an example. Let Then:
Hence
A concept related to Lipschitz continuity is called a contraction.
Definition 1.10 Let
In general, a contraction maps a pair of points into another pair of points that are closer together. A contraction is always continuous.
The ability to discover and apply contraction mappings has considerable theoretical and numerical value. For example, it is possible to prove that stochastic differential equations (SDEs) have unique solutions by the application of fixed point theorems:
Brouwer's fixed point theorem
Kakutani's fixed point theorem
Banach's fixed point theorem
Schauder's fixed point theorem
Our interest here lies in the following fixed point theorem.
Theorem 1.2 (Banach Fixed Point Theorem) Let T be a contraction of a complete metric space (X, d) into itself:
Then T has a unique fixed-point
In general, we assume that X is a Banach space and that T is a linear or non-linear mapping of X into itself. We then say that x is a fixed point of T if
1.7 SUMMARY AND CONCLUSIONS
In this chapter we gave an introduction to a number of relevant mathematical concepts from real analysis that are used throughout this book, directly or indirectly. We also have introduced other relevant topics in other chapters. To summarise:
Chapter 1: Real analysis
Chapter 4: Finite dimensional vector spaces
Chapter 5: Numerical linear algebra
Chapter 16: Complex analysis
In this way we hope that this book becomes more self-contained than otherwise.
CHAPTER 2 Ordinary Differential Equations (ODEs), Part 1
It is better to solve one problem five different ways, than to solve five problems one way.
George Pólya.
2.1 INTRODUCTION AND OBJECTIVES
In this chapter we introduce a class of differential equations in which the highest order derivative is one. Furthermore, these equations have a single independent variable (which in nearly all applications plays the role of time). In short, these are termed ordinary differential equations (ODEs) precisely because of the dependence on a single variable.
ODEs crop up in many application areas, such as mechanics, biology, engineering, dynamical systems, economics and finance, to name just a few. It is for this reason that we devote two dedicated chapters to them.
The following topics are discussed in this chapter:
Motivational examples of ODEs
Qualitative properties of ODEs
Common finite difference schemes for initial value problems for ODEs
Some theoretical foundations.
