Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Then,

and, hence, which completes the proof.

Using the fact that , the following corollary follows immediately from Lemma 1.30.

Corollary 1.31: Let . Then the sets

are finite for every .

Thus, we have the next result which can be found in [126, Proposition I.2.10].

Proposition 1.32: Let . Then the set of points of discontinuity of is countable.

Let us define

A proof that equipped with the variation norm, , is complete can be found in [126, Theorem I.2.11]. We reproduce it in the next theorem.

Theorem 1.33: , equipped with the variation norm, is a Banach space.

Proof. We know that , with the variation norm, is a Banach space. Let be a sequence in converging to in the variation norm. Then, since for every and every , , we obtain

which tends to zero as . Hence, .

We end this section with the Helly's choice principle for Banach space‐valued functions due to C. S. Hönig. See [127, Theorem I.5.8].

Theorem 1.34 (Theorem of Helly): Let be a BT and consider a sequence of elements of , with , for all , and such that there exists , with for all and all . Then, and . Moreover, if , with , for all , then and .

Proof. Consider a division and let , with , for . Then, for all , we have

where the first member on the right‐hand side of the inequality is smaller than , since . Moreover, by hypothesis, given , there is such that, for all , , for Hence, for all ,