Countable systems of ordinary differential equations

Abstract

This chapter discusses countable systems of ordinary differential equations. These equations refer to systems of the type x i ' = f i ( t , x 1 , x 2 , …) for every i ≥ 1 or with ordinary differential operators L of finite order instead of the first derivative. The chapter highlights that the history of such systems dates back to the origins of functional analysis around 1900 at least, and the purpose of the first papers was to show that the techniques available during this period—namely, Picard's successive approximation method and Hubert's quadratic forms on 1 2 —may be applied to obtain existence and uniqueness theorems for the initial value problem x i ' = f i ( t , x 1 , x 2 , …). The chapter highlights that an interesting specific problem for countable systems is the convergence of various truncation methods. It also analyzes the existence of maximal and minimal solutions to the initial value problem.