd 2 → Crd by
$$ L_2 y(t) = y^{\Delta \Delta } (t) + p(t)y^\Delta (t) + q(t)y(t) $$
for \( t \in \mathbb{T}^{\kappa ^2 } \) , then (3.1) can be rewritten as L2y = f. If y ∊ C rd 2 and L2y(t) - f(t) for all \( t \in \mathbb{T}^{\kappa ^2 } \) , then we say y is a solution of L2y = f on T. The fact that L2 is a linear operator (see Theorem 3.1) is why we call equation (3.1) a linear equation. If f(t) = 0 for all \( t \in \mathbb{T}^{\kappa ^2 } \) , then we get the homogeneous dynamic equation L2y = 0. Otherwise we say the equation L2y = f is nonhomogeneous. The following principle of superposition is easy to prove and is left as an exercise.
Martin Bohner, Allan Peterson
Chapter 4. Self-Adjoint Equations
Abstract
In this chapter we are concerned with the self-adjoint dynamic equation of second order
$$ Lx = 0, where Lx(t) = (px^\Delta )^\Delta (t) + q(t)x^\sigma (t). $$
(4.1)
Martin Bohner, Allan Peterson
Chapter 5. Linear Systems and Higher Order Equations
Abstract
Definition 5.1. Let A be an m × n-matrix-valued function on \( \mathbb{T} \). We say that A is rd-continuous on \( \mathbb{T} \) if each entry of A is rd-continuous \( \mathbb{T} \) , and the class of all such rd-continuous onm × n-matrix-valued functions on \( \mathbb{T} \) is denoted, similar to the scalar case (see Definition 1.58), by
$$ C_{rd} = C_{rd} (\mathbb{T}) = C_{rd} (\mathbb{T},\mathbb{R}^{m \times n} ). $$
Martin Bohner, Allan Peterson
Chapter 6. Dynamic Inequalities
Abstract
We start this section with a comparison theorem. Throughout we let t0 ∊ \( t_0 \in \mathbb{T} \) .
Martin Bohner, Allan Peterson
Chapter 7. Linear Symplectic Dynamic Systems
Abstract
In this chapter we investigate so-called symplectic systems of dynamic equations, which have a variety of important equations as their special cases, e.g., linear Hamiltonian dynamic systems or Sturm-Liouville dynamic equations of higher (even) order. Many of the results in this chapter can be found in DošlÝ and Hilscher [121], and are extensions of results by Bohner and DošlÝ [76]. Throughout this chapter we denote by \( \mathcal{J} \) the 2n × 2n-matrix
$$ \mathcal{J} = \left( {\begin{array}{*{20}c} 0 & I \\ { - I} & 0 \\ \end{array} } \right). $$
We start by recalling the concepts of symplectic and Hamiltonian matrices.
Martin Bohner, Allan Peterson
Chapter 8. Extensions
Abstract
In this last chapter we present some possible forms of extensions of the theory of time scales. One such extension is the model of a measure chain as developed by Stefan Hilger in [160], which will be discussed in the first section of this chapter. We also present so-called alpha derivatives as introduced by Ahlbrandt, Bohner, and Ridenhour in [24]. Many results presented in this book can be derived for these more general models, and research in these areas continues.
Martin Bohner, Allan Peterson
Backmatter
Metadaten
Titel
Dynamic Equations on Time Scales
verfasst von
Martin Bohner
Allan Peterson
Copyright-Jahr
2001
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0201-1
Print ISBN
978-1-4612-6659-4
DOI
https://doi.org/10.1007/978-1-4612-0201-1