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Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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tau Subscript i Baseline right-parenthesis parallel-to 2nd Row 1st Column Blank 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 q Baseline left-parenthesis t 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 k Baseline left-parenthesis tau Subscript i Baseline right-parenthesis minus z Subscript i Baseline parallel-to 3rd Row 1st Column Blank 2nd Column plus 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 plus StartFraction epsilon Over 4 EndFraction plus StartFraction epsilon Over 4 EndFraction plus StartFraction epsilon Over 4 EndFraction equals epsilon period EndLayout"/>

      Hence, for every t element-of left-bracket a comma b right-bracket, left-brace f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t right-parenthesis right-brace Subscript k element-of double-struck upper N Baseline subset-of upper X satisfies the Cauchy condition. Due to the fact that upper X is a complete space and because left-brace f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t right-parenthesis right-brace Subscript k element-of double-struck upper N is a Cauchy sequence, the limit limit Underscript k right-arrow infinity Endscripts f Subscript n Sub Subscript k Baseline left-parenthesis t right-parenthesis exists.

      We conclude by considering f 0 left-parenthesis t right-parenthesis equals limit Underscript k right-arrow infinity Endscripts f Subscript n Sub Subscript k Subscript Baseline left-parenthesis t right-parenthesis period Then, f Subscript n Sub Subscript k Baseline right-arrow f 0 on left-bracket a comma b right-bracket, by Lemma 1.13. Hence, f 0 is the uniform limit of the subsequence left-brace f Subscript n Sub Subscript k Subscript Baseline right-brace in upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis. Finally, any sequence left-brace f Subscript n Baseline right-brace Subscript n element-of double-struck upper N Baseline subset-of script í’œ admits a converging subsequence which, in turn, implies that script í’œ is a relatively compact set, and the proof is finished.

      We end this subsection by mentioning an Arzelà–Ascoli‐type theorem for regulated functions taking values in double-struck upper R Superscript n. A slightly different version of it can be found in [96].

      Corollary 1.19: The following conditions are equivalent:

      1 a set is relatively compact;

      2 the set is bounded, and there are an increasing continuous function , with , and a nondecreasing function such that, for every ,for

      3  is equiregulated, and for every , the set is bounded.

      We point out in [96, Theorem 2.17], item (ii), it is required that v is an increasing function. However, it is not difficult to see that if u is a nondecreasing function, then taking v left-parenthesis t right-parenthesis equals u left-parenthesis t right-parenthesis plus t yields v is an increasing function. Therefore, Corollary 1.19 follows as an immediate consequence of [96, Theorem 2.17].

      Definition 1.20: A bilinear triple (we write BT) is a set of three vector spaces upper E, upper F, and upper G, where upper F and upper G are normed spaces with a bilinear mapping script upper B colon upper E times upper F right-arrow upper G. For x element-of upper E and y element-of upper F, we write x y equals script upper B left-parenthesis x comma y right-parenthesis, and we denote the BT by left-parenthesis upper E comma upper F comma upper G right-parenthesis Subscript script upper B or simply by left-parenthesis upper E comma upper F comma upper G right-parenthesis. A topological BT is a BT left-parenthesis upper E comma upper F comma upper G right-parenthesis, where upper E is also a normed space and script upper B is continuous.

      Ifupper E and upper F are normed spaces, then we denote by upper L left-parenthesis upper E comma upper F right-parenthesis the space of all linear continuous functions from upper E to

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