Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Lemma 1.15: Let be a sequence of functions in . Suppose, for each , the function satisfies

(1.2)

for every , where for each and the sequence is equiregulated. Then, the sequence is equiregulated.

Proof. Let be given. Since the sequence is equiregulated, it follows from Theorem 1.11that there is a division such that for every and , for . Thus, by (1.2), for every and every interval , with . Finally, Theorem 1.11ensures the fact that the sequence is equiregulated.

A clear outcome of Lemmas 1.13and 1.15follows below:

Corollary 1.16: Let be a sequence of functions from to and suppose the function satisfies condition (1.2) for every and , where and the sequence is equiregulated. If the sequence converges pointwisely to a function , then it also converges uniformly to .

In this subsection, we investigate an extension of the Arzelà–Ascoli theorem for regulated functions taking values in a general Banach space with norm .

Unlike the finite dimensional case, when we consider functions taking values in , the relatively compactness of a set does not come out as a consequence of the equiregulatedness of the set and the boundedness of the set , for each . In the following lines, we present an example, borrowed from [177] which illustrates this fact.

Example 1.17:Let be bounded. Suppose not relatively compact in . Thus, given an arbitrary , there is a sequence of functions in for which

for all and for some constant . Hence, the set

is bounded, once is bounded. Moreover, is equiregulated and is relatively compact in . On the other hand, is not relatively compact in .