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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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Hence,

The next result, borrowed from [7, Lemma 2.3], specifies the supremum of a function

Proposition 1.6: Let . Then where either , for some , or , for some , or for some .

Proof. Let

. Since

. By the definition of the supremum, for all

, one can choose

such that

which implies

Since

, there exists a subsequence

such that

. Since

is regulated,

belongs to StartSet parallel-to f left-parenthesis sigma right-parenthesis parallel-to comma parallel-to f left-parenthesis sigma Superscript minus Baseline right-parenthesis parallel-to comma parallel-to f left-parenthesis sigma Superscript plus Baseline right-parenthesis parallel-to EndSet

StartSet parallel-to f left-parenthesis sigma right-parenthesis parallel-to comma parallel-to f left-parenthesis sigma Superscript minus Baseline right-parenthesis parallel-to comma parallel-to f left-parenthesis sigma Superscript plus Baseline right-parenthesis parallel-to EndSet

and the proof is complete.

The composition of regulated functions may not be a regulated function as shown by the next example proposed by Dieudonné as an exercise. See, for instance, [58, Problem 2, p. 140].

Example 1.7: Consider, for instance, functions f comma g colon left-bracket 0 comma 1 right-bracket right-arrow double-struck upper R given by f left-parenthesis t right-parenthesis equals t sine StartFraction 1 Over t EndFraction , for t element-of left-parenthesis 0 comma 1 right-bracket , f left-parenthesis 0 right-parenthesis equals 0 and g left-parenthesis t right-parenthesis equals sgn t , that is, is the sign function. Both and belong to upper G left-parenthesis left-bracket 0 comma 1 right-bracket comma double-struck upper R right-parenthesis . However, the composition g ring f does not.

The next result, borrowed from [209, Theorem 10.11], gives us an interesting property of left‐continuous regulated functions. Such result will be used in Chapters 8 and 11. We state it here without any proof.

Proposition 1.8: Let . If for every , there exists such that for every , we have , then

f left-parenthesis s right-parenthesis minus f left-parenthesis a right-parenthesis less-than-or-slanted-equals g left-parenthesis s right-parenthesis minus g left-parenthesis a right-parenthesis comma s element-of left-bracket a comma b right-bracket period

We end this first section by introducing some notation for certain spaces of regulated functions defined on unbounded intervals of the real line double-struck upper R . Given t 0 element-of double-struck upper R , we denote by upper G left-parenthesis left-bracket t 0 comma infinity right-parenthesis comma upper X right-parenthesis the space of regulated functions from to upper X . In order to obtain a Banach space, we can intersect the space Скачать книгу