Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

such that for each

,

, for all

. We denote by

the set of all step functions

.

      It is clear that

Moreover, we have the following important result which is a general version of the result presented in [127, Theorem I.3.1].

      Theorem 1.4: Let and consider a function . The assertions below are equivalent:

      1  is the uniform limit of step functions , with ;

      2 ;

      3 given , there exists a division such that

      Proof. We will prove (i)

(ii), (ii)

(iii) and, then, (iii)

(i).

        (ii) Note that for all . We need to show that , see Remark 1.2. Let . We will only prove that exists, because the existence of follows analogously. Consider a sequence in such that , that is, , for every , and converges to as . Consider the sequence of step functions from to such that uniformly as . Then, given , there exists such that , for all . In addition, since is a step function, there exists such that , for all . Therefore, for , we haveThen, once is a Banach space, exists.

        (iii) Let be given. Since , it follows that (see Remark 1.2). Thus, for every , there exists such thatSimilarly, there are such thatNotice that the set of intervals is an open cover of the interval and, hence, there is a division of , with , such that is a finite subcover of for and, moreover,

        (i). Given , let , , be a division of such thatand , . Definewhere denotes the characteristic function of a measurable set . Note that for all and all . Moreover, is a sequence of step functions which converge uniformly to , as .

      It is a consequence of Theorem 1.4 (with

) that the closure of

is

. Therefore,

is a Banach space when equipped with the usual supremum norm

      See also [127, Theorem I.3.6].

If

denotes the Banach space of bounded functions from

to

, equipped with the supremum norm, then the inclusion

      follows from Theorem 1.4, items (i) and (ii), taking the limit of step functions which are constant on each subinterval of continuity.

      Recently, D. Franková established a fourth assertion equivalent to those assertions of Theorem 1.4 in the case where

. See [97, Theorem 2.3]. One can note, however, that such result also holds for any open set

. This is the content of the next lemma.

      Lemma 1.5: Let and be a function. Then the assertions of Theorem 1.4 are also equivalent to the following assertion:

      1 (iv) for every , there is a division such that

      Proof. Note that condition (iii) from Theorem 1.4 implies condition (iv). Now, assume that condition (iv) holds. Given

, there is a division

such that

, for all

and

According to [97, Theorem 2.3], take

and consider a step function

given by