Fundamental Theorems of Ordinary Differential Equations

Abstract

In this chapter, we explain the fundamental problems of the existence and uniqueness of the initial-value problemP$$\frac{{d\vec y}}{{dt}} = \vec f\left( {t,\vec y} \right), \vec y\left( {{t_0}} \right) = {\vec c_0}$$in the case when the entries of (P)$$\frac{{d\vec{y}}}{{dt}} = f\left( {t,\vec{y}} \right),{\text{ }}\vec{y}\left( {{{t}_{0}}} \right) = {{\vec{c}}_{0}}$$are real-valued and continuous in the variable $$\left( {t,\vec{y}} \right)$$,where t is a real independent variable and$$\vec{y}$$is an unknown quantity in ℝn. Here, ℝ is the real line andℝnis the set of all n-column vectors with real entries. In I-1, we treat the problem when$$\vec{f}\left( {t,\vec{y}} \right)$$, satisfies the Lipschitz condition in$$\vec{y}$$. The main tools are successive approximations and Gronwall's inequality (Lemma I-1–5). In I-2, we treat the problem without the Lipschitz condition. In this case, approximating$$\left( {t,\vec{y}} \right)$$by smooth functions, e-approximate solutions are constructed. In order to find a convergent sequence of approximate solutions, we use ArzelàAscoli's lemma concerning a bounded and equicontinuous set of functions (Lemma I-2-3). The existence Theorem I-2-5 is due to A. L. Cauchy and G. Peano [Peal] and the existence and uniqueness Theorem I-1-4 is due to É. Picard [Pi] and E. Lindelöf [Lindl, Lind2]. The extension of these local solutions to a larger interval is explained in I-3, assuming some basic requirements for such an extension. In I-4, using successive approximations, we explain the power series expansion of a solution in the case when$$\left( {t,\vec{y}} \right)$$, is analytic in$$\left( {t,\vec{y}} \right)$$, In each section, examples and remarks are given for the benefit of the reader. In particular, remarks concerning other methods of proving these fundamental theorems are given at the end of I-2.