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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

Equiregulated Sets

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 script í’œ subset-of upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis is called equiregulated, if it has the following property: for every epsilon greater-than 0 and every sigma element-of left-bracket a comma b right-bracket , there exists delta greater-than 0 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) parallel-to f left-parenthesis t Superscript prime Baseline right-parenthesis minus f left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals epsilon comma

for every and , for .

Proof. left-parenthesis right double arrow right-parenthesis Let epsilon greater-than 0 be given and let upper B be the set of all gamma element-of left-parenthesis a comma b right-bracket such that there is a division d prime element-of upper D Subscript left-bracket a comma gamma right-bracket , that is, d prime colon a equals t 0 less-than t 1 less-than midline-horizontal-ellipsis less-than t Subscript StartAbsoluteValue d Sub Superscript prime Subscript EndAbsoluteValue Baseline equals gamma for which (1.1) holds with StartAbsoluteValue d prime EndAbsoluteValue instead of StartAbsoluteValue d EndAbsoluteValue . Since script í’œ is equiregulated, there is delta 1 element-of left-parenthesis 0 comma b minus a right-bracket such that parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis a Superscript plus Baseline right-parenthesis parallel-to less-than-or-slanted-equals StartFraction epsilon Over 2 EndFraction comma for every f element-of script í’œ and t element-of left-parenthesis a comma a plus delta 1 right-parenthesis . Let a 1 equals a plus delta 1 and a equals t 0 less-than t 1 equals a 1 . Thus, for left-bracket t comma t Superscript prime Baseline right-bracket subset-of left-parenthesis a comma a 1 right-parenthesis and f element-of script í’œ , we have

parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis t Superscript prime Baseline right-parenthesis parallel-to less-than-or-slanted-equals parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis a Superscript plus Baseline right-parenthesis parallel-to plus parallel-to f left-parenthesis t Superscript prime Baseline right-parenthesis minus f left-parenthesis a Superscript plus Baseline right-parenthesis parallel-to less-than-or-slanted-equals epsilon period

Hence, a 1 element-of upper B .

Let c overTilde be the supremum of the set upper B . Since f element-of script í’œ , is regulated. Thus, there exists delta greater-than 0 such that parallel-to f left-parenthesis c overTilde Superscript minus Baseline right-parenthesis minus f left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals StartFraction epsilon Over 2 EndFraction for every f element-of script í’œ and t element-of left-parenthesis c overTilde minus delta comma c overTilde right-parenthesis intersection left-bracket a comma b right-bracket . Take c element-of upper B intersection left-parenthesis c overTilde minus delta comma c overTilde right-parenthesis and a division d double-prime element-of upper D Subscript left-bracket a comma c right-bracket , say, d double-prime colon a equals t 0 less-than t 1 less-than midline-horizontal-ellipsis less-than t Subscript StartAbsoluteValue d Sub Superscript double-prime Subscript EndAbsoluteValue Baseline equals c , such that (1.1) holds with StartAbsoluteValue d double-prime EndAbsoluteValue instead of StartAbsoluteValue d EndAbsoluteValue . Denote . Then, for left-bracket t comma t Superscript prime Baseline right-bracket subset-of left-parenthesis t Subscript StartAbsoluteValue d Sub Superscript double-prime Subscript EndAbsoluteValue Baseline comma t Subscript StartAbsoluteValue d Sub Superscript double-prime Subscript EndAbsoluteValue plus 1 Baseline right-parenthesis and f element-of script í’œ , we have

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