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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

tilde left-parenthesis a right-parenthesis minus integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis ModifyingAbove f With tilde left-parenthesis t right-parenthesis period"/>

The next theorem is due to C. S. Hönig (see [129]), and it concerns multipliers for Perron–Stieltjes integrals.

Theorem 1.57: Suppose and . Then, and Eqs. (1.3) and (1.4) hold.

Since upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper K left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis subset-of upper S upper V left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis , it is immediate that if f element-of upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and alpha element-of upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis , then alpha f element-of upper K left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis . As a matter of fact, the next result gives us information about the multipliers for the Henstock vector integral. See [72, Theorem 7].

Theorem 1.58: Assume that and . Then, and equalities (1.3) and (1.4) hold.

Proof. Since f element-of upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis , f overTilde is continuous by Theorem 1.49. Thus, given epsilon greater-than 0 , there exists delta Superscript asterisk Baseline greater-than 0 such that

omega left-parenthesis f overTilde comma left-bracket c comma d right-bracket right-parenthesis equals sup left-brace parallel-to ModifyingAbove f With tilde left-parenthesis t right-parenthesis minus ModifyingAbove f With tilde left-parenthesis s right-parenthesis parallel-to colon t comma s element-of left-bracket c comma d right-bracket right-brace less-than epsilon comma

whenever 0 less-than d minus c less-than delta Superscript asterisk , where left-bracket c comma d right-bracket subset-of left-bracket a comma b right-bracket . Moreover, there is a gauge delta on , with delta left-parenthesis t right-parenthesis less-than StartFraction delta Superscript asterisk Baseline Over 2 EndFraction for t element-of left-bracket a comma b right-bracket , such that for every delta ‐fine 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 ,

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 less-than epsilon period

Thus,

StartLayout 1st Row 1st Column Blank 2nd Column sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus times times times of alpha alpha left-parenthesis right-parenthesis xi i 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 times times of alpha alpha left-parenthesis right-parenthesis t of ff left-parenthesis right-parenthesis t separator d separator t 2nd 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 times times of alpha alpha left-parenthesis right-parenthesis xi i left-bracket right-bracket 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 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 times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis t of alpha alpha left-parenthesis right-parenthesis xi i of ff left-parenthesis right-parenthesis t separator d separator t 3rd Row 1st Column Blank 2nd Column less-than vertical-bar vertical-bar vertical-bar vertical-bar alpha Subscript infinity Baseline epsilon plus 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 times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis t of alpha alpha left-parenthesis right-parenthesis xi i of ff left-parenthesis right-parenthesis t separator d separator t period EndLayout

But by Corollary 1.56, item (ii), alpha f element-of upper K left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis and

integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis d t equals integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d ModifyingAbove f With tilde left-parenthesis t right-parenthesis equals alpha left-parenthesis b right-parenthesis ModifyingAbove f With tilde left-parenthesis b right-parenthesis minus alpha left-parenthesis a right-parenthesis ModifyingAbove f With tilde left-parenthesis a right-parenthesis minus integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis ModifyingAbove f With tilde left-parenthesis t right-parenthesis

and a similar formula also holds for every subinterval contained in left-bracket a comma b right-bracket . Hence, for Скачать книгу