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

      Taking approximated Riemannian‐type sums for the integrals and , we obtain

      On the other hand, when and , we have

      Then, taking , , , and , we get

      because the Saks‐Henstock lemma (Lemma 1.45) yields , for every .

      

      Corollary 1.69: Consider functions , , , and . Then, we have , , and Eq. (1.12) holds.

      Proof. Theorem 1.47, item (i), yields . Then, the statement follows from Theorem 1.68.

      The next result gives us an integration by parts formula for Perron–Stieltjes integrals. A proof of it can be found in [212, Theorem 13].

      Proposition 1.70: Suppose and or and . Then, the Perron–Stieltjes integrals and exist, and the following equality holds:

       where , , , and

      As an immediate consequence of the previous proposition, we have the following result.

      Corollary 1.71: If and is a nondecreasing function, then the integral exists.

      We end this subsection by presenting a result, borrowed from [172] and [179, Theorem 5.4.5], which gives us a change of variable formula for Perron–Stieltjes integrals.

      

      Theorem 1.72: Suppose is increasing and