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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

, the function is increasing.

3 (SB3) Given and , .

Definition 1.24: Consider the BT left-parenthesis upper L left-parenthesis upper X comma upper Y right-parenthesis comma upper X comma upper Y right-parenthesis . Then, instead of upper S upper B left-parenthesis alpha right-parenthesis and , we write simply upper S upper V left-parenthesis alpha right-parenthesis and , respectively. Hence,

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 equals upper S upper B left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis

and we call any element 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 a function of bounded semivariation.

Definition 1.25: Given a function alpha colon left-bracket a comma b right-bracket right-arrow upper E , upper E a normed space, and a division d equals left-parenthesis t Subscript i Baseline right-parenthesis element-of upper D Subscript left-bracket a comma b right-bracket , we define

v a r Subscript d Baseline left-parenthesis alpha right-parenthesis equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus of alpha alpha left-parenthesis right-parenthesis ti of alpha alpha left-parenthesis right-parenthesis t minus minus i 1

and the variation of alpha is given by

v a r left-parenthesis alpha right-parenthesis equals v a r Subscript a Superscript b Baseline left-parenthesis alpha right-parenthesis equals sup left-brace right-brace colon times times vard left-parenthesis right-parenthesis alpha colon element-of element-of dD left-bracket right-bracket comma a comma b period

If v a r left-parenthesis alpha right-parenthesis less-than infinity , then alpha is called a function of bounded variation, in which case, we write alpha element-of upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper E right-parenthesis .

It is not difficult to prove that upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper E comma upper F 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 E comma upper F right-parenthesis right-parenthesis and .

Moreover (see [127, Corollary I.3.4]), upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis .

The space upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis is complete when equipped with the variation norm, parallel-to dot parallel-to Subscript upper B upper V Baseline , given by

parallel-to f parallel-to equals parallel-to f left-parenthesis a right-parenthesis parallel-to plus v a r Subscript a Superscript b Baseline left-parenthesis f right-parenthesis comma

for f element-of upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis period When there is no room for misunderstanding, we may use the notation parallel-to dot parallel-to instead of parallel-to dot parallel-to Subscript upper B upper V Baseline .

Remark 1.26: Consider a BT left-parenthesis upper E comma upper F comma upper G right-parenthesis . The definition of variation of a function alpha colon left-bracket a comma b right-bracket right-arrow upper E , where upper E is a normed space, can also be considered as a particular case of the script upper B ‐variation of alpha in two different ways.

1 Let , or and . By the definition of the norm in , we haveThus, when we consider the BT , we write and instead of and , respectively.

2 Let , or and . By the Hahn–Banach Theorem, we haveand, hence,

Definition 1.27: Given c element-of left-bracket a comma b right-bracket , we define the spaces

StartLayout 1st Row 1st Column Blank 2nd Column upper B upper V Subscript c Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis equals StartSet f element-of upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis colon f left-parenthesis c right-parenthesis equals 0 EndSet and 2nd Row 1st Column Blank 2nd Column upper S upper V Subscript c Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis equals StartSet alpha element-of upper S upper V left-parenthesis left-bracket <hr><noindex><a href=

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