Caratheodory Conditions

Abstract

In Section 2, we defined a functional differential equation for continuous f: R × C → Rⁿ. On the other hand, it was then shown that the initial value problem was equivalent to

$$\begin{array}{*{20}c} {{\rm{x}}_{\rm{\sigma }} = {\rm{\phi }}} \\ {{\rm{x}}\left( {\rm{t}} \right) = {\rm{\phi }}\left( {\rm{0}} \right) + \int_{\rm{\sigma }}^{\rm{t}} {{\rm{f}}\left( {{\rm{s}},{\rm{x}}_{\rm{s}} } \right){\rm{ds,t}} \ge \sigma {\rm{.}}} } \\ \end{array}$$

((7.1))

The equation is certainly meaningful for a more general class of functions f if it is not required that x(t) has a continuous first derivative for t > σ. We give in this section the appropriate generalization to functional differential equations of the well known Caratheodory conditions of ordinary differential equations.