Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Take and consider such that . Therefore, either , for some , in which case, we have

or , for some , in which case, we have

Hence, for every , satisfies the Cauchy condition. Due to the fact that is a complete space and because is a Cauchy sequence, the limit exists.

We conclude by considering Then, on , by Lemma 1.13. Hence, is the uniform limit of the subsequence in . Finally, any sequence admits a converging subsequence which, in turn, implies that is a relatively compact set, and the proof is finished.

We end this subsection by mentioning an Arzelà–Ascoli‐type theorem for regulated functions taking values in . A slightly different version of it can be found in [96].

Corollary 1.19: The following conditions are equivalent:

1 a set is relatively compact;

2 the set is bounded, and there are an increasing continuous function , with , and a nondecreasing function such that, for every ,for

3 is equiregulated, and for every , the set is bounded.

We point out in [96, Theorem 2.17], item (ii), it is required that is an increasing function. However, it is not difficult to see that if is a nondecreasing function, then taking yields is an increasing function. Therefore, Corollary 1.19follows as an immediate consequence of [96, Theorem 2.17].

The concept of a function of bounded ‐variation generalizes the concepts of a function of semivariation and of a function of bounded variation, as we will see in the sequel.

Definition 1.20:A bilinear triple (we write BT) is a set of three vector spaces , , and , where and are normed spaces with a bilinear mapping . For and , we write , and we denote the BT by or simply by . A topological BT is a BT , where is also a normed space and is continuous.