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

(1.7) hold.

      The next result is a Substitution Formula for Perron–Stieltjes integrals. A similar result holds for Riemann–Stieltjes integrals. For a proof of it, see [72, Theorem 11].

      Theorem 1.64: Consider functions , , and . Let

       Then, if and only if , in which case, we have

      (1.8)

      (1.9)

Using Theorem 1.53, one can prove the next corollary. See [72, Corollary 8]. From now on, and are Banach spaces.

      Corollary 1.65: Consider functions , , and , and define

       Then, and equality (1.8) and inequality (1.9) hold.

      Another substitution formula for Perron–Stieltjes integrals is presented next. Its proof uses a very nice trick provided by Professor C. S. Hönig while advising M. Federson's Master Thesis. Such result is borrowed from [72, Theorem 12].

      Theorem 1.66: Consider functions , and , that is,

       Then, , if and only if and

      (1.10)

      Proof. Since , given , there exists a gauge on such that, for every ‐fine ,

      Taking approximated sums for and , we obtain

      But, if and , then