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

      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,

But by Corollary 1.56, item (ii), and

      and a similar formula also holds for every subinterval contained in . Hence, for