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

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      Corollary 1.55: Let and be such that . Then, and (1.3) holds.

      A second corollary of Theorem 1.54 follows by the fact that Riemann–Stieltjes integrals are special cases of Perron–Stieltjes integrals. Then, it suffices to apply Theorems 1.49 and 1.53.

      Corollary 1.56: Suppose the following conditions hold:

      1 either and , with ;

      2 or and .

       Then, , equality (1.3) holds, and we have

      (1.4)

      The next theorem is due to C. S. Hönig (see [129]), and it concerns multipliers for Perron–Stieltjes integrals.

      Theorem 1.57: Suppose and . Then, and Eqs. (1.3) and (1.4) hold.

      Since and , it is immediate that if and , then . As a matter of fact, the next result gives us information about the multipliers for the Henstock vector integral. See [72, Theorem 7].

      Theorem 1.58: Assume that and . Then, and equalities (1.3) and (1.4) hold.

      Proof. Since , is continuous by Theorem 1.49. Thus, given , there exists such that

      whenever , where . Moreover, there is a gauge on , with for , such that for every ‐fine ,

      Thus,

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