Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Taking approximated sums for and , we obtain

But, if and , then

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 3in order to guarantee the existence of some Perron–Stieltjes integrals. As a matter of fact, the corollary following Theorem 1.68will 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

Taking approximated Riemannian‐type sums for the integrals and , we obtain

On the other hand, when and , we have

Then, taking , , , and , we get