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

      Such spaces are complete when endowed, respectively, with the norm given by the variation and the norm given by the semivariation

      where

      and is a division of .

The following properties are not difficult to prove:

      1 (V1) Every is bounded and , .

      2 (V2) Given and , we have .

      Remark 1.28: Note that property (V1) above implies for all .

      For more details about the spaces in Definition 1.27, the reader may want to consult [127]. The next results are borrowed from [126]. We include the proofs here since this reference is not easily available. Lemmas 1.29 and 1.30 below are, respectively, Theorems I.2.7 and I.2.8 from [126].

      Lemma 1.29: Let . Then,

      1 For all , there exists .

      2 For all , there exists .

      Proof. We only prove item (i), because item (ii) follows analogously. Consider an increasing sequence in converging to . Then,

      for all Therefore, we have and, hence, as Thus, is a Cauchy sequence, since for any given , we have

      for sufficiently large . Finally, note that the limit of is independent of the choice of , and we finish the proof.

      It comes from Lemma 1.29 that all functions of bounded variation are also regulated functions (see, e.g. [127, Corollary 3.4]) which, in turn, are Darboux integrable [127, Theorem 3.6].

      Lemma 1.30: Let . For every , let . Then,

      1 , ;

      2 , .

Proof. By property (SB2), is increasing and, hence, and exist. By Lemma 1.29,

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