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

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

href="#fb3_img_img_9b3e8e46-2bc8-5209-a7e9-da40bde6d467.png" alt="f left-parenthesis t right-parenthesis element-of l 2 left-parenthesis double-struck upper N times double-struck upper N right-parenthesis"/>, with t element-of left-bracket 0 comma 1 right-bracket

t element-of left-bracket 0 comma 1 right-bracket

, such that limit Underscript n right-arrow infinity Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus fnf Subscript upper A Baseline equals 0

limit Underscript n right-arrow infinity Endscripts vertical-bar vertical-bar vertical-bar vertical-bar minus minus fnf Subscript upper A Baseline equals 0

The next result follows from Theorem 1.80. A proof of it can be found in [75, Theorem 5].

Theorem 1.85: Suppose is nonconstant on any nondegenerate subinterval of . Then, the mapping

alpha element-of 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 right-arrow from bar alpha overTilde Subscript f Baseline element-of upper C Subscript a Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis

is an isometry, that is onto a dense subspace of .

The next result, known as straddle Lemma, will be useful to prove that the space upper G left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis of regulated functions from to is dense in 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 in the Alexiewicz norm vertical-bar vertical-bar vertical-bar vertical-bar dot Subscript upper A comma f . For a proof of the straddle Lemma, the reader may want to consult [130, 3.4] or [119].

Lemma 1.86 (Straddle Lemma): Suppose are functions such that is differentiable, with , for all . Then, given , there exists such that

vertical-bar vertical-bar vertical-bar vertical-bar minus minus minus of FF left-parenthesis right-parenthesis t of FF left-parenthesis right-parenthesis s times times of ff left-parenthesis right-parenthesis xi left-parenthesis right-parenthesis minus minus ts less-than epsilon left-parenthesis t minus s right-parenthesis comma

whenever .

The next result is adapted from [75, Theorem 8].

Proposition 1.87: Suppose is differentiable and nonconstant on any nondegenerate subinterval of . Then, the Banach space is dense in under the Alexiewicz norm .

Proof. Assume that alpha element-of 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 let epsilon greater-than 0 be given. We need to find a function beta element-of upper G left-parenthesis left-bracket a comma b right-bracket comma upper L left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis such that vertical-bar vertical-bar vertical-bar vertical-bar minus minus beta alpha Subscript upper A comma f Baseline less-than epsilon , or equivalently,

(1.13)

By Corollary 1.50, alpha overTilde Subscript f Baseline element-of upper C Subscript a Baseline left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis equals StartSet x element-of upper C left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis colon x left-parenthesis a right-parenthesis equals 0 EndSet . Let us denote by upper C Subscript a Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis the subspace of upper C Subscript a Baseline left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis of functions which are differentiable with continuous derivative. Hence, there is a function h element-of upper C Subscript a Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis left-bracket a comma b right-bracket comma upper Y right-parenthesis such that

(1.14) vertical-bar vertical-bar vertical-bar vertical-bar minus minus h alpha tilde f Subscript infinity Baseline less-than epsilon period

Let beta colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis be defined by beta left-parenthesis t right-parenthesis x equals h prime left-parenthesis t right-parenthesis , for all x element-of upper X such that x not-equals 0 , and by . In particular, whenever f prime left-parenthesis t right-parenthesis not-equals 0 . Therefore, for almost every t element-of left-bracket a comma b right-bracket , since f colon left-bracket a comma b right-bracket right-arrow upper X is differentiable and nonconstant

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