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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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sums concerning delta

‐fine tagged divisions of an interval left-bracket a comma b right-bracket

, we bring up an important result which guarantees the existence of a delta

‐fine tagged division for a given gauge delta

. This result is known as Cousin Lemma, and a proof of it can be found in [120, Theorem 4.1].

Lemma 1.36 (Cousin Lemma): Given a gauge of , there exists a ‐fine tagged division of .

Definition 1.37: We say that alpha is Kurzweil ‐integrable (or Kurzweil integrable with respect to ), if there exists upper I element-of upper Y 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 ,

vertical-bar vertical-bar vertical-bar vertical-bar minus minus sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d 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 upper I less-than epsilon period

In this case, we write upper I equals integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis and alpha element-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 .

Analogously, we define the Kurzweil integral of f colon left-bracket a comma b right-bracket right-arrow upper X with respect to a function alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis .

Definition 1.38: We say that is Kurzweil alpha minus integrable (or Kurzweil integrable with respect to alpha ), if there exists upper I element-of upper Y such that given epsilon greater-than 0 , there is a gauge delta on left-bracket a comma b right-bracket such that

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. In this case, we write upper I equals integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis and f element-of upper K Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis .

Suppose the Kurzweil vector integral integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis exists. Then, we define

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

An analogous consideration holds for the Kurzweil vector integral integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis .

If the gauge delta in the definition of alpha element-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 is a constant function, then we obtain the Riemann–Stieltjes integral integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis , and we write alpha element-of 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 . Similarly, when we consider only constant gauges delta in the definition of f element-of upper K Superscript alpha Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis , we obtain the Riemann–Stieltjes integral Скачать книгу