On Carathéodory's conditions revisited (Q1852343)

Consider the initial value problem for systems \[ x'(t)=f \bigl( t,x(t)\bigr) \text{ in }I=[t_0,t_0+a] \text{ and }x(t_0)=x_0. \tag{1} \] Peano's existence theorem assumes that \(f(t,x)\) is continuous. Carathéodory's generalization assumes \((C)\): \(f(t,x)\) is measurable in \(t\) and continuous in \(x\) plus a boundedness condition. Z. Grande generalized this by replacing the continuity of \(f(t,x)\) in \(x\) by two assumptions (i) \(f(t,x(t)\) is measurable for any \(x(t)\in C(I)\) and another rather involved condition (ii). In the present paper, the author shows that (ii) is equivalent to the propertv (ii'): If a sequence \((x_n(t))\) in \(C(I)\) converges uniformly in \(I\) to \(x(t)\), then \(\int^t_{t_0} f(s,x_n)(s))ds\) tends to \(\int^t_{t_0}f (s,x(s))ds\) as \(n\to \infty\). With this new version, Grande's theorem can now more easily applied.