Fundamental Theorems of Ordinary Differential Equations

Abstract

In this chapter, we explain the fundamental problems of the existence and uniqueness of the initial-value problem $$ \frac{{d\vec y}}{{dt}} = \vec f\left( {t,\vec y} \right), \vec y\left( {{t_0}} \right) = {\vec c_0} $$ (P) in the case when the entries of $$\frac{{d\vec{y}}}{{dt}} = f\left( {t,\vec{y}} \right),{\text{ }}\vec{y}\left( {{{t}_{0}}} \right) = {{\vec{c}}_{0}}$$ ((P)) 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 and ℝn is 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.KeywordsRational NumberSuccessive ApproximationLipschitz ConditionFundamental TheoremInfinite SubsequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.