Carathéodory Solutions of an Initial Value Problem

In the setting of Integrals over Subintervals of a Closed Interval: Notation, assume the axiom of dependent choice. Let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a function and let $\xi\in\mathbb{R}^{n}$.

(i) Carathéodory solutions. A Carathéodory solution on $[a,b]$ of the integral equation with right-hand side $F$ and initial value $\xi$ is a function $x:[a,b]\to\mathbb{R}^{n}$ such that the function $r\mapsto F(r,x(r))$ on $[a,b]$ lies in $\mathcal{L}^{1}([a,b];\mathbb{R}^{n})$ and $x$ satisfies the integral equation

$$x(t)=\xi+\int_{[a,t]}F\bigl(r,x(r)\bigr)\,dr\qquad\bigl(t\in[a,b]\bigr).$$

(ii) The solution set. $\mathrm{Sol}(F,\xi)$ is the set of those elements of the set $(\mathbb{R}^{n})^{[a,b]}$ of functions from $[a,b]$ to $\mathbb{R}^{n}$ that are Carathéodory solutions on $[a,b]$ of this equation (a subset of a set by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii), the condition being predicative as it is written with the functions of Functions and Function-Valued Operators Given by Integrals over Subintervals (iv) and Functions and Function-Valued Operators Given by Integrals over Subintervals (iii); by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i), $x\in\mathrm{Sol}(F,\xi)$ if and only if $x$ is such a solution).