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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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

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an analogous way, we define the Henstock alpha‐integrability of f colon left-bracket a comma b right-bracket right-arrow upper X.

      Definition 1.41: We say that f is Henstock alpha‐integrable (or Henstock variationally integrable with respect to alpha), if there exists a function upper F Subscript alpha Baseline colon left-bracket a comma b right-bracket right-arrow upper Y (called the associate function of f) such that for every epsilon greater-than 0, there is a gauge delta on 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 left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis ti of alpha alpha left-parenthesis right-parenthesis t minus minus i 1 of ff left-parenthesis right-parenthesis xi i left-bracket right-bracket minus minus of FF alpha left-parenthesis right-parenthesis ti of FF alpha left-parenthesis right-parenthesis t minus minus i 1 less-than epsilon period

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

      Next, we define indefinite vector integrals.

      Definition 1.42 (Indefinite Vector Integrals):

      1 Given and , we define the indefinite integral of with respect to byIf, in addition, , then

      2 Given and , we define the indefinite integral of with respect to byIf, moreover, , then

      Note that Definition 1.42 yields the inclusions

StartLayout 1st Row 1st Column Blank 2nd Column 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 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 H 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

      If in item (ii) of Definition 1.42, we consider the particular case, where upper X equals upper Y, and for every t element-of left-bracket a comma b right-bracket, alpha left-parenthesis t right-parenthesis is the identity in upper X, then instead of upper K Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis, upper R Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis, upper H Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and f overTilde Superscript alpha, we write simply

upper K left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis comma upper R left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis comma upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and f overTilde

      respectively, where

ModifyingAbove f With tilde left-parenthesis t right-parenthesis equals integral Subscript a Superscript t Baseline f left-parenthesis s right-parenthesis d s comma t element-of left-bracket a comma b right-bracket period

      If in item (i) of Definition 1.42, we have upper X equals upper Y equals double-struck upper R, then one can identify the isomorphic spaces upper L left-parenthesis double-struck upper R comma double-struck upper R right-parenthesis and double-struck upper R and, hence, the spaces upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis double-struck upper R right-parenthesis right-parenthesis, upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis, upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis double-struck upper R right-parenthesis right-parenthesis, and upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis can also be identified, because one has

upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis period

      Indeed. It is clear that upper H Subscript f Baseline left-parenthesis <hr><noindex><a href=Скачать книгу