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

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(1.13)

      By Corollary 1.50, . Let us denote by the subspace of of functions which are differentiable with continuous derivative. Hence, there is a function such that

(1.14)

      Let be defined by , for all such that , and by . In particular, whenever . Therefore, for almost every , since is differentiable and nonconstant on any nondegenerate subinterval of . Hence, . Then, the Riemann integral exists and

(1.15)

where we applied the Fundamental Theorem of Calculus for the Riemann integral in order to obtain the last equality. Thus, replacing (1.15) in (1.14), we obtain

(1.16)

Now, in view of (1.13), it remains to prove that the Perron–Stieltjes integral exists and

(1.17)

      Let be the gauge on from the definition of . Take , and for every , let be such that if , then, by the Straddle Lemma (Lemma 1.86), we have

(1.18) Скачать книгу