Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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.

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

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 based on Riemannian sums concerning ‐fine tagged divisions of an interval , we bring up an important result which guarantees the existence of a ‐fine tagged division for a given gauge . This result is known as Cousin Lemma, and a proof of it can be found in [120, Theorem 4.1].

Lemma 1.36 (Cousin Lemma): Given a gauge of , there exists a ‐fine tagged division of .

Definition 1.37:We say that is Kurzweil ‐ integrable (or Kurzweil integrable with respect to ), if there exists such that for every , there is a gauge on such that for every ‐fine ,

In this case, we write and .

Analogously, we define the Kurzweil integral of with respect to a function .

Definition 1.38:We say that is Kurzweil integrable (or Kurzweil integrable with respect to ), if there exists such that given , there is a gauge on such that

whenever is ‐fine. In this case, we write and .