Generalized Ordinary Differential Equations in Abstract Spaces and Applications

is a division of and, to each subinterval , with , we associate a point called “tag” of the subinterval . We denote such semitagged division by and, by , we mean the set of all semitagged divisions of the interval . But what is the difference between a semitagged division and a tagged division? Well, in a semitagged division , it is not required that a tag belongs to its associated subinterval . In fact, neither the subintervals need to contain their corresponding tags. Nevertheless, likewise for tagged divisions, given a gauge of , in order for a semitagged division to be -fine, we need to require that

This simple modification provides an elegant characterization of the Lebesgue integral through Riemann sums (see [174]).

Let us denote by the space of all real-valued Kurzweil–McShane integrable functions , that is, is integrable in the sense of Kurzweil with the modification of McShane. Formally, we have the next definition which can be extended straightforwardly to Banach space-valued functions.

Definition 1.91:We say that is Kurzweil–McShane integrable , and we write if and only if there exists such that for every , there is a gauge on such that

whenever is -fine. We denote the Kurzweil–McShane integral of a function by .

The following inclusions hold

Moreover, as one can note by the next classical example.

Example 1.92:Let be defined by if , and , and consider . Since is Riemann improper integrable, , because the Kurzweil–Henstock (or Perron) integral contains its improper integrals (see Theorem 2.9, [158], or [213]). However, , since is not absolutely integrable (see also [227]).