Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

is an equiregulated set. Indeed, by Lemma 1.13, converges uniformly to .

      Since the function is regulated, its lateral limits exist. Then, for every and every , there is such that

      for every .

      But the hypotheses say that we can find such that, for , we have

      When satisfies , we have, for every ,

On the other hand, when satisfies , we get, for ,

      But this yields the fact that is an equiregulated sequence, and the proof is complete.

      The next lemma guarantees that, if a sequence of functions is bounded by an equiregulated sequence of functions, then is also equiregulated.

      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 1.11 that there is a division such that for every and