Initial value problems

Abstract

We begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to

$$ \left\{ {\begin{array}{*{20}{c}}{y' = f(t,y)} \\ {y(0) = r}\end{array}} \right. $$

(1.1)

where f: I × R ⁿ → R ⁿ and I = [0, b]. We shall seek solutions that are defined either locally or globally on I, according to the assumptions imposed on f. Notice that (1.1) is a system of first order equations because f takes its values in R ⁿ. In section 3.2 we will first establish some basic existence theorems which guarantee that a solution to (1.1) exists for t > 0 and near zero. Familiar examples show that the interval of existence can be arbitrarily short, depending on the initial value r and the nonlinear behaviour of f. As a result we will also examine in section 3.2 the dependence of the interval of existence on f and r. We mention in passing that, in the results which follow, the interval I can be replaced by any bounded interval and the initial value can be specified at any point in I. The reasoning needed to cover this slightly more general situation requires minor modifications on the arguments given here. In this chapter we also present the notion of upper and lower solution for initial value problems.