Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Thus,

But by Corollary 1.56, item (ii), and

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

since for every , we have

The proof is then complete.

A proof of the next result, borrowed from [72, Theorem 8], follows from the definitions of the integrals.

Theorem 1.59: Let and . If is bounded, then and

(1.5)

If, in addition, , then .

Corollary 1.60: Suppose with and . Then, and (1.5) holds.

Proof. By Theorem 1.49, . Then, the result follows from Theorem 1.47, since .

The next corollaries follow from Theorems 1.49and 1.53.

Corollary 1.61: Suppose with and . Then, , and we have

(1.6)

and the following integration by parts formula holds

(1.7)

Corollary 1.62: Consider functions and . Then, and equalities (1.6) and (1.7) hold.

The next two theorems generalize Corollary 1.62. For their proofs, the reader may want to consult [72].

Theorem 1.63: Consider . If respectively, , then respectively, and both (1.6) and (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 ,