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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

t comma t Superscript prime Baseline right-bracket subset-of left-parenthesis t Subscript j minus 1 Baseline comma t Subscript j Baseline right-parenthesis"/>, for j equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue

j equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue

. Thus, by (1.2), parallel-to f Subscript k Baseline left-parenthesis t Superscript prime Baseline right-parenthesis minus f Subscript k Baseline left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals epsilon comma

parallel-to f Subscript k Baseline left-parenthesis t Superscript prime Baseline right-parenthesis minus f Subscript k Baseline left-parenthesis t right-parenthesis parallel-to less-than-or-slanted-equals epsilon comma

for every

and every interval left-bracket t comma t Superscript prime Baseline right-bracket subset-of left-parenthesis t Subscript j minus 1 Baseline comma t Subscript j Baseline right-parenthesis

left-bracket t comma t Superscript prime Baseline right-bracket subset-of left-parenthesis t Subscript j minus 1 Baseline comma t Subscript j Baseline right-parenthesis

, with

. Finally, Theorem 1.11 ensures the fact that the sequence left-brace f Subscript k Baseline right-brace Subscript k element-of double-struck upper N

left-brace f Subscript k Baseline right-brace Subscript k element-of double-struck upper N

is equiregulated.

A clear outcome of Lemmas 1.13 and 1.15 follows below:

Corollary 1.16: Let be a sequence of functions from to and suppose the function satisfies condition (1.2) for every and , where and the sequence is equiregulated. If the sequence converges pointwisely to a function , then it also converges uniformly to .

1.1.4 Relatively Compact Sets

In this subsection, we investigate an extension of the Arzelà–Ascoli theorem for regulated functions taking values in a general Banach space upper X with norm parallel-to dot parallel-to .

Unlike the finite dimensional case, when we consider functions taking values in upper X , the relatively compactness of a set script í’œ subset-of upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis does not come out as a consequence of the equiregulatedness of the set and the boundedness of the set StartSet f left-parenthesis t right-parenthesis colon f element-of script í’œ EndSet subset-of upper X , for each t element-of left-bracket a comma b right-bracket . In the following lines, we present an example, borrowed from [177] which illustrates this fact.

Example 1.17: Let upper Z subset-of upper X be bounded. Suppose upper Z not relatively compact in upper X . Thus, given an arbitrary epsilon greater-than 0 , there is a sequence of functions left-brace z Subscript n Baseline right-brace Subscript n element-of double-struck upper N in upper Z for which

parallel-to z Subscript n Baseline parallel-to less-than-or-slanted-equals upper K and parallel-to z Subscript n Baseline minus z Subscript m Baseline parallel-to greater-than-or-slanted-equals epsilon comma

for all n not-equals m and for some constant upper K greater-than 0 . Hence, the set

upper B equals left-brace y Subscript n Baseline colon left-bracket 0 comma 1 right-bracket right-arrow upper X colon y Subscript n Baseline left-parenthesis t right-parenthesis equals t z Subscript n Baseline comma n element-of double-struck upper N right-brace

is bounded, once left-brace z Subscript n Baseline right-brace Subscript n element-of double-struck upper N is bounded. Moreover, upper B is equiregulated and left-brace y Subscript n Baseline left-parenthesis 0 right-parenthesis right-brace Subscript n element-of double-struck upper N is relatively compact in upper X . On the other hand, upper B is not relatively compact in upper G left-parenthesis left-bracket 0 comma 1 right-bracket comma upper X right-parenthesis .

At this point, it is important to say that, in order to guarantee that a set script í’œ subset-of upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis is relatively compact, one needs an additional condition. It is clear that, if one assumes that, for each t element-of left-bracket a comma b right-bracket , the set StartSet f left-parenthesis t right-parenthesis colon f element-of script í’œ EndSet is relatively compact in upper X , then becomes relatively compact in upper G left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis . This is precisely what the next result says, and we refer to it as the Arzelà–Ascoli theorem for Banach space‐valued regulated functions. Such important result can be found in [97] and [177] as well.

Theorem 1.18: Suppose is equiregulated and, for every , is relatively compact in . Then, is relatively compact in .

Proof. Take a sequence of functions left-brace f Subscript n Baseline right-brace Subscript n element-of double-struck upper N Baseline subset-of script í’œ . The set Скачать книгу