Generalized Ordinary Differential Equations in Abstract Spaces and Applications

The next result is a consequence of the fact that and Lemmas 1.97and 1.98. A proof of it can be found in [132, Theorem 10.3].

Corollary 1.99: All functions of are absolutely.integrable

The reader can find a proof of the next lemma in [35, Theorem 9].

Lemma 1.100: All functions of are.measurable

Finally, we can prove the following inclusion.

Theorem 1.101: .

Proof. The result follows from the facts that all functions of and, hence, of are measurable ( Lemma 1.100), and all functions of are absolutely integrable by Corollary 1.99.

As we mentioned before, the inclusion always holds. However, when is an infinite dimensional Banach space, then for sure , as shown by the next result due to C. S. Hönig (personal communication by him to his students in 1990 at the University of São Paulo) and presented in [73].

Proposition 1.102 (Hönig): If is an infinite dimensional Banach space, then there exists .

Proof. Let denote the dimension of . If , then the Theorem of Dvoretsky–Rogers (see [60] and also [57]) implies there exists a sequence in which is summable but not absolutely summable. Thus, if we define a function by , for , then whenever the integral exists. However, , since

and this completes the proof.

In the next example, borrowed from [73, Example 3.4], we exhibit a Banach space-valued function which is integrable in the variational Henstock sense and also in the sense of Kurzweil–McShane. Nevertheless, it is not absolutely integrable.

Example 1.103:Let be given by

Then,

which is summable in . Since the Henstock integral contains its improper integrals (and the same applies to the Kurzweil integral), we have . However, because the sequence is non-summable in . By the Monotone Convergence Theorem for the Kurzweil–McShane integral (which follows the ideas of [71] with obvious adaptations), . But , since is not bounded, where by we denote the space of Riemann–McShane integrable functions from to .