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

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Название Generalized Ordinary Differential Equations in Abstract Spaces and Applications
Автор произведения Группа авторов
Жанр Математика
Серия
Издательство Математика
Год выпуска 0
isbn 9781119655008
Generalized Ordinary Differential Equations in Abstract Spaces and Applications - Группа авторов
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alt="script 풜"/> is equiregulated by hypothesis. Then, for every epsilon greater-than 0, there exists a division d equals left-parenthesis t Subscript i Baseline right-parenthesis element-of upper D Subscript left-bracket a comma b right-bracket fulfilling

parallel-to f Subscript n Baseline left-parenthesis t Superscript prime Baseline right-parenthesis minus f Subscript n Baseline left-parenthesis t right-parenthesis parallel-to less-than StartFraction epsilon Over 4 EndFraction comma

      for every n element-of double-struck upper N and t Subscript i minus 1 Baseline less-than t less-than t prime less-than t Subscript i, i equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue.

      On the other hand, the sets left-brace f Subscript n Baseline left-parenthesis t Subscript i Baseline right-parenthesis right-brace Subscript n element-of double-struck upper N and left-brace f Subscript n Baseline left-parenthesis tau Subscript i Baseline right-parenthesis right-brace Subscript n element-of double-struck upper N are relatively compact in upper X for every i equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue, where t Subscript i minus 1 Baseline less-than tau Subscript i Baseline less-than t Subscript i. Thus, there is a subsequence of indexes left-brace n Subscript k Baseline right-brace Subscript k element-of double-struck upper N Baseline subset-of double-struck upper N, with n Subscript k plus 1 Baseline greater-than n Subscript k, for which left-brace f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t Subscript i Baseline right-parenthesis right-brace Subscript k element-of double-struck upper N and left-brace f Subscript n Sub Subscript k Subscript Baseline left-parenthesis tau Subscript i Baseline right-parenthesis right-brace Subscript k element-of double-struck upper N are also relatively compact sets of upper X, for every i.

      This last statement implies that there exist left-brace y Subscript i Baseline colon i equals 0 comma 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue right-brace subset-of upper X and left-brace z Subscript i Baseline colon i equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue right-brace subset-of upper X satisfying

y Subscript i Baseline equals limit Underscript k right-arrow infinity Endscripts f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t Subscript i Baseline right-parenthesis and z Subscript i Baseline equals limit Underscript k right-arrow infinity Endscripts f Subscript n Sub Subscript k Subscript Baseline left-parenthesis tau Subscript i Baseline right-parenthesis period

      Thus, there exists upper N element-of double-struck upper N such that

vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ffnk left-parenthesis right-parenthesis tiyi less-than StartFraction epsilon Over 4 EndFraction and vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ffnk left-parenthesis right-parenthesis tau izi less-than StartFraction epsilon Over 4 EndFraction comma parallel-to f Subscript n Sub Subscript q Subscript Baseline left-parenthesis t Subscript i Baseline right-parenthesis minus y Subscript i Baseline parallel-to less-than StartFraction epsilon Over 4 EndFraction and parallel-to f Subscript n Sub Subscript q Subscript Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus z Subscript i Baseline parallel-to less-than StartFraction epsilon Over 4 EndFraction comma

      for i equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue.

      Take t element-of left-bracket a comma b right-bracket and consider q element-of double-struck upper N such that q greater-than k. Therefore, either t equals t Subscript i, for some i element-of StartSet 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue EndSet, in which case, we have

parallel-to f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t right-parenthesis minus f Subscript n Sub Subscript q Subscript Baseline left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals parallel-to f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t Subscript i Baseline right-parenthesis minus y Subscript i Baseline parallel-to plus parallel-to f Subscript n Sub Subscript q Subscript Baseline left-parenthesis t Subscript i Baseline right-parenthesis minus y Subscript i Baseline parallel-to less-than StartFraction epsilon Over 2 EndFraction comma

      or t element-of left-parenthesis t Subscript i minus 1 Baseline comma t Subscript i Baseline right-parenthesis, for some i element-of StartSet 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue EndSet, in which case, we have

StartLayout 1st Row 1st Column parallel-to f Subscript n Sub Subscript k Baseline left-parenthesis t right-parenthesis minus f Subscript n Sub Subscript q Baseline left-parenthesis t right-parenthesis parallel-to 2nd Column less-than-or-slanted-equals parallel-to f Subscript n Sub Subscript k Baseline left-parenthesis t right-parenthesis minus f Subscript n Sub Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to plus parallel-to f Subscript n Sub Subscript k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus f Subscript n Sub Subscript q Baseline left-parenthesis tau Subscript i Baseline right-parenthesis parallel-to plus parallel-to f Subscript n Sub Subscript q Baseline left-parenthesis t right-parenthesis minus f Subscript n Sub Subscript </div> </div> </div> <hr> <div class=
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