Generalized Ordinary Differential Equations in Abstract Spaces and Applications

where

and is a division of .

The following properties are not difficult to prove:

1 (V1) Every is bounded and , .

2 (V2) Given and , we have .

Remark 1.28:Note that property (V1) above implies for all .

For more details about the spaces in Definition 1.27, the reader may want to consult [127]. The next results are borrowed from [126]. We include the proofs here since this reference is not easily available. Lemmas 1.29and 1.30below are, respectively, Theorems I.2.7 and I.2.8 from [126].

Lemma 1.29: Let . Then,

1 For all , there exists .

2 For all , there exists .

Proof. We only prove item (i), because item (ii) follows analogously. Consider an increasing sequence in converging to . Then,

for all Therefore, we have and, hence, as Thus, is a Cauchy sequence, since for any given , we have

for sufficiently large . Finally, note that the limit of is independent of the choice of , and we finish the proof.

It comes from Lemma 1.29that all functions of bounded variation are also regulated functions (see, e.g. [127, Corollary 3.4]) which, in turn, are Darboux integrable [127, Theorem 3.6].

Lemma 1.30: Let . For every , let . Then,

1 , ;

2 , .

Proof. By property (SB2), is increasing and, hence, and exist. By Lemma 1.29, and also exist. We prove (i). The proof of (ii) follows analogously.

Suppose . Then, property (V2) implies Thus,

and, hence,

Conversely, for any given , let . Then for every , there exists such that and , and there exists a division such that