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

Hence, taking , , and , we obtain

      where we applied the Saks–Henstock lemma (Lemma 1.45) to obtain

      for every .

      A proof of the next proposition follows similarly as the proof of Theorem 1.66.

      

      Proposition 1.67: Let be any interval of the real line and , with . Consider functions and of locally bounded variation. Assume that is locally Perron–Stieltjes integrable with respect to , that is, the Perron–Stieltjes integral exists, for every compact interval . Assume, further, that , defined by

       is also of locally bounded variation. Then, the Perron–Stieltjes integrals and exist and

      (1.11)

      Yet another substitution formula for Perron–Stieltjes integrals, borrowed from [72, Theorem 11], is brought up here and, again, another interesting trick provided by Professor Hönig is used in its proof. Such substitution formula will be used in Chapter 3 in order to guarantee the existence of some Perron–Stieltjes integrals. As a matter of fact, the corollary following Theorem 1.68 will do the job.

      Theorem 1.68: Consider functions , , , that is,

       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

Скачать книгу