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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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parallel-to less-than StartFraction epsilon Over 3 EndFraction"/> and parallel-to f Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus f 0 left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to less-than StartFraction epsilon Over 3 EndFraction comma for i equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue period

      Take an arbitrary t element-of left-bracket a comma b right-bracket. Then, either t equals t Subscript i for some i, or t element-of left-parenthesis t Subscript i minus 1 Baseline comma t Subscript i Baseline right-parenthesis for some i. In the former case, parallel-to f Subscript k Baseline left-parenthesis t right-parenthesis minus f 0 left-parenthesis t right-parenthesis parallel-to less-than StartFraction epsilon Over 3 EndFraction period The other case yields

parallel-to f Subscript k Baseline left-parenthesis t right-parenthesis minus f 0 left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals parallel-to f Subscript k Baseline left-parenthesis t right-parenthesis minus f Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to plus parallel-to f Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus f 0 left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to plus parallel-to f 0 left-parenthesis tau Subscript i Baseline right-parenthesis minus f 0 left-parenthesis t right-parenthesis parallel-to less-than epsilon period

      Then, parallel-to f Subscript k Baseline minus f 0 parallel-to less-than epsilon and, therefore, f Subscript k Baseline right-arrow f 0 uniformly on left-bracket a comma b right-bracket.

      

      Lemma 1.14: Let be a sequence in . The following assertions hold:

      1 if the sequence of functions converges uniformly to as on , then , for , and , for ;

      2 if the sequence of functions converges pointwisely to as on and , for , and , for , where , then the sequence converges uniformly to as .

      Proof. We start by proving left-parenthesis normal i right-parenthesis. By hypothesis, the sequence left-brace f Subscript k Baseline right-brace Subscript k element-of double-struck upper N converges uniformly to f 0. Then, Moore–Osgood theorem (see, e.g., [19]) implies

limit Underscript k right-arrow infinity Endscripts limit Underscript s right-arrow t Superscript minus Baseline Endscripts f Subscript k Baseline left-parenthesis s right-parenthesis equals limit Underscript s right-arrow t Superscript minus Baseline Endscripts limit Underscript k right-arrow infinity Endscripts f Subscript k Baseline left-parenthesis s right-parenthesis comma t element-of left-parenthesis a comma b right-bracket period

      Therefore, f Subscript k Baseline left-parenthesis t Superscript minus Baseline right-parenthesis right-arrow f 0 left-parenthesis t Superscript minus Baseline right-parenthesis, for t element-of left-parenthesis a comma b right-bracket. In a similar way, one can show that f Subscript k Baseline left-parenthesis t Superscript plus Baseline right-parenthesis right-arrow f 0 left-parenthesis t Superscript plus Baseline right-parenthesis, for every t element-of left-bracket a comma b right-parenthesis.

      Now, we prove left-parenthesis i i right-parenthesis. It suffices to show that left-brace f Subscript k Baseline colon left-bracket a comma b right-bracket right-arrow upper X colon k element-of double-struck upper N right-brace is an equiregulated set. Indeed, by Lemma 1.13, f Subscript k converges uniformly to f 0.

      Since the function f 0 is regulated, its lateral limits exist. Then, for every t 0 element-of left-bracket a comma b right-bracket and every epsilon greater-than 0, there is delta greater-than 0 such that

StartLayout 1st Row 1st Column parallel-to f 0 left-parenthesis t 0 Superscript minus Baseline right-parenthesis minus f 0 left-parenthesis t right-parenthesis parallel-to 2nd Column less-than epsilon comma for t 0 minus delta less-than-or-slanted-equals t less-than t 0 comma 2nd Row 1st Column parallel-to f 0 left-parenthesis t right-parenthesis minus f 0 left-parenthesis t 0 Superscript plus Baseline right-parenthesis parallel-to 2nd Column less-than epsilon comma for t 0 less-than t less-than-or-slanted-equals t 0 plus delta comma EndLayout

      for every t element-of left-bracket a comma b right-bracket.

      But the hypotheses say that we can find n 0 element-of double-struck upper N such that, for n greater-than-or-slanted-equals n 0, we have

parallel-to f Subscript n Baseline left-parenthesis t 0 minus delta right-parenthesis minus f 0 left-parenthesis t 0 minus delta right-parenthesis parallel-to less-than epsilon comma parallel-to f Subscript n Baseline left-parenthesis t 0 right-parenthesis minus f 0 left-parenthesis t 0 right-parenthesis parallel-to less-than epsilon comma parallel-to f Subscript n Baseline left-parenthesis t 0 Superscript plus Baseline right-parenthesis minus f 0 left-parenthesis t 0 Superscript plus Baseline right-parenthesis parallel-to less-than epsilon comma

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