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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

alt="upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis minus script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis not-equals empty-set"/> as one can note by the next classical example.

Example 1.92: Let upper F colon left-bracket 0 comma 1 right-bracket right-arrow double-struck upper R be defined by upper F left-parenthesis t right-parenthesis equals t squared sine left-parenthesis t Superscript negative 2 Baseline right-parenthesis comma if 0 less-than t less-than-or-slanted-equals 1 , and upper F left-parenthesis 0 right-parenthesis equals 0 , and consider f equals upper F prime . Since is Riemann improper integrable, f element-of upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals upper H left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis , because the Kurzweil–Henstock (or Perron) integral contains its improper integrals (see Theorem 2.9, [158], or [213]). However, f not-an-element-of script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis , since is not absolutely integrable (see also [227]).

Example 1.92 tells us that the elements of upper K left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis equals upper H left-parenthesis left-bracket a comma b right-bracket comma double-struck upper R right-parenthesis are not absolutely integrable.

When McShane's idea is applied to Kurzweil and Henstock vector integrals, the story changes. In fact, the modification of McShane applied to the Kurzweil vector integral originates an integral which encompasses the Bochner–Lebesgue integral (see Example 1.74). On the other hand, when McShane's idea is used to modify the variational Henstock integral, we obtain exactly the Bochner–Lebesgue integral (see [[47] and [131]]). Thus, if italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis denotes the space of Henstock–McShane integrable functions f colon left-bracket a comma b right-bracket right-arrow upper X , that is, f element-of italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis is integrable in the sense of Henstock with the modification of McShane, then

italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis equals script upper L 1 left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis period

We will prove this equality in the sequel. Furthermore, italic upper H upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper H left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis , italic upper K upper M upper S 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 comma and italic RMS left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis subset-of upper R left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis , where we use the notation italic upper K upper M upper S left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis and italic RMS left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis to denote, respectively, the spaces of Kurzweil–McShane and Riemann–McShane integrable functions from to upper X . For other interesting results, the reader may want to consult [55].

Our aim in the remaining of this chapter is to show that the integrals of Bochner–Lebesgue and Henstock–McShane coincide. See [132, Theorem 10.4]. The next results are due to C. S. Hönig. They belong to a brochure of a series of lectures Professor Hönig gave in Rio de Janeiro in 1993. We include the proofs here, once the brochure is in Portuguese.

Lemma 1.93: Let be a sequence in and be a function. Suppose there exists

limit Underscript n right-arrow infinity Endscripts left-parenthesis italic upper K upper M upper S right-parenthesis integral Subscript a Superscript b Baseline vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ffn left-parenthesis right-parenthesis t of ff left-parenthesis right-parenthesis t d t equals 0 period

Then, and

Proof. Given epsilon greater-than 0 , take n Subscript epsilon such that for m comma n greater-than-or-slanted-equals n Subscript epsilon Baseline ,

left-parenthesis italic upper K upper M upper S right-parenthesis integral Subscript a Superscript b Baseline vertical-bar vertical-bar vertical-bar vertical-bar minus minus of ffn left-parenthesis right-parenthesis t of ffm left-parenthesis right-parenthesis t d t less-than epsilon

and take a gauge delta on left-bracket a comma b right-bracket such that for every Скачать книгу