Читать онлайн книгу - Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов. Математика. LiveLib

Новинки Лучшее Рекомендации

Информация о книге:

Название:

Автор:

Жанр:

Серия:

Издательство:

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

Скачать книгу

integrable;

2 .

Proof. (i) (ii). Suppose is absolutely integrable. Since the variation of , , 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) (i). Suppose . We prove that the integral exists and . Given , we need to find a gauge on 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,

(1.A.4)

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

Скачать книгу

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

Чтение книги онлайн.

Читать онлайн книгу Generalized Ordinary Differential Equations in Abstract Spaces and Applications - Группа авторов страница 56

Информация о книге: