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Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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alt="n less-than-or-slanted-equals t less-than n plus 1"/>, then 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 equals sigma-summation Underscript n element-of double-struck upper N Endscripts x Subscript n Baseline element-of upper X comma whenever the integral exists. However, f not-an-element-of script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis, since

integral Subscript a Superscript b Baseline vertical-bar vertical-bar vertical-bar vertical-bar of ff left-parenthesis right-parenthesis t d t equals vertical-bar vertical-bar vertical-bar vertical-bar x 1 plus vertical-bar vertical-bar vertical-bar vertical-bar x 2 plus vertical-bar vertical-bar vertical-bar vertical-bar x 3 plus midline-horizontal-ellipsis equals infinity period

      and this completes the proof.

      In the next example, borrowed from [73, Example 3.4], we exhibit a Banach space-valued function which is integrable in the variational Henstock sense and also in the sense of Kurzweil–McShane. Nevertheless, it is not absolutely integrable.

      Example 1.103: Let f colon left-bracket 0 comma 1 right-bracket right-arrow l 2 left-parenthesis double-struck upper N right-parenthesis be given by

f left-parenthesis t right-parenthesis equals StartFraction 2 Superscript i Baseline Over i EndFraction e Subscript i Baseline comma for StartFraction 1 Over 2 Superscript i Baseline EndFraction less-than-or-slanted-equals t less-than StartFraction 1 Over 2 Superscript i minus 1 Baseline EndFraction and i element-of double-struck upper N period

      Then,

integral Subscript StartFraction 1 Over 2 Superscript i Baseline EndFraction Superscript StartFraction 1 Over 2 Superscript i minus 1 Baseline EndFraction Baseline StartFraction 2 Superscript i Baseline Over i EndFraction e Subscript i Baseline d t equals StartFraction 1 Over i EndFraction e Subscript i

      which is summable in l 2 left-parenthesis double-struck upper N right-parenthesis. Since the Henstock integral contains its improper integrals (and the same applies to the Kurzweil integral), we have f element-of upper H left-parenthesis left-bracket 0 comma 1 right-bracket comma l 2 left-parenthesis double-struck upper N right-parenthesis right-parenthesis. However, f not-an-element-of script upper L 1 left-parenthesis left-bracket 0 comma 1 right-bracket comma l 2 left-parenthesis double-struck upper N right-parenthesis right-parenthesis because the sequence left-brace StartFraction 1 Over i EndFraction e Subscript i Baseline right-brace Subscript i element-of double-struck upper N is non-summable in script upper L 1 left-parenthesis left-bracket 0 comma 1 right-bracket comma l 2 left-parenthesis double-struck upper N right-parenthesis right-parenthesis. By the Monotone Convergence Theorem for the Kurzweil–McShane integral (which follows the ideas of [71] with obvious adaptations), f element-of italic upper K upper M upper S left-parenthesis left-bracket 0 comma 1 right-bracket comma l 2 left-parenthesis double-struck upper N right-parenthesis right-parenthesis. But f not-an-element-of italic RMS left-parenthesis left-bracket 0 comma 1 right-bracket comma l 2 left-parenthesis double-struck upper N right-parenthesis right-parenthesis, since f is not bounded, where by italic RMS left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis we denote the space of Riemann–McShane integrable functions from left-bracket a comma b right-bracket to upper X.

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