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Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis colon alpha left-parenthesis c right-parenthesis equals 0 EndSet period EndLayout"/>

      Such spaces are complete when endowed, respectively, with the norm given by the variation v a r Subscript a Superscript b Baseline left-parenthesis f right-parenthesis and the norm given by the semivariation

upper S upper V left-parenthesis alpha right-parenthesis equals sup Underscript d element-of upper D Subscript left-bracket a comma b right-bracket Baseline Endscripts upper S upper V Subscript d Baseline left-parenthesis alpha right-parenthesis comma

      where

upper S upper V Subscript d Baseline left-parenthesis alpha right-parenthesis equals sup Underscript parallel-to x Subscript i Baseline parallel-to less-than-or-slanted-equals 1 Endscripts vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d 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 xi comma

      and d colon a equals t 0 less-than t 1 less-than midline-horizontal-ellipsis less-than t Subscript StartAbsoluteValue d EndAbsoluteValue Baseline equals b is a division of left-bracket a comma b right-bracket.

      1 (V1) Every is bounded and , .

      2 (V2) Given and , we have .

      Remark 1.28: Note that property (V1) above implies parallel-to alpha parallel-to less-than-or-slanted-equals parallel-to alpha parallel-to for all alpha element-of upper B upper V left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis.

      For more details about the spaces in Definition 1.27, the reader may want to consult [127]. The next results are borrowed from [126]. We include the proofs here since this reference is not easily available. Lemmas 1.29 and 1.30 below are, respectively, Theorems I.2.7 and I.2.8 from [126].

      Lemma 1.29: Let . Then,

      1 For all , there exists .

      2 For all , there exists .

      Proof. We only prove item (i), because item (ii) follows analogously. Consider an increasing sequence left-brace t Subscript n Baseline right-brace Subscript n element-of double-struck upper N in left-bracket a comma t right-parenthesis converging to t. Then,

sigma-summation Underscript i equals 1 Overscript n Endscripts parallel-to alpha left-parenthesis t Subscript i Baseline right-parenthesis minus alpha left-parenthesis t Subscript i minus 1 Baseline right-parenthesis parallel-to less-than-or-slanted-equals v a r Subscript a Superscript t Baseline left-parenthesis alpha right-parenthesis comma

      for all n element-of double-struck upper N period Therefore, we have sigma-summation Underscript i equals 1 Overscript infinity Endscripts parallel-to alpha left-parenthesis t Subscript i Baseline right-parenthesis minus alpha left-parenthesis t Subscript i minus 1 Baseline right-parenthesis parallel-to less-than-or-slanted-equals v a r Subscript a Superscript t Baseline left-parenthesis alpha right-parenthesis and, hence, sigma-summation Underscript i equals j Overscript infinity Endscripts parallel-to alpha left-parenthesis t Subscript i Baseline right-parenthesis minus alpha left-parenthesis t Subscript i minus 1 Baseline right-parenthesis parallel-to right-arrow 0 comma as j right-arrow infinity period Thus, left-brace alpha left-parenthesis t Subscript n Baseline right-parenthesis right-brace Subscript n element-of double-struck upper N is a Cauchy sequence, since for any given epsilon greater-than 0, we have

parallel-to alpha left-parenthesis t Subscript m Baseline right-parenthesis minus alpha left-parenthesis t Subscript n Baseline right-parenthesis parallel-to less-than-or-slanted-equals sigma-summation Underscript i equals n plus 1 Overscript m Endscripts parallel-to alpha left-parenthesis t Subscript i Baseline right-parenthesis minus alpha left-parenthesis t Subscript i minus 1 Baseline right-parenthesis parallel-to less-than-or-slanted-equals epsilon comma

      for sufficiently large m comma n. Finally, note that the limit alpha left-parenthesis t Superscript minus Baseline right-parenthesis of left-brace alpha left-parenthesis t Subscript n Baseline right-parenthesis right-brace Subscript n element-of double-struck upper N is independent of the choice of left-brace t Subscript n Baseline right-brace Subscript n element-of double-struck upper N, and we finish the proof.

      It comes from Lemma 1.29 that all functions x colon left-bracket a comma b right-bracket right-arrow upper X of bounded variation are also regulated functions (see, e.g. [127, Corollary 3.4]) which, in turn, are Darboux integrable [127, Theorem 3.6].

      Lemma 1.30: Let . For every , let . Then,

      1 , ;

      2 , .