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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

t Subscript n Sub Subscript i Subscript minus 1 Baseline right-parenthesis parallel-to less-than StartFraction epsilon Over n 2 Superscript n Baseline EndFraction period"/>

Consider a gauge delta of left-bracket a comma b right-bracket such that delta left-parenthesis xi right-parenthesis equals delta Subscript n Baseline left-parenthesis xi right-parenthesis , whenever xi element-of upper E Subscript n , and can assume any value in left-parenthesis 0 comma infinity right-parenthesis , otherwise. Then, for every delta -fine d equals left-parenthesis xi Subscript i Baseline comma left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-parenthesis element-of upper T upper D Subscript left-bracket a comma b right-bracket , we have

StartLayout 1st Row 1st Column sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts parallel-to alpha left-parenthesis xi Subscript i Baseline right-parenthesis left-bracket f left-parenthesis t Subscript i Baseline right-parenthesis minus f left-parenthesis t Subscript i minus 1 Baseline right-parenthesis right-bracket parallel-to less-than-or-slanted-equals 2nd Column sigma-summation Underscript n element-of double-struck upper N Endscripts sigma-summation Underscript xi Subscript i Baseline element-of upper E Subscript n Endscripts parallel-to alpha left-parenthesis xi Subscript i Baseline right-parenthesis parallel-to parallel-to f left-parenthesis t Subscript i Baseline right-parenthesis minus f left-parenthesis t Subscript i minus 1 Baseline right-parenthesis parallel-to 2nd Row 1st Column less-than-or-slanted-equals 2nd Column sigma-summation Underscript n element-of double-struck upper N Endscripts n sigma-summation Underscript xi Subscript i Baseline element-of upper E Subscript n Baseline Endscripts parallel-to f left-parenthesis t Subscript i Baseline right-parenthesis minus f left-parenthesis t Subscript i minus 1 Baseline right-parenthesis parallel-to less-than epsilon EndLayout

and we complete the proof.

Given a function f colon left-bracket a comma b right-bracket right-arrow upper X , since upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis subset-of upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis , Theorem 1.81 holds for instead of upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis . Then, next proposition follows easily (see, also, [70, Corollary after Theorem 5]).

Proposition 1.82: Suppose and . Assume, in addition, that is such that almost everywhere in . Then, and , for every . If, moreover, , then .

In view of Proposition 1.82, we can define equivalence classes of nonabsolute vector integrable functions.

Definition 1.83: Let us assume that f element-of upper S upper L left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis . Two functions beta comma alpha element-of upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis are equivalent if and only if their indefinite integrals coincide, that is, beta overTilde Subscript f Baseline equals alpha overTilde Subscript f . By bold upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis and bold upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis we mean, respectively, the spaces of equivalence classes of functions of upper K Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis and of upper H Subscript f Baseline left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis , and we endow these spaces with an Alexiewicz-type norm

vertical-bar vertical-bar vertical-bar vertical-bar alpha Subscript upper A comma f Baseline equals vertical-bar vertical-bar vertical-bar vertical-bar alpha tilde f Subscript infinity Baseline equals sup left-brace right-brace colon vertical-bar vertical-bar vertical-bar vertical-bar integral integral at of alpha alpha left-parenthesis right-parenthesis s times times d of ff left-parenthesis right-parenthesis s colon element-of element-of t left-bracket right-bracket comma a comma b period

From Example 1.74, we know that although g comma f element-of upper R left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper K left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis may belong to the same equivalence class, that is, integral Subscript a Superscript t Baseline g left-parenthesis s right-parenthesis d s equals integral Subscript a Superscript t Baseline f left-parenthesis s right-parenthesis d s for all t element-of left-bracket a comma b right-bracket , one cannot conclude that g equals f almost everywhere in . It is known, however, that the space of all equivalence classes of real-valued Perron integrable functions f colon left-bracket a comma b right-bracket right-arrow double-struck upper R , equipped with the usual

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