On the method of upper and lower solutions in Banach spaces under Carathéodory assumptions (Q5942923)

The authors find sufficient conditions to ensure the existence of a solution between a lower and an upper solution for an initial value problem \(x'(t)=f(t,x(t))\), \(t\in I\), \(x(t_0)=x_0\), where \(f:I\times E \to E\) is a Carathéodory function and \(E\) is a Banach space partially ordered by a cone with nonempty interior.