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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

alt="script í’œ"/> 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

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 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 are also relatively compact sets of upper X , for every .

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

provided k greater-than upper N . In particular, for q greater-than k , we have

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 <hr><noindex><a href=

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