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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

equals left-bracket alpha left-parenthesis t Subscript i Baseline right-parenthesis minus alpha left-parenthesis xi Subscript i Baseline right-parenthesis right-bracket ModifyingAbove f With tilde left-parenthesis t Subscript i Baseline right-parenthesis"/> and beta Subscript t Sub Subscript i minus 1 Baseline equals left-bracket alpha left-parenthesis t Subscript i minus 1 Baseline right-parenthesis minus alpha left-parenthesis xi Subscript i Baseline right-parenthesis right-bracket ModifyingAbove f With tilde left-parenthesis t Subscript i minus 1 Baseline right-parenthesis

beta Subscript t Sub Subscript i minus 1 Baseline equals left-bracket alpha left-parenthesis t Subscript i minus 1 Baseline right-parenthesis minus alpha left-parenthesis xi Subscript i Baseline right-parenthesis right-bracket ModifyingAbove f With tilde left-parenthesis t Subscript i minus 1 Baseline right-parenthesis

, we have

StartLayout 1st Row 1st Column Blank 2nd Column sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar integral integral t minus minus i 1 ti times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis t of alpha alpha left-parenthesis right-parenthesis xi i of ff left-parenthesis right-parenthesis t separator d separator t equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus minus beta ti beta t minus minus i 1 integral integral t minus minus i 1 ti times times times d of alpha alpha left-parenthesis right-parenthesis t of ff tilde left-parenthesis right-parenthesis t 2nd Row 1st Column Blank 2nd Column equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus beta ti integral integral xi iti minus minus minus times times times d of alpha alpha left-parenthesis right-parenthesis t of ff tilde left-parenthesis right-parenthesis t beta t minus minus i 1 integral integral t minus minus i 1 xi i times times times d of alpha alpha left-parenthesis right-parenthesis t of ff tilde left-parenthesis right-parenthesis t 3rd Row 1st Column Blank 2nd Column equals sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts vertical-bar vertical-bar vertical-bar vertical-bar integral integral xi iti plus plus times times times d of alpha alpha left-parenthesis right-parenthesis t left-bracket right-bracket minus minus of ff tilde left-parenthesis right-parenthesis ti of ff tilde left-parenthesis right-parenthesis t integral integral t minus minus i 1 xi i times times times d of alpha alpha left-parenthesis right-parenthesis t left-bracket right-bracket minus minus of ff tilde left-parenthesis right-parenthesis t minus minus i 1 of ff tilde left-parenthesis right-parenthesis t less-than-or-slanted-equals epsilon v a r Subscript a Superscript b Baseline left-parenthesis alpha right-parenthesis comma EndLayout

since for every t element-of left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket , we have

StartLayout 1st Row 1st Column Blank 2nd Column vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis ti of ff tilde left-parenthesis right-parenthesis t less-than-or-slanted-equals sup left-brace parallel-to ModifyingAbove f With tilde left-parenthesis t right-parenthesis minus ModifyingAbove f With tilde left-parenthesis s right-parenthesis parallel-to colon t comma s element-of left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-brace and 2nd Row 1st Column Blank 2nd Column vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis t minus minus i 1 of ff tilde left-parenthesis right-parenthesis t less-than-or-slanted-equals sup left-brace parallel-to ModifyingAbove f With tilde left-parenthesis t right-parenthesis minus ModifyingAbove f With tilde left-parenthesis s right-parenthesis parallel-to colon t comma s element-of left-bracket t Subscript i minus 1 Baseline comma t Subscript i Baseline right-bracket right-brace period EndLayout

The proof is then complete.

A proof of the next result, borrowed from [72, Theorem 8], follows from the definitions of the integrals.

Theorem 1.59: Let and . If is bounded, then and

(1.5) integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis d t equals integral Subscript a Superscript b Baseline d ModifyingAbove alpha With tilde left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis period

If, in addition, , then .

Corollary 1.60: Suppose with and . Then, and (1.5) holds.

Proof. By Theorem 1.49, alpha overTilde element-of upper C left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis . Then, the result follows from Theorem 1.47, since .

The next corollaries follow from Theorems 1.49 and 1.53.

Corollary 1.61: Suppose with and . Then, , and we have

(1.6) integral Subscript a Superscript b Baseline alpha left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis d t equals integral Subscript a Superscript b Baseline d ModifyingAbove alpha With tilde left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis

and the following integration by parts formula holds

(1.7)

Corollary 1.62: Consider functions and . Then, and equalities (1.6) and (1.7) hold.

The next two theorems generalize Corollary 1.62. For their proofs, the reader may want to consult [72].

Theorem 1.63: Consider . If respectively, , then respectively, and both (1.6) and

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