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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

t right-parenthesis f left-parenthesis t right-parenthesis"/>, and we write f element-of upper R Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis

f element-of upper R Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis

. Hence, if we denote by upper R Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis

upper R Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis

the set of all functions alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis

alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis

which are Riemann integrable with respect to f colon left-bracket a comma b right-bracket right-arrow upper X

and by

upper R Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis

the set of all functions f colon left-bracket a comma b right-bracket right-arrow upper X

which are Riemann integrable with respect to alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis

, then we have

StartLayout 1st Row 1st Column Blank 2nd Column upper R Subscript f Baseline 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 K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis and 2nd Row 1st Column Blank 2nd Column upper R Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper K Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis period EndLayout

The next very important remark concerns the terminology we adopt from now on in this book concerning Kurzweil vector integrals given by Definitions 1.37 and 1.38.

Remark 1.39: We refer to the vector integrals from Definitions 1.37 and 1.38, namely,

integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis and integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis comma

where f colon left-bracket a comma b right-bracket right-arrow upper X and alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis , as Perron–Stieltjes integrals. For the particular case where is the identity in and , we refer to the integral

integral Subscript a Superscript b Baseline f left-parenthesis t right-parenthesis d t

simply as a Perron integral. Our choice to use this terminology is due to the fact that, in Chapter 2, we deal with a more general definition of the Kurzweil integral which encompasses all integrals presented here. Moreover, since the same notation for the integrals

integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis comma integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis comma and integral Subscript a Superscript b Baseline f left-parenthesis t right-parenthesis d t

are used for Riemann–Stieltjes integrals, we will specify which integral we are dealing with whenever there is possibility for an ambiguous interpretation.

The vector integral of Henstock, which we define in the sequel, is more restrictive than the Kurzweil vector integral for integrands taking values in infinite dimensional Banach spaces.

Again, consider functions f colon left-bracket a comma b right-bracket right-arrow upper X and alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis .

Definition 1.40: We say that alpha is Henstock ‐integrable (or Henstock variationally integrable with respect to ), if there exists a function upper A Subscript f Baseline colon left-bracket a comma b right-bracket right-arrow upper Y (called the associate function of alpha ) such that for every epsilon greater-than 0 , there is a gauge delta on 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 alpha alpha left-parenthesis right-parenthesis xi i left-bracket right-bracket minus minus of ff left-parenthesis right-parenthesis ti of ff left-parenthesis right-parenthesis t minus minus i 1 left-bracket right-bracket minus minus of AAf left-parenthesis right-parenthesis ti of AAf left-parenthesis right-parenthesis t minus minus i 1 less-than epsilon period

We write alpha element-of upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis in this case.

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