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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

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

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target="_blank" rel="nofollow" href="#fb3_img_img_b07cbf0a-311f-545c-a19b-c10bb6e04b00.png" alt="upper X"/> which are Lebesgue integrable with finite integral. See the appendix of this chapter. As a matter of fact, the Fundamental Theorem of Calculus for the Henstock integral (see Theorem 1.73) yields that f not-an-element-of upper H left-parenthesis left-bracket 0 comma 1 right-bracket comma upper X right-parenthesis. Optionally, one can verify that f not-an-element-of upper H left-parenthesis left-bracket 0 comma 1 right-bracket comma upper X right-parenthesis simply by noticing that

vertical-bar vertical-bar vertical-bar vertical-bar minus minus times times of ff left-parenthesis right-parenthesis xi i left-parenthesis right-parenthesis minus minus tit minus minus i 1 integral integral t minus minus i 1 ti of ff left-parenthesis right-parenthesis t separator d separator t greater-than-or-slanted-equals one half left-parenthesis t Subscript i Baseline minus t Subscript i minus 1 Baseline right-parenthesis comma

      for every 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 0 comma 1 right-bracket.

      Claim. f element-of italic RMS left-parenthesis left-bracket 0 comma 1 right-bracket comma upper X right-parenthesis, that is, f is Riemann–McShane integrable (see the appendix of this chapter).

      Is is sufficient to prove that, given epsilon greater-than 0, we can find delta greater-than 0 such that 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 S upper T upper D Subscript left-bracket 0 comma 1 right-bracket (the reader may want to check the notation upper S upper T upper D Subscript left-bracket a comma b right-bracket in the appendix of this chapter),

vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis 1 sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times of ff left-parenthesis right-parenthesis xi i left-parenthesis right-parenthesis minus minus tit minus minus i 1 less-than epsilon period

       If and , then . Therefore, and, hence,

       If and , then . Therefore,and we obtain

      Finally, we get

StartLayout 1st Row 1st Column vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ff tilde left-parenthesis right-parenthesis 1 sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times of ff left-parenthesis right-parenthesis xi i left-parenthesis right-parenthesis minus minus tit minus minus i 1 Subscript infinity Baseline equals 2nd Column sup Underscript 0 less-than-or-slanted-equals s less-than-or-slanted-equals 1 Endscripts StartAbsoluteValue ModifyingAbove f With tilde left-parenthesis 1 right-parenthesis left-parenthesis s right-parenthesis minus sigma-summation Underscript i equals 1 Overscript StartAbsoluteValue d EndAbsoluteValue Endscripts f left-parenthesis xi Subscript i Baseline right-parenthesis left-parenthesis s right-parenthesis left-parenthesis t Subscript i Baseline minus t Subscript i minus 1 Baseline right-parenthesis EndAbsoluteValue 2nd Row 1st Column equals 2nd Column sup Underscript 0 less-than-or-slanted-equals s less-than-or-slanted-equals 1 Endscripts StartAbsoluteValue s minus sigma-summation Underscript xi Subscript i Baseline less-than-or-slanted-equals s Endscripts left-parenthesis t Subscript i Baseline minus t Subscript i minus 1 Baseline right-parenthesis EndAbsoluteValue less-than delta less-than epsilon EndLayout

      and the Claim is proved.

      A less restrict version of the Fundamental Theorem of Calculus is stated next. A proof of it follows as in [108, Theorem 9.6].

      Theorem 1.75 (Fundamental Theorem of Calculus): Suppose is a continuous function such that there exists the derivative , for nearly everywhere on i.e. except for a countable subset of . Then, and

integral Subscript a Superscript t Baseline f left-parenthesis s right-parenthesis d s equals upper F left-parenthesis t right-parenthesis minus upper F left-parenthesis a right-parenthesis comma t element-of left-bracket a comma b right-bracket period

      Now, we present a class of functions f colon left-bracket a comma b right-bracket right-arrow upper X, laying between absolute continuous and continuous functions, for which we can obtain a version of the Fundamental Theorem of Calculus for Henstock vector integrals. Let m denote the Lebesgue measure.