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

      The next result concerns the existence of Perron–Stieltjes integrals. A proof of its item (i) can be found in [210, Theorem 15]. A proof of item (ii) follows similarly as the proof of item (i). See also [212, Proposition 7].

      Theorem 1.47: The following assertions hold.

      1 If and , then .

      2 If and , then we have .

The following consequence of Theorem 1.47 will be used later in many chapters. The inequalities follow after some calculations. See, for instance, [210, Proposition 10].

      

      Corollary 1.48: The following assertions hold.

      1 If and , then the Perron–Stieltjes integral exists, and we haveSimilarly, if and is nondecreasing, then

      2 If and , then the Perron–Stieltjes integral exists, and we have

      The next result, borrowed from [74, Theorem 1.2], gives us conditions for indefinite Perron–Stieltjes integrals to be regulated functions.

      Theorem 1.49: The following assertions hold:

      1 if and , then ;

      2 if and , then .

      Proof. We prove (i). Item (ii) follows similarly. For item (i), it is enough to show that

      because, in this case, the equality

      follows in an analogous way. By hypothesis, . Hence, given , there is a gauge on such that for every ‐fine division ,

Now, let . Since , there exists , for every . In particular, there exists such that

      If and , then by the Saks–Henstock lemma (Lemma 1.45)

      Thus,