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

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

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StartSet t element-of left-bracket a comma b right-bracket colon StartAbsoluteValue t minus xi Subscript i Baseline EndAbsoluteValue less-than delta left-parenthesis xi Subscript i Baseline right-parenthesis EndSet for all i equals 1 comma 2 comma ellipsis"/>

      This simple modification provides an elegant characterization of the Lebesgue integral through Riemann sums (see [174]).

      Definition 1.91: We say that f colon left-bracket a comma b right-bracket right-arrow double-struck upper R is Kurzweil–McShane integrable, and we write f element-of italic upper K upper M upper S left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis if and only if there exists upper I element-of double-struck upper R such that for every epsilon greater-than 0, there is a gauge delta on left-bracket a comma b right-bracket such that

StartAbsoluteValue upper I minus sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts f left-parenthesis xi Subscript i Baseline right-parenthesis left-parenthesis t Subscript i Baseline minus t Subscript i minus 1 Baseline right-parenthesis EndAbsoluteValue less-than epsilon comma

      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 S upper T upper D Subscript left-bracket a comma b right-bracket is delta-fine. We denote the Kurzweil–McShane integral of a function f colon left-bracket a comma b right-bracket right-arrow double-struck upper R by left-parenthesis italic upper K upper M upper S right-parenthesis integral Subscript a Superscript b Baseline f left-parenthesis t right-parenthesis d t.

      The following inclusions hold

upper R left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis subset-of script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals italic upper K upper M upper S left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis subset-of upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals upper H left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis period

      Moreover, upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis minus script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis not-equals empty-set as one can note by the next classical example.

      Example 1.92: Let upper F colon left-bracket 0 comma 1 right-bracket right-arrow double-struck upper R be defined by upper F left-parenthesis t right-parenthesis equals t squared sine left-parenthesis t Superscript negative 2 Baseline right-parenthesis comma if 0 less-than t less-than-or-slanted-equals 1, and upper F left-parenthesis 0 right-parenthesis equals 0, and consider f equals upper F prime. Since f is Riemann improper integrable, f element-of upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals upper H left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis, because the Kurzweil–Henstock (or Perron) integral contains its improper integrals (see Theorem 2.9, [158], or [213]). However, f not-an-element-of script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis, since f is not absolutely integrable (see also [227]).

      Example 1.92 tells us that the elements of upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals upper H left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis are not absolutely integrable.

      When McShane's idea is applied to Kurzweil and Henstock vector integrals, the story changes. In fact, the modification of McShane applied to the Kurzweil vector integral originates an integral which encompasses the Bochner–Lebesgue integral (see Example 1.74). On the other hand, when McShane's idea is used to modify the variational Henstock integral, we obtain exactly the Bochner–Lebesgue integral (see [[47] and [131]]). Thus, if italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis denotes the space of Henstock–McShane integrable functions f colon left-bracket a comma b right-bracket right-arrow upper X, that is, f element-of italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis is integrable in the sense of Henstock with the modification of McShane, then

italic upper H upper M upper S left-parenthesis left-bracket <noindex><p style= Скачать книгу


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