Generalized Ordinary Differential Equations in Abstract Spaces and Applications

The next example, borrowed from [73], exhibits a Cauchy sequence of Henstock integrable functions which is not convergent in the usual Alexiewicz norm, .

Example 1.84:Consider functions , defined by , where , whenever , , and otherwise.

Hence,

Then,

and, hence,

By induction, one can show that

for every . Then,

which goes to zero for sufficiently large , with . Thus, is a -Cauchy sequence. On the other hand,

for every . Hence, there is no function , with , such that .

The next result follows from Theorem 1.80. A proof of it can be found in [75, Theorem 5].

Theorem 1.85: Suppose is nonconstant on any nondegenerate subinterval of . Then, the mapping

is an isometry, that is onto a dense subspace of .

The next result, known as straddle Lemma , will be useful to prove that the space of regulated functions from to is dense in in the Alexiewicz norm . For a proof of the straddle Lemma, the reader may want to consult [130, 3.4] or [119].

Lemma 1.86 (Straddle Lemma): Suppose are functions such that is differentiable, with , for all . Then, given , there exists such that

whenever .

The next result is adapted from [75, Theorem 8].

Proposition 1.87: Suppose is differentiable and nonconstant on any nondegenerate subinterval of . Then, the Banach space is dense in under the Alexiewicz norm .

Proof. Assume that and let be given. We need to find a function such that , or equivalently,