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

       and assume that . Thus, if and only if , in which case, we have

      (1.12)

      Proof. By hypothesis, . Therefore, for every , there is a gauge of such that for every ‐fine , we have

      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

Скачать книгу