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

, , we obtain

      which tends to zero as . Hence, .

      We end this section with the Helly's choice principle for Banach space‐valued functions due to C. S. Hönig. See [127, Theorem I.5.8].

      

      Theorem 1.34 (Theorem of Helly): Let be a BT and consider a sequence of elements of , with , for all , and such that there exists , with for all and all . Then, and . Moreover, if , with , for all , then and .

      Proof. Consider a division and let , with , for . Then, for all , we have

      where the first member on the right‐hand side of the inequality is smaller than , since . Moreover, by hypothesis, given , there is such that, for all , , for Hence, for all ,

      and we conclude the proof of the first part. The second part follows analogously.

      For more details about functions of bounded variation, the reader may want to consult [68], for instance.

1.3 Kurzweil and Henstock Vector Integrals

      Throughout this section, we consider functions and , where and are Banach spaces.

      1.3.1 Definitions

We start by recalling some definitions of vector integrals in the sense of J. Kurzweil and R. Henstock. At first, we need some auxiliary concepts, namely tagged division, gauge, and ‐fine tagged division.

      

      Definition 1.35: Let be a compact interval.

      1 Any set of point‐interval pairs such that and for , is called a tagged division of . In this case, we write , where denotes the set of all tagged divisions of .

      2 Any subset of a tagged division of is a tagged partial division of and, in this case, we write .

      3 A gauge on a set is any function . Given a gauge on , we say that is a ‐fine tagged partial division, whenever for , that is,whenever

      Before presenting the definition of any integral