Generalized Ordinary Differential Equations in Abstract Spaces and Applications

If we denote by the space of all absolutely continuous functions from to , then it is not difficult to prove that

In , we consider the usual supremum norm, , induced by .

The next two versions of the Fundamental Theorem of Calculus for Henstock vector integrals, as described in Definition 1.41, are borrowed from [70, Theorems 1 and 2]. We use the term almost everywhere in the sense of the Lebesgue measure .

Theorem 1.77: If and are both differentiable and is such that for almost every , then and , that is,

Theorem 1.78: If is differentiable and is bounded, then , and there exists the derivative for almost every , that is,

Corollary 1.79: Suppose is differentiable and nonconstant on any nondegenerate subinterval of and is bounded and such that . Then, almost everywhere in .

From Corollary 1.50, we know that if and , then . For the Henstock vector integral, we have the following analogue whose proof can be found in [70, Theorem 7].

Theorem 1.80: If and , then we have .

The next result is borrowed from [70, Theorem 5]. We reproduce its proof here.

Theorem 1.81: Suppose and is such that almost everywhere on . Then, and , that is,

Proof. Consider the sets

By hypothesis, , where denotes the Lebesgue measure. Hence, for every . In addition, . Then, for every and every , there exists a gauge on such that, for every -fine tagged partial division , with , for , we have

Consider a gauge of such that , whenever , and can assume any value in , otherwise. Then, for every -fine , we have