Generalized Ordinary Differential Equations in Abstract Spaces and Applications

The next result shows that is a Banach space with respect to a special norm. This result, whose proof follows ideas similar to those of [124, 220], will be largely used in. Chapters 5and 8

Proposition 1.9: The space , equipped with the norm

is a Banach space.

Proof. Let be the linear mapping defined by

for all and .

Claim . is an isometric isomorphism. Indeed, is an isometry because

for all . Moreover, if , then defined by

is such that and , since

Therefore, is onto and the Claim is proved.

Once is an isometric isomorphism and is a Banach space, we conclude that is also a Banach space.

In this subsection, our goal is to investigate important properties of equiregulated sets. In addition to [97], the reference [40] also deals with a characterization of subsets of equiregulated functions.

Definition 1.10:A set is called equiregulated , if it has the following property: for every and every , there exists such that

1 if , and , then ;

2 if , and , then .

The next result, which can be found in [97, Proposition 3.2], gives a characterization of equiregulated functions taking values in a Banach space.

Theorem 1.11: A set is equiregulated if and only if for every , there is a division of such that

(1.1)

for every and , for .

Proof. Let be given and let be the set of all such that there is a division , that is, for which (1.1)holds with instead of . Since is equiregulated, there is such that for every and . Let and . Thus, for and , we have