Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

-fine ,

(1.A.2)

      The limit exists, since for ,

      Hence, if , then

Thus, the first summand on the right-hand side of the last inequality is smaller than by (1.A.2), the third summand is smaller than by the definition of and, if we refine the gauge , we may suppose, by the definition of , that the second summand is smaller than , and the proof is complete.

We show next that Lemma 1.93 remains valid if we replace by , that is, if instead of the space of Kurzweil–McSchane integrable functions, we consider the space of Henstock–McSchane integrable functions.

      Lemma 1.94: Let be a sequence in and be a function. If , then and

      Proof. By Lemma 1.93, , and we have the convergence of the integrals. It remains to prove that , that is, for every , there exists a gauge on such that for every -fine ,

      However,