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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

semitagged division d equals left-parenthesis xi Subscript i Baseline comma left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-parenthesis

d equals left-parenthesis xi Subscript i Baseline comma left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-parenthesis

(1.A.3)

Let s 0 less-than s 1 less-than midline-horizontal-ellipsis less-than s Subscript m be any division of left-bracket xi minus delta left-parenthesis xi right-parenthesis comma xi plus delta left-parenthesis xi right-parenthesis right-bracket . If we take xi Subscript j Baseline equals xi for j equals 1 comma 2 comma ellipsis comma m , then the point-interval pair left-parenthesis xi Subscript j Baseline comma left-bracket s Subscript j minus 1 Baseline comma s Subscript j Baseline right-bracket right-parenthesis is a delta -fine tagged partial division of left-bracket xi minus delta left-parenthesis xi right-parenthesis comma xi plus delta left-parenthesis xi right-parenthesis right-bracket and, therefore, from (1.A.3) and since we can assume, without loss of generality (see comments in the paragraph before the statement), that f left-parenthesis xi Subscript j Baseline right-parenthesis equals f left-parenthesis xi right-parenthesis equals 0 for j equals 1 comma 2 comma ellipsis comma m , we have

sigma-summation Underscript j equals 1 Overscript m Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis sj of ff tilde left-parenthesis right-parenthesis s minus minus j 1 less-than epsilon

and the proof is complete.

Lemma 1.98: Suppose . The following properties are equivalent:

1 is absolutely integrable;

2 .

Proof. (i) right double arrow (ii). Suppose is absolutely integrable. Since the variation of f overTilde , v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis , is given by

v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis equals sup left-brace right-brace colon sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis ti of ff tilde left-parenthesis right-parenthesis t minus minus i 1 colon equals equals d element-of element-of left-parenthesis right-parenthesis tiD left-bracket right-bracket comma a comma b

we have

sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis ti of ff tilde left-parenthesis right-parenthesis t minus minus i 1 equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar integral integral t minus minus i 1 ti of ff left-parenthesis right-parenthesis t separator d separator t less-than-or-slanted-equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts integral Subscript t Subscript i minus 1 Baseline Superscript t Subscript i Baseline Baseline vertical-bar vertical-bar vertical-bar vertical-bar of ff left-parenthesis right-parenthesis t d t equals integral Subscript a Superscript b Baseline vertical-bar vertical-bar vertical-bar vertical-bar of ff left-parenthesis right-parenthesis t d t period

(ii) right double arrow (i). Suppose f overTilde element-of upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis . We prove that the integral integral Subscript a Superscript b Baseline vertical-bar vertical-bar vertical-bar vertical-bar of ff left-parenthesis right-parenthesis t d t exists and . Given epsilon greater-than 0 , we need to find a gauge delta on left-bracket a comma b right-bracket such that

StartAbsoluteValue sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar of ff left-parenthesis right-parenthesis xi i left-parenthesis t Subscript i Baseline minus t Subscript i minus 1 Baseline right-parenthesis minus v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis EndAbsoluteValue less-than epsilon comma

whenever d equals left-parenthesis xi Subscript i Baseline comma left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-parenthesis element-of upper T upper D Subscript left-bracket a comma b right-bracket is delta -fine. However,

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