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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t minus minus i 1 integral integral t minus minus i 1 ti of alpha alpha left-parenthesis right-parenthesis t times times d of ff left-parenthesis right-parenthesis t less-than epsilon period"/>

      Taking approximated Riemannian‐type sums for the integrals integral Subscript a Superscript b Baseline gamma left-parenthesis t right-parenthesis alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis and integral Subscript a Superscript b Baseline gamma left-parenthesis t right-parenthesis d beta left-parenthesis t right-parenthesis, we obtain

StartLayout 1st Row 1st Column Blank 2nd Column 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 times of gamma gamma left-parenthesis right-parenthesis xi i 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 sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times of gamma gamma left-parenthesis right-parenthesis xi i left-bracket right-bracket minus minus of beta beta left-parenthesis right-parenthesis ti of beta beta left-parenthesis right-parenthesis t minus minus i 1 2nd Row 1st Column Blank 2nd Column equals vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times of gamma gamma left-parenthesis right-parenthesis xi i left-brace right-brace minus minus 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 integral integral minus minus times times ti 1 ti of alpha alpha left-parenthesis right-parenthesis t times times d of ff left-parenthesis right-parenthesis t equals upper I period EndLayout

      On the other hand, when gamma Subscript i Baseline element-of upper L left-parenthesis upper X comma upper Y right-parenthesis and x Subscript i Baseline element-of upper X, we have

sigma-summation Underscript i equals i Overscript n Endscripts gamma Subscript i Baseline x Subscript i Baseline equals sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis gamma Subscript j Baseline minus gamma Subscript j minus 1 Baseline right-parenthesis left-parenthesis sigma-summation Underscript i equals j Overscript n Endscripts x Subscript i Baseline right-parenthesis plus gamma 0 left-parenthesis sigma-summation Underscript i equals j Overscript n Endscripts x Subscript i Baseline right-parenthesis comma n element-of double-struck upper N period

      Then, taking x Subscript i Baseline equals alpha left-parenthesis xi Subscript i Baseline right-parenthesis left-bracket f left-parenthesis t Subscript i Baseline right-parenthesis minus f left-parenthesis t Subscript i minus 1 Baseline right-parenthesis right-bracket minus integral Subscript t i minus 1 Superscript t Subscript i Baseline Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis, gamma Subscript i Baseline equals gamma left-parenthesis xi Subscript i Baseline right-parenthesis, gamma 0 equals gamma left-parenthesis a right-parenthesis, and n equals StartAbsoluteValue d EndAbsoluteValue, we get

upper I equals vertical-bar vertical-bar vertical-bar vertical-bar plus plus sigma-summation sigma-summation equals equals j 1 vertical-bar vertical-bar d times times left-bracket right-bracket minus minus of gamma gamma left-parenthesis right-parenthesis xi j of gamma gamma left-parenthesis right-parenthesis xi minus minus j 1 left-parenthesis right-parenthesis sigma-summation sigma-summation equals equals ij vertical-bar vertical-bar dxi of gamma gamma 0 left-parenthesis right-parenthesis sigma-summation sigma-summation equals equals ij vertical-bar vertical-bar dxi less-than-or-slanted-equals upper S upper V left-parenthesis gamma right-parenthesis epsilon plus parallel-to gamma left-parenthesis a right-parenthesis parallel-to epsilon comma

      because the Saks‐Henstock lemma (Lemma 1.45) yields vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals ij vertical-bar vertical-bar dxi less-than-or-slanted-equals epsilon, for every j element-of StartSet 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue EndSet.

      

      Corollary 1.69: Consider functions , , , and . Then, we have , , and Eq. (1.12) holds.

      Proof. Theorem 1.47, item (i), yields gamma element-of upper K Subscript g Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper W comma upper Y right-parenthesis right-parenthesis. Then, the statement follows from Theorem 1.68.

      The next result gives us an integration by parts formula for Perron–Stieltjes integrals. A proof of it can be found in [212, Theorem 13].

      Proposition 1.70: Suppose and or and . Then, the Perron–Stieltjes integrals and exist, and the following equality holds:

StartLayout 1st Row 1st Column integral Subscript a Superscript b Baseline d alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis plus integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis d f left-parenthesis t right-parenthesis equals 2nd Column alpha left-parenthesis b right-parenthesis f left-parenthesis b right-parenthesis minus alpha left-parenthesis a right-parenthesis f left-parenthesis a right-parenthesis 2nd Row 1st Column Blank 2nd Column minus sigma-summation Underscript a less-than-or-slanted-equals tau less-than b Endscripts normal upper Delta Superscript plus Baseline alpha left-parenthesis tau right-parenthesis normal upper Delta Superscript plus Baseline f left-parenthesis tau right-parenthesis plus sigma-summation Underscript a less-than-or-slanted-equals tau less-than b Endscripts normal upper Delta Superscript minus Baseline alpha left-parenthesis tau right-parenthesis normal upper Delta Superscript minus Baseline f left-parenthesis tau right-parenthesis comma EndLayout

       where , , , and

      As an immediate consequence of the previous proposition, we have the following result.

      Corollary 1.71: If and is a nondecreasing function, then the integral exists.

      We end this subsection by presenting a result, borrowed from [172] and [179, Theorem 5.4.5], which gives us a change of variable formula for Perron–Stieltjes integrals.

      

      Theorem 1.72: Suppose is increasing and

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