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

and every , we have

      Then, .

      Let . The equiregulatedness of implies that there exists such that for every and . Take . Thus, for every ,

      which, together with the hypotheses, yield

      Hence, .

      Let . Since is equiregulated, there exists such that, for every , , for all Therefore, for every , we have

      for , where . Note that which contradicts the fact that . Hence, and the statement follows.

      1.1.3 Uniform Convergence

      This subsection brings a few results borrowed from [177]. In particular, Lemma 1.13 describes an interesting and useful property of equiregulated converging sequences of Banach space‐valued functions and it is used later in the proof of a version of Arzelà–Ascoli theorem for Banach space‐valued regulated functions.

      Lemma 1.13: Let be a sequence of functions from to . If the sequence converges pointwisely to and is equiregulated, then it converges uniformly to .

      Proof. By hypothesis, the sequence of functions is equiregulated. Then, Theorem 1.11 yields that, for every