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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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 2nd Row 1st Column Blank 2nd Column less-than-or-slanted-equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts StartAbsoluteValue 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 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 EndAbsoluteValue plus StartAbsoluteValue 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 minus v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis EndAbsoluteValue 3rd Row 1st Column Blank 2nd Column less-than-or-slanted-equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus times times of ff left-parenthesis right-parenthesis xi i left-parenthesis right-parenthesis minus minus tit minus minus i 1 integral integral t minus minus i 1 ti of ff left-parenthesis right-parenthesis t separator d separator t plus StartAbsoluteValue 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 minus v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis EndAbsoluteValue period EndLayout"/>

By the definition of v a r Subscript a Superscript b Baseline left-parenthesis f overTilde right-parenthesis , we may take left-parenthesis t Subscript i Baseline right-parenthesis element-of upper D Subscript left-bracket a comma b right-bracket such that the last summand in (1.A.4) is smaller than StartFraction epsilon Over 2 EndFraction . Because f element-of upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis , we may take a gauge delta such that for every delta -fine 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 , the first summand in (1.A.4) is also smaller than StartFraction epsilon Over 2 EndFraction (and we may suppose that the points chosen for the second summand are the points of the delta -fine tagged division left-parenthesis xi Subscript i Baseline comma left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-parenthesis ).

The next result is a consequence of the fact that italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and Lemmas 1.97 and 1.98. A proof of it can be found in [132, Theorem 10.3].

Corollary 1.99: All functions of are absolutely.integrable

The reader can find a proof of the next lemma in [35, Theorem 9].

Lemma 1.100: All functions of are.measurable

Finally, we can prove the following inclusion.

Theorem 1.101: italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis .

Proof. The result follows from the facts that all functions of upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and, hence, of italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis are measurable (Lemma 1.100), and all functions of are absolutely integrable by Corollary 1.99.

As we mentioned before, the inclusion script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of italic upper K upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis always holds. However, when upper X is an infinite dimensional Banach space, then for sure , as shown by the next result due to C. S. Hönig (personal communication by him to his students in 1990 at the University of São Paulo) and presented in [73].

Proposition 1.102 (Hönig): If is an infinite dimensional Banach space, then there exists .

Proof. Let dimension upper X denote the dimension of upper X . If dimension upper X equals infinity , then the Theorem of Dvoretsky–Rogers (see [60] and also [57]) implies there exists a sequence left-brace x Subscript n Baseline right-brace Subscript n element-of double-struck upper N in upper X which is summable but not absolutely summable. Thus, if we define a function Скачать книгу